Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.6% → 99.2%

Time: 19.4s

Alternatives: 24

Speedup: 7.6×

• 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: 70.6% 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: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\\ t_1 := \frac{t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_2 := \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\\ \mathbf{if}\;x1 \leq -2.6 \cdot 10^{+50}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{t\_2}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 2.55 \cdot 10^{+69}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(t\_1, 4, -6\right), \frac{\left(-3 + t\_1\right) \cdot \left(\left(x1 \cdot 2\right) \cdot t\_0\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot t\_1, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), t\_2\right), 6 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right), x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))))
        (t_1 (/ t_0 (fma x1 x1 1.0)))
        (t_2 (fma 4.0 (fma x2 2.0 -3.0) 9.0)))
   (if (<= x1 -2.6e+50)
     (+ x1 (* (pow x1 4.0) (+ 6.0 (/ (- (/ t_2 x1) 3.0) x1))))
     (if (<= x1 2.55e+69)
       (+
        x1
        (fma
         (/ (- (* x1 (* x1 3.0)) (fma 2.0 x2 x1)) (fma x1 x1 1.0))
         3.0
         (fma
          (fma x1 x1 1.0)
          (fma
           x1
           (* x1 (fma t_1 4.0 -6.0))
           (/ (* (+ -3.0 t_1) (* (* x1 2.0) t_0)) (fma x1 x1 1.0)))
          (fma x1 (* (* x1 3.0) t_1) (fma x1 (* x1 x1) x1)))))
       (fma
        x1
        (fma x1 (fma x1 (fma x1 6.0 -3.0) t_2) (* 6.0 (fma x2 2.0 -3.0)))
        x1)))))

C

double code(double x1, double x2) {
	double t_0 = fma(2.0, x2, fma(x1, (x1 * 3.0), -x1));
	double t_1 = t_0 / fma(x1, x1, 1.0);
	double t_2 = fma(4.0, fma(x2, 2.0, -3.0), 9.0);
	double tmp;
	if (x1 <= -2.6e+50) {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + (((t_2 / x1) - 3.0) / x1)));
	} else if (x1 <= 2.55e+69) {
		tmp = x1 + fma((((x1 * (x1 * 3.0)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), fma(x1, (x1 * fma(t_1, 4.0, -6.0)), (((-3.0 + t_1) * ((x1 * 2.0) * t_0)) / fma(x1, x1, 1.0))), fma(x1, ((x1 * 3.0) * t_1), fma(x1, (x1 * x1), x1))));
	} else {
		tmp = fma(x1, fma(x1, fma(x1, fma(x1, 6.0, -3.0), t_2), (6.0 * fma(x2, 2.0, -3.0))), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1)))
	t_1 = Float64(t_0 / fma(x1, x1, 1.0))
	t_2 = fma(4.0, fma(x2, 2.0, -3.0), 9.0)
	tmp = 0.0
	if (x1 <= -2.6e+50)
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(t_2 / x1) - 3.0) / x1))));
	elseif (x1 <= 2.55e+69)
		tmp = Float64(x1 + fma(Float64(Float64(Float64(x1 * Float64(x1 * 3.0)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(t_1, 4.0, -6.0)), Float64(Float64(Float64(-3.0 + t_1) * Float64(Float64(x1 * 2.0) * t_0)) / fma(x1, x1, 1.0))), fma(x1, Float64(Float64(x1 * 3.0) * t_1), fma(x1, Float64(x1 * x1), x1)))));
	else
		tmp = fma(x1, fma(x1, fma(x1, fma(x1, 6.0, -3.0), t_2), Float64(6.0 * fma(x2, 2.0, -3.0))), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]}, If[LessEqual[x1, -2.6e+50], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(t$95$2 / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.55e+69], N[(x1 + N[(N[(N[(N[(x1 * N[(x1 * 3.0), $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[(x1 * N[(x1 * N[(t$95$1 * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-3.0 + t$95$1), $MachinePrecision] * N[(N[(x1 * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(6.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.6000000000000002e50

   1. Initial program 24.9%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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. Applied rewrites 100.0%

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

   if -2.6000000000000002e50 < x1 < 2.54999999999999999e69

   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

   4. Applied rewrites 99.7%

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

   if 2.54999999999999999e69 < x1

   1. Initial program 39.2%

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

   3. Taylor expanded in x1 around -inf

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

      1. 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. Applied rewrites 100.0%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.6 \cdot 10^{+50}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 2.55 \cdot 10^{+69}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(-3 + \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 6 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := -1 - x1 \cdot x1\\ t_5 := \frac{t\_2}{t\_4}\\ t_6 := x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_5\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_5\right)\right) \cdot t\_4 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ t_7 := x1 + x2 \cdot \left(x1 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{+173}:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+228}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (- -1.0 (* x1 x1)))
        (t_5 (/ t_2 t_4))
        (t_6
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_5)))
                (* (* (* x1 2.0) t_3) (+ 3.0 t_5)))
               t_4)
              (* t_1 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0)))))
        (t_7 (+ x1 (* x2 (* x1 (* x2 8.0))))))
   (if (<= t_6 -2e+173)
     t_7
     (if (<= t_6 5e+228)
       (- (* x2 -6.0) x1)
       (if (<= t_6 INFINITY) t_7 (* x1 (fma x1 9.0 -1.0)))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = -1.0 - (x1 * x1);
	double t_5 = t_2 / t_4;
	double t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double t_7 = x1 + (x2 * (x1 * (x2 * 8.0)));
	double tmp;
	if (t_6 <= -2e+173) {
		tmp = t_7;
	} else if (t_6 <= 5e+228) {
		tmp = (x2 * -6.0) - x1;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = t_7;
	} else {
		tmp = x1 * fma(x1, 9.0, -1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(-1.0 - Float64(x1 * x1))
	t_5 = Float64(t_2 / t_4)
	t_6 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_5))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_5))) * t_4) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	t_7 = Float64(x1 + Float64(x2 * Float64(x1 * Float64(x2 * 8.0))))
	tmp = 0.0
	if (t_6 <= -2e+173)
		tmp = t_7;
	elseif (t_6 <= 5e+228)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	elseif (t_6 <= Inf)
		tmp = t_7;
	else
		tmp = Float64(x1 * fma(x1, 9.0, -1.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 + N[(N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -2e+173], t$95$7, If[LessEqual[t$95$6, 5e+228], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], If[LessEqual[t$95$6, Infinity], t$95$7, N[(x1 * N[(x1 * 9.0 + -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)))))) < -2e173 or 5e228 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   3. 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. Applied rewrites 58.8%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 64.1

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

   8. Applied rewrites 64.1%

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

   if -2e173 < (+.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)))))) < 5e228

   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 x2 around 0

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 82.4%

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

   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. lower-*.f64 81.3

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

   11. Applied rewrites 81.3%

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

   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 x2 around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. distribute-lft-out N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. associate-+l+ N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 88.4

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

   11. Applied rewrites 88.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\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 \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\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^{+173}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\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 \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\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^{+228}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\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 \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\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 + x2 \cdot \left(x1 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := -1 - x1 \cdot x1\\ t_5 := \frac{t\_2}{t\_4}\\ t_6 := x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_5\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_5\right)\right) \cdot t\_4 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{+173}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+59}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 6, 9\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(x1, 6, -3\right), x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (- -1.0 (* x1 x1)))
        (t_5 (/ t_2 t_4))
        (t_6
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_5)))
                (* (* (* x1 2.0) t_3) (+ 3.0 t_5)))
               t_4)
              (* t_1 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_6 -2e+173)
     (+ x1 (* x2 (* x1 (* x2 8.0))))
     (if (<= t_6 5e+59)
       (+ x1 (fma x2 -6.0 (* x1 (fma x1 (fma x2 6.0 9.0) -2.0))))
       (fma (* x1 x1) (* x1 (fma x1 6.0 -3.0)) x1)))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = -1.0 - (x1 * x1);
	double t_5 = t_2 / t_4;
	double t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_6 <= -2e+173) {
		tmp = x1 + (x2 * (x1 * (x2 * 8.0)));
	} else if (t_6 <= 5e+59) {
		tmp = x1 + fma(x2, -6.0, (x1 * fma(x1, fma(x2, 6.0, 9.0), -2.0)));
	} else {
		tmp = fma((x1 * x1), (x1 * fma(x1, 6.0, -3.0)), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(-1.0 - Float64(x1 * x1))
	t_5 = Float64(t_2 / t_4)
	t_6 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_5))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_5))) * t_4) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_6 <= -2e+173)
		tmp = Float64(x1 + Float64(x2 * Float64(x1 * Float64(x2 * 8.0))));
	elseif (t_6 <= 5e+59)
		tmp = Float64(x1 + fma(x2, -6.0, Float64(x1 * fma(x1, fma(x2, 6.0, 9.0), -2.0))));
	else
		tmp = fma(Float64(x1 * x1), Float64(x1 * fma(x1, 6.0, -3.0)), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 + N[(N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -2e+173], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 5e+59], N[(x1 + N[(x2 * -6.0 + N[(x1 * N[(x1 * N[(x2 * 6.0 + 9.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -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)))))) < -2e173

   1. Initial program 99.8%

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

   3. 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. Applied rewrites 82.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   if -2e173 < (+.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)))))) < 4.9999999999999997e59

   1. Initial program 99.2%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 91.9%

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

   if 4.9999999999999997e59 < (+.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 46.1%

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

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - 3 \cdot \frac{1}{x1}\right)} \]
   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 81.7

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

   5. Applied rewrites 81.7%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + \frac{-3}{x1}\right)} \]
   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 81.7

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

      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. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 81.7

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

   7. Applied rewrites 81.7%

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

   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. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 81.7

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

   10. Applied rewrites 81.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2 \cdot 10^{+173}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+59}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 6, 9\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(x1, 6, -3\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := -1 - x1 \cdot x1\\ t_5 := \frac{t\_2}{t\_4}\\ t_6 := x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_5\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_5\right)\right) \cdot t\_4 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{+173}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{elif}\;t\_6 \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, x1 \cdot -19, -2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(x1, 6, -3\right), x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (- -1.0 (* x1 x1)))
        (t_5 (/ t_2 t_4))
        (t_6
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_5)))
                (* (* (* x1 2.0) t_3) (+ 3.0 t_5)))
               t_4)
              (* t_1 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_6 -2e+173)
     (+ x1 (* x2 (* x1 (* x2 8.0))))
     (if (<= t_6 2e+42)
       (+ x1 (fma x1 (fma x1 (* x1 -19.0) -2.0) (* x2 -6.0)))
       (fma (* x1 x1) (* x1 (fma x1 6.0 -3.0)) x1)))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = -1.0 - (x1 * x1);
	double t_5 = t_2 / t_4;
	double t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_6 <= -2e+173) {
		tmp = x1 + (x2 * (x1 * (x2 * 8.0)));
	} else if (t_6 <= 2e+42) {
		tmp = x1 + fma(x1, fma(x1, (x1 * -19.0), -2.0), (x2 * -6.0));
	} else {
		tmp = fma((x1 * x1), (x1 * fma(x1, 6.0, -3.0)), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(-1.0 - Float64(x1 * x1))
	t_5 = Float64(t_2 / t_4)
	t_6 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_5))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_5))) * t_4) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_6 <= -2e+173)
		tmp = Float64(x1 + Float64(x2 * Float64(x1 * Float64(x2 * 8.0))));
	elseif (t_6 <= 2e+42)
		tmp = Float64(x1 + fma(x1, fma(x1, Float64(x1 * -19.0), -2.0), Float64(x2 * -6.0)));
	else
		tmp = fma(Float64(x1 * x1), Float64(x1 * fma(x1, 6.0, -3.0)), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 + N[(N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -2e+173], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2e+42], N[(x1 + N[(x1 * N[(x1 * N[(x1 * -19.0), $MachinePrecision] + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -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)))))) < -2e173

   1. Initial program 99.8%

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

   3. 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. Applied rewrites 82.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   if -2e173 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 2.00000000000000009e42

   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 x2 around 0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(x1 \cdot \left(-19 \cdot x1 + 3 \cdot \left(3 - -2 \cdot x2\right)\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(x1 \cdot \left(-19 \cdot x1 + 3 \cdot \left(3 - -2 \cdot x2\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, x1 \cdot \left(-19 \cdot x1 + 3 \cdot \left(3 - -2 \cdot x2\right)\right) - 2, -6 \cdot x2\right)} \]

   8. Applied rewrites 94.1%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -19, \mathsf{fma}\left(x2, 6, 9\right)\right), -2\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x1 around inf

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{-19 \cdot x1}, -2\right), x2 \cdot -6\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{x1 \cdot -19}, -2\right), x2 \cdot -6\right) \]

       2. lower-*.f64 93.1

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{x1 \cdot -19}, -2\right), x2 \cdot -6\right) \]

   11. Applied rewrites 93.1%

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{x1 \cdot -19}, -2\right), x2 \cdot -6\right) \]

   if 2.00000000000000009e42 < (+.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 47.5%

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

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - 3 \cdot \frac{1}{x1}\right)} \]
   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 80.9

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

   5. Applied rewrites 80.9%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + \frac{-3}{x1}\right)} \]
   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 80.9

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

      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. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 80.9

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

   7. Applied rewrites 80.9%

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

   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. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 80.9

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

   10. Applied rewrites 80.9%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2 \cdot 10^{+173}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, x1 \cdot -19, -2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(x1, 6, -3\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := -1 - x1 \cdot x1\\ t_5 := \frac{t\_2}{t\_4}\\ t_6 := x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_5\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_5\right)\right) \cdot t\_4 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{+173}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{elif}\;t\_6 \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(x1, 6, -3\right), x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (- -1.0 (* x1 x1)))
        (t_5 (/ t_2 t_4))
        (t_6
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_5)))
                (* (* (* x1 2.0) t_3) (+ 3.0 t_5)))
               t_4)
              (* t_1 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_6 -2e+173)
     (+ x1 (* x2 (* x1 (* x2 8.0))))
     (if (<= t_6 2e+42)
       (- (* x2 -6.0) x1)
       (fma (* x1 x1) (* x1 (fma x1 6.0 -3.0)) x1)))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = -1.0 - (x1 * x1);
	double t_5 = t_2 / t_4;
	double t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_6 <= -2e+173) {
		tmp = x1 + (x2 * (x1 * (x2 * 8.0)));
	} else if (t_6 <= 2e+42) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = fma((x1 * x1), (x1 * fma(x1, 6.0, -3.0)), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(-1.0 - Float64(x1 * x1))
	t_5 = Float64(t_2 / t_4)
	t_6 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_5))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_5))) * t_4) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_6 <= -2e+173)
		tmp = Float64(x1 + Float64(x2 * Float64(x1 * Float64(x2 * 8.0))));
	elseif (t_6 <= 2e+42)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	else
		tmp = fma(Float64(x1 * x1), Float64(x1 * fma(x1, 6.0, -3.0)), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 + N[(N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -2e+173], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2e+42], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -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)))))) < -2e173

   1. Initial program 99.8%

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

   3. 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. Applied rewrites 82.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   if -2e173 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 2.00000000000000009e42

   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 x2 around 0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 93.8%

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

   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. lower-*.f64 93.0

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

   11. Applied rewrites 93.0%

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

   if 2.00000000000000009e42 < (+.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 47.5%

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

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - 3 \cdot \frac{1}{x1}\right)} \]
   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 80.9

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

   5. Applied rewrites 80.9%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + \frac{-3}{x1}\right)} \]
   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 80.9

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

      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. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 80.9

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

   7. Applied rewrites 80.9%

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

   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. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 80.9

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

   10. Applied rewrites 80.9%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2 \cdot 10^{+173}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(x1, 6, -3\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := -1 - x1 \cdot x1\\ t_5 := \frac{t\_2}{t\_4}\\ t_6 := x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_5\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_5\right)\right) \cdot t\_4 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{+173}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+59}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(6, \left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right), x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (- -1.0 (* x1 x1)))
        (t_5 (/ t_2 t_4))
        (t_6
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_5)))
                (* (* (* x1 2.0) t_3) (+ 3.0 t_5)))
               t_4)
              (* t_1 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_6 -2e+173)
     (+ x1 (* x2 (* x1 (* x2 8.0))))
     (if (<= t_6 5e+59)
       (- (* x2 -6.0) x1)
       (fma 6.0 (* (* x1 x1) (* x1 x1)) x1)))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = -1.0 - (x1 * x1);
	double t_5 = t_2 / t_4;
	double t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_6 <= -2e+173) {
		tmp = x1 + (x2 * (x1 * (x2 * 8.0)));
	} else if (t_6 <= 5e+59) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = fma(6.0, ((x1 * x1) * (x1 * x1)), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(-1.0 - Float64(x1 * x1))
	t_5 = Float64(t_2 / t_4)
	t_6 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_5))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_5))) * t_4) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_6 <= -2e+173)
		tmp = Float64(x1 + Float64(x2 * Float64(x1 * Float64(x2 * 8.0))));
	elseif (t_6 <= 5e+59)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	else
		tmp = fma(6.0, Float64(Float64(x1 * x1) * Float64(x1 * x1)), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 + N[(N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -2e+173], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 5e+59], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], N[(6.0 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $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)))))) < -2e173

   1. Initial program 99.8%

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

   3. 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. Applied rewrites 82.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   if -2e173 < (+.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)))))) < 4.9999999999999997e59

   1. Initial program 99.2%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 91.9%

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

   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. lower-*.f64 90.9

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

   11. Applied rewrites 90.9%

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

   if 4.9999999999999997e59 < (+.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 46.1%

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

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - 3 \cdot \frac{1}{x1}\right)} \]
   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 81.7

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

   5. Applied rewrites 81.7%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + \frac{-3}{x1}\right)} \]
   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 81.7

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

      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. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 81.7

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

   7. Applied rewrites 81.7%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 81.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2 \cdot 10^{+173}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+59}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(6, \left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. 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 100.0

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

   5. Applied rewrites 100.0%

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

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

      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. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 100.0%

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

4. Final simplification 99.6%

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

Alternative 8: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := -1 - x1 \cdot x1\\ t_5 := \frac{t\_2}{t\_4}\\ \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_5\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_5\right)\right) \cdot t\_4 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right) \leq 2 \cdot 10^{+255}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (- -1.0 (* x1 x1)))
        (t_5 (/ t_2 t_4)))
   (if (<=
        (+
         x1
         (+
          (+
           x1
           (+
            (+
             (*
              (+
               (* (* x1 x1) (+ 6.0 (* 4.0 t_5)))
               (* (* (* x1 2.0) t_3) (+ 3.0 t_5)))
              t_4)
             (* t_1 t_3))
            (* x1 (* x1 x1))))
          (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))
        2e+255)
     (- (* x2 -6.0) x1)
     (* x1 (fma x1 9.0 -1.0)))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = -1.0 - (x1 * x1);
	double t_5 = t_2 / t_4;
	double tmp;
	if ((x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)))) <= 2e+255) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = x1 * fma(x1, 9.0, -1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(-1.0 - Float64(x1 * x1))
	t_5 = Float64(t_2 / t_4)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_5))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_5))) * t_4) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0)))) <= 2e+255)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	else
		tmp = Float64(x1 * fma(x1, 9.0, -1.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$4), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+255], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], N[(x1 * N[(x1 * 9.0 + -1.0), $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)))))) < 1.99999999999999998e255

   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 x2 around 0

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 68.8%

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

   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. lower-*.f64 64.5

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

   11. Applied rewrites 64.5%

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

   if 1.99999999999999998e255 < (+.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 37.2%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 24.4%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 52.8%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. distribute-lft-out N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. associate-+l+ N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 57.2

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

   11. Applied rewrites 57.2%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 2 \cdot 10^{+255}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := -1 - x1 \cdot x1\\ t_5 := \frac{t\_2}{t\_4}\\ \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_5\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_5\right)\right) \cdot t\_4 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right) \leq 4 \cdot 10^{+305}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;\left(x1 \cdot x2\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (- -1.0 (* x1 x1)))
        (t_5 (/ t_2 t_4)))
   (if (<=
        (+
         x1
         (+
          (+
           x1
           (+
            (+
             (*
              (+
               (* (* x1 x1) (+ 6.0 (* 4.0 t_5)))
               (* (* (* x1 2.0) t_3) (+ 3.0 t_5)))
              t_4)
             (* t_1 t_3))
            (* x1 (* x1 x1))))
          (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))
        4e+305)
     (- (* x2 -6.0) x1)
     (* (* x1 x2) 12.0))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = -1.0 - (x1 * x1);
	double t_5 = t_2 / t_4;
	double tmp;
	if ((x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)))) <= 4e+305) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = (x1 * x2) * 12.0;
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (x1 * x1) + 1.0d0
    t_1 = x1 * (x1 * 3.0d0)
    t_2 = (t_1 + (2.0d0 * x2)) - x1
    t_3 = t_2 / t_0
    t_4 = (-1.0d0) - (x1 * x1)
    t_5 = t_2 / t_4
    if ((x1 + ((x1 + ((((((x1 * x1) * (6.0d0 + (4.0d0 * t_5))) + (((x1 * 2.0d0) * t_3) * (3.0d0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0d0 * (((t_1 - (2.0d0 * x2)) - x1) / t_0)))) <= 4d+305) then
        tmp = (x2 * (-6.0d0)) - x1
    else
        tmp = (x1 * x2) * 12.0d0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = -1.0 - (x1 * x1);
	double t_5 = t_2 / t_4;
	double tmp;
	if ((x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)))) <= 4e+305) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = (x1 * x2) * 12.0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * x1) + 1.0
	t_1 = x1 * (x1 * 3.0)
	t_2 = (t_1 + (2.0 * x2)) - x1
	t_3 = t_2 / t_0
	t_4 = -1.0 - (x1 * x1)
	t_5 = t_2 / t_4
	tmp = 0
	if (x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)))) <= 4e+305:
		tmp = (x2 * -6.0) - x1
	else:
		tmp = (x1 * x2) * 12.0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(-1.0 - Float64(x1 * x1))
	t_5 = Float64(t_2 / t_4)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_5))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_5))) * t_4) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0)))) <= 4e+305)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	else
		tmp = Float64(Float64(x1 * x2) * 12.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * x1) + 1.0;
	t_1 = x1 * (x1 * 3.0);
	t_2 = (t_1 + (2.0 * x2)) - x1;
	t_3 = t_2 / t_0;
	t_4 = -1.0 - (x1 * x1);
	t_5 = t_2 / t_4;
	tmp = 0.0;
	if ((x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)))) <= 4e+305)
		tmp = (x2 * -6.0) - x1;
	else
		tmp = (x1 * x2) * 12.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$4), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+305], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], N[(N[(x1 * x2), $MachinePrecision] * 12.0), $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)))))) < 3.9999999999999998e305

   1. Initial program 99.2%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 66.1%

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

   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. lower-*.f64 61.8

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

   11. Applied rewrites 61.8%

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

   if 3.9999999999999998e305 < (+.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 34.2%

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

   3. Taylor expanded in x1 around -inf

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

      1. 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. Applied rewrites 91.8%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 12.3

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

   8. Applied rewrites 12.3%

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

   9. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(x1 \cdot x2\right) \cdot 12} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(x1 \cdot x2\right) \cdot 12} \]

       3. lower-*.f64 11.6

          \[\leadsto \color{blue}{\left(x1 \cdot x2\right)} \cdot 12 \]

   11. Applied rewrites 11.6%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 4 \cdot 10^{+305}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;\left(x1 \cdot x2\right) \cdot 12\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 6 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right), x1\right)\\ \mathbf{if}\;x1 \leq -2.3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \left(3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + 4 \cdot \left(x2 \cdot \left(x1 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (fma
          x1
          (fma
           x1
           (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))
           (* 6.0 (fma x2 2.0 -3.0)))
          x1)))
   (if (<= x1 -2.3)
     t_0
     (if (<= x1 740000000.0)
       (+
        x1
        (+
         (* 3.0 (/ (- (- (* x1 (* x1 3.0)) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))
         (+ x1 (+ (* x1 (* x1 x1)) (* 4.0 (* x2 (* x1 (fma x2 2.0 -3.0))))))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma(x1, fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), (6.0 * fma(x2, 2.0, -3.0))), x1);
	double tmp;
	if (x1 <= -2.3) {
		tmp = t_0;
	} else if (x1 <= 740000000.0) {
		tmp = x1 + ((3.0 * ((((x1 * (x1 * 3.0)) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))) + (x1 + ((x1 * (x1 * x1)) + (4.0 * (x2 * (x1 * fma(x2, 2.0, -3.0)))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(x1, fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), Float64(6.0 * fma(x2, 2.0, -3.0))), x1)
	tmp = 0.0
	if (x1 <= -2.3)
		tmp = t_0;
	elseif (x1 <= 740000000.0)
		tmp = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(Float64(Float64(x1 * Float64(x1 * 3.0)) - Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0))) + Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(4.0 * Float64(x2 * Float64(x1 * fma(x2, 2.0, -3.0))))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2.3], t$95$0, If[LessEqual[x1, 740000000.0], N[(x1 + N[(N[(3.0 * N[(N[(N[(N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(x2 * N[(x1 * N[(x2 * 2.0 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.2999999999999998 or 7.4e8 < x1

   1. Initial program 43.4%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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. Applied rewrites 94.6%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 94.6%

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

   if -2.2999999999999998 < x1 < 7.4e8

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.3:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 6 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right), x1\right)\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \left(3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + 4 \cdot \left(x2 \cdot \left(x1 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 6 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 6 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right), x1\right)\\ \mathbf{if}\;x1 \leq -2.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \left(3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (fma
          x1
          (fma
           x1
           (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))
           (* 6.0 (fma x2 2.0 -3.0)))
          x1)))
   (if (<= x1 -2.5)
     t_0
     (if (<= x1 740000000.0)
       (+
        x1
        (+
         (* 3.0 (/ (- (- (* x1 (* x1 3.0)) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))
         (+ x1 (* 4.0 (* x2 (* x1 (fma x2 2.0 -3.0)))))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma(x1, fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), (6.0 * fma(x2, 2.0, -3.0))), x1);
	double tmp;
	if (x1 <= -2.5) {
		tmp = t_0;
	} else if (x1 <= 740000000.0) {
		tmp = x1 + ((3.0 * ((((x1 * (x1 * 3.0)) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))) + (x1 + (4.0 * (x2 * (x1 * fma(x2, 2.0, -3.0))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(x1, fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), Float64(6.0 * fma(x2, 2.0, -3.0))), x1)
	tmp = 0.0
	if (x1 <= -2.5)
		tmp = t_0;
	elseif (x1 <= 740000000.0)
		tmp = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(Float64(Float64(x1 * Float64(x1 * 3.0)) - Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0))) + Float64(x1 + Float64(4.0 * Float64(x2 * Float64(x1 * fma(x2, 2.0, -3.0)))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2.5], t$95$0, If[LessEqual[x1, 740000000.0], N[(x1 + N[(N[(3.0 * N[(N[(N[(N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(4.0 * N[(x2 * N[(x1 * N[(x2 * 2.0 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.5 or 7.4e8 < x1

   1. Initial program 43.4%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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. Applied rewrites 94.6%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 94.6%

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

   if -2.5 < x1 < 7.4e8

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.5:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 6 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right), x1\right)\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \left(3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 6 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 6 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right), x1\right)\\ \mathbf{if}\;x1 \leq -2.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, 2 \cdot x2, -12\right), -2\right)\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (fma
          x1
          (fma
           x1
           (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))
           (* 6.0 (fma x2 2.0 -3.0)))
          x1)))
   (if (<= x1 -2.5)
     t_0
     (if (<= x1 740000000.0)
       (+
        x1
        (fma
         x1
         (fma
          x1
          (fma
           x2
           -4.0
           (fma
            2.0
            (fma x2 -2.0 3.0)
            (fma 3.0 (fma x2 2.0 3.0) (fma x2 14.0 -6.0))))
          (fma x2 (fma 4.0 (* 2.0 x2) -12.0) -2.0))
         (* x2 -6.0)))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma(x1, fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), (6.0 * fma(x2, 2.0, -3.0))), x1);
	double tmp;
	if (x1 <= -2.5) {
		tmp = t_0;
	} else if (x1 <= 740000000.0) {
		tmp = x1 + fma(x1, fma(x1, fma(x2, -4.0, fma(2.0, fma(x2, -2.0, 3.0), fma(3.0, fma(x2, 2.0, 3.0), fma(x2, 14.0, -6.0)))), fma(x2, fma(4.0, (2.0 * x2), -12.0), -2.0)), (x2 * -6.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(x1, fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), Float64(6.0 * fma(x2, 2.0, -3.0))), x1)
	tmp = 0.0
	if (x1 <= -2.5)
		tmp = t_0;
	elseif (x1 <= 740000000.0)
		tmp = Float64(x1 + fma(x1, fma(x1, fma(x2, -4.0, fma(2.0, fma(x2, -2.0, 3.0), fma(3.0, fma(x2, 2.0, 3.0), fma(x2, 14.0, -6.0)))), fma(x2, fma(4.0, Float64(2.0 * x2), -12.0), -2.0)), Float64(x2 * -6.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2.5], t$95$0, If[LessEqual[x1, 740000000.0], N[(x1 + N[(x1 * N[(x1 * N[(x2 * -4.0 + N[(2.0 * N[(x2 * -2.0 + 3.0), $MachinePrecision] + N[(3.0 * N[(x2 * 2.0 + 3.0), $MachinePrecision] + N[(x2 * 14.0 + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x2 * N[(4.0 * N[(2.0 * x2), $MachinePrecision] + -12.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.5 or 7.4e8 < x1

   1. Initial program 43.4%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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. Applied rewrites 94.6%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 94.6%

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

   if -2.5 < x1 < 7.4e8

   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. Applied rewrites 87.4%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.5:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 6 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right), x1\right)\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, 2 \cdot x2, -12\right), -2\right)\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 6 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 6 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right), x1\right)\\ \mathbf{if}\;x1 \leq -2.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (fma
          x1
          (fma
           x1
           (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))
           (* 6.0 (fma x2 2.0 -3.0)))
          x1)))
   (if (<= x1 -2.5)
     t_0
     (if (<= x1 740000000.0)
       (+ x1 (fma x2 -6.0 (* x1 (fma x2 (fma x2 8.0 -12.0) -2.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma(x1, fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), (6.0 * fma(x2, 2.0, -3.0))), x1);
	double tmp;
	if (x1 <= -2.5) {
		tmp = t_0;
	} else if (x1 <= 740000000.0) {
		tmp = x1 + fma(x2, -6.0, (x1 * fma(x2, fma(x2, 8.0, -12.0), -2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(x1, fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), Float64(6.0 * fma(x2, 2.0, -3.0))), x1)
	tmp = 0.0
	if (x1 <= -2.5)
		tmp = t_0;
	elseif (x1 <= 740000000.0)
		tmp = Float64(x1 + fma(x2, -6.0, Float64(x1 * fma(x2, fma(x2, 8.0, -12.0), -2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2.5], t$95$0, If[LessEqual[x1, 740000000.0], N[(x1 + N[(x2 * -6.0 + N[(x1 * N[(x2 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.5 or 7.4e8 < x1

   1. Initial program 43.4%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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. Applied rewrites 94.6%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 94.6%

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

   if -2.5 < x1 < 7.4e8

   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. Applied rewrites 87.4%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 86.9

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

   8. Applied rewrites 86.9%

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

Alternative 14: 86.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right) \cdot \left(6 + \frac{-3}{x1}\right)\\ \mathbf{if}\;x1 \leq -1.18 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (* (* x1 (* x1 (* x1 x1))) (+ 6.0 (/ -3.0 x1))))))
   (if (<= x1 -1.18e+45)
     t_0
     (if (<= x1 740000000.0)
       (+ x1 (fma x2 -6.0 (* x1 (fma x2 (fma x2 8.0 -12.0) -2.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + ((x1 * (x1 * (x1 * x1))) * (6.0 + (-3.0 / x1)));
	double tmp;
	if (x1 <= -1.18e+45) {
		tmp = t_0;
	} else if (x1 <= 740000000.0) {
		tmp = x1 + fma(x2, -6.0, (x1 * fma(x2, fma(x2, 8.0, -12.0), -2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64(Float64(x1 * Float64(x1 * Float64(x1 * x1))) * Float64(6.0 + Float64(-3.0 / x1))))
	tmp = 0.0
	if (x1 <= -1.18e+45)
		tmp = t_0;
	elseif (x1 <= 740000000.0)
		tmp = Float64(x1 + fma(x2, -6.0, Float64(x1 * fma(x2, fma(x2, 8.0, -12.0), -2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[(x1 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(6.0 + N[(-3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.18e+45], t$95$0, If[LessEqual[x1, 740000000.0], N[(x1 + N[(x2 * -6.0 + N[(x1 * N[(x2 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.17999999999999993e45 or 7.4e8 < x1

   1. Initial program 38.6%

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

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - 3 \cdot \frac{1}{x1}\right)} \]
   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 91.8

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

   5. Applied rewrites 91.8%

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 91.8

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

   7. Applied rewrites 91.8%

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

   if -1.17999999999999993e45 < x1 < 7.4e8

   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. Applied rewrites 84.2%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 83.7

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

   8. Applied rewrites 83.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.18 \cdot 10^{+45}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right) \cdot \left(6 + \frac{-3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right) \cdot \left(6 + \frac{-3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 86.5% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 + \frac{-3}{x1}\right), x1, x1\right)\\ \mathbf{if}\;x1 \leq -1.18 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* (* x1 (* x1 x1)) (+ 6.0 (/ -3.0 x1))) x1 x1)))
   (if (<= x1 -1.18e+45)
     t_0
     (if (<= x1 740000000.0)
       (+ x1 (fma x2 -6.0 (* x1 (fma x2 (fma x2 8.0 -12.0) -2.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma(((x1 * (x1 * x1)) * (6.0 + (-3.0 / x1))), x1, x1);
	double tmp;
	if (x1 <= -1.18e+45) {
		tmp = t_0;
	} else if (x1 <= 740000000.0) {
		tmp = x1 + fma(x2, -6.0, (x1 * fma(x2, fma(x2, 8.0, -12.0), -2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(Float64(x1 * Float64(x1 * x1)) * Float64(6.0 + Float64(-3.0 / x1))), x1, x1)
	tmp = 0.0
	if (x1 <= -1.18e+45)
		tmp = t_0;
	elseif (x1 <= 740000000.0)
		tmp = Float64(x1 + fma(x2, -6.0, Float64(x1 * fma(x2, fma(x2, 8.0, -12.0), -2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 + N[(-3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1 + x1), $MachinePrecision]}, If[LessEqual[x1, -1.18e+45], t$95$0, If[LessEqual[x1, 740000000.0], N[(x1 + N[(x2 * -6.0 + N[(x1 * N[(x2 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.17999999999999993e45 or 7.4e8 < x1

   1. Initial program 38.6%

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

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - 3 \cdot \frac{1}{x1}\right)} \]
   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 91.8

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

   5. Applied rewrites 91.8%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + \frac{-3}{x1}\right)} \]
   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 91.8

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

      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. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 91.8

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

   7. Applied rewrites 91.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 91.8

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

   9. Applied rewrites 91.8%

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

   if -1.17999999999999993e45 < x1 < 7.4e8

   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. Applied rewrites 84.2%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 83.7

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

   8. Applied rewrites 83.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.18 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 + \frac{-3}{x1}\right), x1, x1\right)\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 + \frac{-3}{x1}\right), x1, x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 86.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(x1, 6, -3\right), x1\right)\\ \mathbf{if}\;x1 \leq -1.18 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* x1 x1) (* x1 (fma x1 6.0 -3.0)) x1)))
   (if (<= x1 -1.18e+45)
     t_0
     (if (<= x1 740000000.0)
       (+ x1 (fma x2 -6.0 (* x1 (fma x2 (fma x2 8.0 -12.0) -2.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * x1), (x1 * fma(x1, 6.0, -3.0)), x1);
	double tmp;
	if (x1 <= -1.18e+45) {
		tmp = t_0;
	} else if (x1 <= 740000000.0) {
		tmp = x1 + fma(x2, -6.0, (x1 * fma(x2, fma(x2, 8.0, -12.0), -2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * x1), Float64(x1 * fma(x1, 6.0, -3.0)), x1)
	tmp = 0.0
	if (x1 <= -1.18e+45)
		tmp = t_0;
	elseif (x1 <= 740000000.0)
		tmp = Float64(x1 + fma(x2, -6.0, Float64(x1 * fma(x2, fma(x2, 8.0, -12.0), -2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -1.18e+45], t$95$0, If[LessEqual[x1, 740000000.0], N[(x1 + N[(x2 * -6.0 + N[(x1 * N[(x2 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.17999999999999993e45 or 7.4e8 < x1

   1. Initial program 38.6%

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

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - 3 \cdot \frac{1}{x1}\right)} \]
   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 91.8

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

   5. Applied rewrites 91.8%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + \frac{-3}{x1}\right)} \]
   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 91.8

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

      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. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 91.8

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

   7. Applied rewrites 91.8%

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

   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. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 91.8

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

   10. Applied rewrites 91.8%

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

   if -1.17999999999999993e45 < x1 < 7.4e8

   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. Applied rewrites 84.2%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 83.7

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

   8. Applied rewrites 83.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.18 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(x1, 6, -3\right), x1\right)\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(x1, 6, -3\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.4% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.4 \cdot 10^{+44}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot -19\right)\right)\\ \mathbf{elif}\;x1 \leq 0.19:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 1.7 \cdot 10^{+150}:\\ \;\;\;\;x2 \cdot \frac{x1}{x2}\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.4e+44)
   (* x1 (* x1 (* x1 -19.0)))
   (if (<= x1 0.19)
     (- (* x2 -6.0) x1)
     (if (<= x1 1.7e+150) (* x2 (/ x1 x2)) (* x1 (fma x1 9.0 -1.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.4e+44) {
		tmp = x1 * (x1 * (x1 * -19.0));
	} else if (x1 <= 0.19) {
		tmp = (x2 * -6.0) - x1;
	} else if (x1 <= 1.7e+150) {
		tmp = x2 * (x1 / x2);
	} else {
		tmp = x1 * fma(x1, 9.0, -1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.4e+44)
		tmp = Float64(x1 * Float64(x1 * Float64(x1 * -19.0)));
	elseif (x1 <= 0.19)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	elseif (x1 <= 1.7e+150)
		tmp = Float64(x2 * Float64(x1 / x2));
	else
		tmp = Float64(x1 * fma(x1, 9.0, -1.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.4e+44], N[(x1 * N[(x1 * N[(x1 * -19.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 0.19], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], If[LessEqual[x1, 1.7e+150], N[(x2 * N[(x1 / x2), $MachinePrecision]), $MachinePrecision], N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -2.40000000000000013e44

   1. Initial program 26.0%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 18.5%

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

   6. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(x1 \cdot \left(-19 \cdot x1 + 3 \cdot \left(3 - -2 \cdot x2\right)\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(x1 \cdot \left(-19 \cdot x1 + 3 \cdot \left(3 - -2 \cdot x2\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, x1 \cdot \left(-19 \cdot x1 + 3 \cdot \left(3 - -2 \cdot x2\right)\right) - 2, -6 \cdot x2\right)} \]

   8. Applied rewrites 80.0%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -19, \mathsf{fma}\left(x2, 6, 9\right)\right), -2\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{-19 \cdot {x1}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{x1}^{3} \cdot -19} \]

       2. cube-mult N/A

          \[\leadsto \color{blue}{\left(x1 \cdot \left(x1 \cdot x1\right)\right)} \cdot -19 \]

       3. unpow2 N/A

          \[\leadsto \left(x1 \cdot \color{blue}{{x1}^{2}}\right) \cdot -19 \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{x1 \cdot \left({x1}^{2} \cdot -19\right)} \]

       5. unpow2 N/A

          \[\leadsto x1 \cdot \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot -19\right) \]

       6. associate-*r* N/A

          \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot -19\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(-19 \cdot x1\right)}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(-19 \cdot x1\right)\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(-19 \cdot x1\right)\right)} \]

       10. *-commutative N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot -19\right)}\right) \]

       11. lower-*.f64 75.3

           \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot -19\right)}\right) \]

   11. Applied rewrites 75.3%

       \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(x1 \cdot -19\right)\right)} \]

   if -2.40000000000000013e44 < x1 < 0.19

   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 x2 around 0

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 68.7%

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

   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. lower-*.f64 68.3

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

   11. Applied rewrites 68.3%

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

   if 0.19 < x1 < 1.69999999999999991e150

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.9

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

   5. Applied rewrites 3.9%

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

   6. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 19.9

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

   8. Applied rewrites 19.9%

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

   9. Taylor expanded in x1 around inf

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

       1. lower-/.f64 20.5

          \[\leadsto x2 \cdot \color{blue}{\frac{x1}{x2}} \]

   11. Applied rewrites 20.5%

       \[\leadsto x2 \cdot \color{blue}{\frac{x1}{x2}} \]

   if 1.69999999999999991e150 < x1

   1. Initial program 3.1%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 72.2%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. distribute-lft-out N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. associate-+l+ N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 97.2

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

   11. Applied rewrites 97.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.4 \cdot 10^{+44}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot -19\right)\right)\\ \mathbf{elif}\;x1 \leq 0.19:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 1.7 \cdot 10^{+150}:\\ \;\;\;\;x2 \cdot \frac{x1}{x2}\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 68.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.4 \cdot 10^{+44}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot -19\right)\right)\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 1.4 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x2, 6, 9\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.4e+44)
   (* x1 (* x1 (* x1 -19.0)))
   (if (<= x1 740000000.0)
     (- (* x2 -6.0) x1)
     (if (<= x1 1.4e+153)
       (* x1 (* x1 (fma x2 6.0 9.0)))
       (* x1 (fma x1 9.0 -1.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.4e+44) {
		tmp = x1 * (x1 * (x1 * -19.0));
	} else if (x1 <= 740000000.0) {
		tmp = (x2 * -6.0) - x1;
	} else if (x1 <= 1.4e+153) {
		tmp = x1 * (x1 * fma(x2, 6.0, 9.0));
	} else {
		tmp = x1 * fma(x1, 9.0, -1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.4e+44)
		tmp = Float64(x1 * Float64(x1 * Float64(x1 * -19.0)));
	elseif (x1 <= 740000000.0)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	elseif (x1 <= 1.4e+153)
		tmp = Float64(x1 * Float64(x1 * fma(x2, 6.0, 9.0)));
	else
		tmp = Float64(x1 * fma(x1, 9.0, -1.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.4e+44], N[(x1 * N[(x1 * N[(x1 * -19.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 740000000.0], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], If[LessEqual[x1, 1.4e+153], N[(x1 * N[(x1 * N[(x2 * 6.0 + 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -2.40000000000000013e44

   1. Initial program 26.0%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 18.5%

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

   6. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(x1 \cdot \left(-19 \cdot x1 + 3 \cdot \left(3 - -2 \cdot x2\right)\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(x1 \cdot \left(-19 \cdot x1 + 3 \cdot \left(3 - -2 \cdot x2\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, x1 \cdot \left(-19 \cdot x1 + 3 \cdot \left(3 - -2 \cdot x2\right)\right) - 2, -6 \cdot x2\right)} \]

   8. Applied rewrites 80.0%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -19, \mathsf{fma}\left(x2, 6, 9\right)\right), -2\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{-19 \cdot {x1}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{x1}^{3} \cdot -19} \]

       2. cube-mult N/A

          \[\leadsto \color{blue}{\left(x1 \cdot \left(x1 \cdot x1\right)\right)} \cdot -19 \]

       3. unpow2 N/A

          \[\leadsto \left(x1 \cdot \color{blue}{{x1}^{2}}\right) \cdot -19 \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{x1 \cdot \left({x1}^{2} \cdot -19\right)} \]

       5. unpow2 N/A

          \[\leadsto x1 \cdot \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot -19\right) \]

       6. associate-*r* N/A

          \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot -19\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(-19 \cdot x1\right)}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(-19 \cdot x1\right)\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(-19 \cdot x1\right)\right)} \]

       10. *-commutative N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot -19\right)}\right) \]

       11. lower-*.f64 75.3

           \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot -19\right)}\right) \]

   11. Applied rewrites 75.3%

       \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(x1 \cdot -19\right)\right)} \]

   if -2.40000000000000013e44 < x1 < 7.4e8

   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 x2 around 0

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 67.2%

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

   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. lower-*.f64 66.8

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

   11. Applied rewrites 66.8%

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

   if 7.4e8 < x1 < 1.39999999999999993e153

   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 x2 around 0

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

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 20.5%

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

   9. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(x1 \cdot x1\right)} \cdot \left(9 + 6 \cdot x2\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(9 + 6 \cdot x2\right)\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(9 + 6 \cdot x2\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + 6 \cdot x2\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(6 \cdot x2 + 9\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto x1 \cdot \left(x1 \cdot \left(\color{blue}{x2 \cdot 6} + 9\right)\right) \]

       7. lower-fma.f64 20.5

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

   11. Applied rewrites 20.5%

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

   if 1.39999999999999993e153 < x1

   1. Initial program 0.0%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 74.2%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. distribute-lft-out N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. associate-+l+ N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.4 \cdot 10^{+44}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot -19\right)\right)\\ \mathbf{elif}\;x1 \leq 740000000:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 1.4 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x2, 6, 9\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 67.4% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.4 \cdot 10^{+44}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot -19\right)\right)\\ \mathbf{elif}\;x1 \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.4e+44)
   (* x1 (* x1 (* x1 -19.0)))
   (if (<= x1 9.5e-11) (- (* x2 -6.0) x1) (* x1 (fma x1 9.0 -1.0)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.4e+44) {
		tmp = x1 * (x1 * (x1 * -19.0));
	} else if (x1 <= 9.5e-11) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = x1 * fma(x1, 9.0, -1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.4e+44)
		tmp = Float64(x1 * Float64(x1 * Float64(x1 * -19.0)));
	elseif (x1 <= 9.5e-11)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	else
		tmp = Float64(x1 * fma(x1, 9.0, -1.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.4e+44], N[(x1 * N[(x1 * N[(x1 * -19.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 9.5e-11], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.40000000000000013e44

   1. Initial program 26.0%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 18.5%

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

   6. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(x1 \cdot \left(-19 \cdot x1 + 3 \cdot \left(3 - -2 \cdot x2\right)\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(x1 \cdot \left(-19 \cdot x1 + 3 \cdot \left(3 - -2 \cdot x2\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, x1 \cdot \left(-19 \cdot x1 + 3 \cdot \left(3 - -2 \cdot x2\right)\right) - 2, -6 \cdot x2\right)} \]

   8. Applied rewrites 80.0%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -19, \mathsf{fma}\left(x2, 6, 9\right)\right), -2\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{-19 \cdot {x1}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{x1}^{3} \cdot -19} \]

       2. cube-mult N/A

          \[\leadsto \color{blue}{\left(x1 \cdot \left(x1 \cdot x1\right)\right)} \cdot -19 \]

       3. unpow2 N/A

          \[\leadsto \left(x1 \cdot \color{blue}{{x1}^{2}}\right) \cdot -19 \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{x1 \cdot \left({x1}^{2} \cdot -19\right)} \]

       5. unpow2 N/A

          \[\leadsto x1 \cdot \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot -19\right) \]

       6. associate-*r* N/A

          \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot -19\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(-19 \cdot x1\right)}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(-19 \cdot x1\right)\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(-19 \cdot x1\right)\right)} \]

       10. *-commutative N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot -19\right)}\right) \]

       11. lower-*.f64 75.3

           \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot -19\right)}\right) \]

   11. Applied rewrites 75.3%

       \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(x1 \cdot -19\right)\right)} \]

   if -2.40000000000000013e44 < x1 < 9.49999999999999951e-11

   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 x2 around 0

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 69.0%

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

   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. lower-*.f64 68.7

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

   11. Applied rewrites 68.7%

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

   if 9.49999999999999951e-11 < x1

   1. Initial program 54.9%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 42.6%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 44.0%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. distribute-lft-out N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. associate-+l+ N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 48.7

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

   11. Applied rewrites 48.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.4 \cdot 10^{+44}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot -19\right)\right)\\ \mathbf{elif}\;x1 \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 28.4% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.8 \cdot 10^{-16}:\\ \;\;\;\;x1 \cdot -17\\ \mathbf{else}:\\ \;\;\;\;x1 + x2 \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.8e-16) (* x1 -17.0) (+ x1 (* x2 -6.0))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.8e-16) {
		tmp = x1 * -17.0;
	} else {
		tmp = x1 + (x2 * -6.0);
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if (x1 <= (-2.8d-16)) then
        tmp = x1 * (-17.0d0)
    else
        tmp = x1 + (x2 * (-6.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.8e-16) {
		tmp = x1 * -17.0;
	} else {
		tmp = x1 + (x2 * -6.0);
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if x1 <= -2.8e-16:
		tmp = x1 * -17.0
	else:
		tmp = x1 + (x2 * -6.0)
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.8e-16)
		tmp = Float64(x1 * -17.0);
	else
		tmp = Float64(x1 + Float64(x2 * -6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if (x1 <= -2.8e-16)
		tmp = x1 * -17.0;
	else
		tmp = x1 + (x2 * -6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.8e-16], N[(x1 * -17.0), $MachinePrecision], N[(x1 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.8000000000000001e-16

   1. Initial program 38.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. 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. Applied rewrites 88.4%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 10.5

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

   8. Applied rewrites 10.5%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 + -18 \cdot x1} \]
   10. Step-by-step derivation

       1. distribute-rgt1-in N/A

          \[\leadsto \color{blue}{\left(-18 + 1\right) \cdot x1} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{x1 \cdot \left(-18 + 1\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{x1 \cdot \left(-18 + 1\right)} \]

       4. metadata-eval 6.8

          \[\leadsto x1 \cdot \color{blue}{-17} \]

   11. Applied rewrites 6.8%

       \[\leadsto \color{blue}{x1 \cdot -17} \]

   if -2.8000000000000001e-16 < x1

   1. Initial program 82.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 34.6

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

   5. Applied rewrites 34.6%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 28.4% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.8 \cdot 10^{-16}:\\ \;\;\;\;x1 \cdot -17\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.8e-16) (* x1 -17.0) (fma x2 -6.0 x1)))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.8e-16) {
		tmp = x1 * -17.0;
	} else {
		tmp = fma(x2, -6.0, x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.8e-16)
		tmp = Float64(x1 * -17.0);
	else
		tmp = fma(x2, -6.0, x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.8e-16], N[(x1 * -17.0), $MachinePrecision], N[(x2 * -6.0 + x1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.8000000000000001e-16

   1. Initial program 38.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. 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. Applied rewrites 88.4%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 10.5

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

   8. Applied rewrites 10.5%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 + -18 \cdot x1} \]
   10. Step-by-step derivation

       1. distribute-rgt1-in N/A

          \[\leadsto \color{blue}{\left(-18 + 1\right) \cdot x1} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{x1 \cdot \left(-18 + 1\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{x1 \cdot \left(-18 + 1\right)} \]

       4. metadata-eval 6.8

          \[\leadsto x1 \cdot \color{blue}{-17} \]

   11. Applied rewrites 6.8%

       \[\leadsto \color{blue}{x1 \cdot -17} \]

   if -2.8000000000000001e-16 < x1

   1. Initial program 82.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 34.6

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

   5. Applied rewrites 34.6%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]
   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 34.6

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

   7. Applied rewrites 34.6%

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

Alternative 22: 39.0% 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 68.7%

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

3. Taylor expanded in x2 around 0

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

5. Applied rewrites 50.6%

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

6. Taylor expanded in x1 around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

8. Applied rewrites 60.9%

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

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. lower-*.f64 34.5

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

11. Applied rewrites 34.5%

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

12. Final simplification 34.5%

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

Alternative 23: 27.0% 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 68.7%

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

3. Taylor expanded in x1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 24.2

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

5. Applied rewrites 24.2%

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

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 23.9

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

8. Applied rewrites 23.9%

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

Alternative 24: 4.6% accurate, 49.7× speedup

Math

\[\begin{array}{l} \\ x1 \cdot -17 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (* x1 -17.0))

C

double code(double x1, double x2) {
	return x1 * -17.0;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = x1 * (-17.0d0)
end function

Java

public static double code(double x1, double x2) {
	return x1 * -17.0;
}

Python

def code(x1, x2):
	return x1 * -17.0

Julia

function code(x1, x2)
	return Float64(x1 * -17.0)
end

MATLAB

function tmp = code(x1, x2)
	tmp = x1 * -17.0;
end

Wolfram

code[x1_, x2_] := N[(x1 * -17.0), $MachinePrecision]

Derivation

1. Initial program 68.7%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
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. Applied rewrites 54.1%

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

6. Taylor expanded in x1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 9.2

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

8. Applied rewrites 9.2%

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

9. Taylor expanded in x2 around 0

   \[\leadsto \color{blue}{x1 + -18 \cdot x1} \]
10. Step-by-step derivation

    1. distribute-rgt1-in N/A

       \[\leadsto \color{blue}{\left(-18 + 1\right) \cdot x1} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{x1 \cdot \left(-18 + 1\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \color{blue}{x1 \cdot \left(-18 + 1\right)} \]

    4. metadata-eval 4.6

       \[\leadsto x1 \cdot \color{blue}{-17} \]

11. Applied rewrites 4.6%

    \[\leadsto \color{blue}{x1 \cdot -17} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))))