Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.9% → 99.7%

Time: 22.0s

Alternatives: 19

Speedup: 7.8×

• 46.6% 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: 70.9% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   4. Applied rewrites 99.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 63.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \left(\left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_4 := x1 \cdot x1 - -1\\ t_5 := \frac{t\_3}{t\_4}\\ t_6 := x1 - \left(\left(\left(\left(\frac{t\_3}{t\_2} \cdot t\_0 - t\_2 \cdot \left(\left(3 - t\_5\right) \cdot \left(t\_5 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_5 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_4} \cdot 3\right)\\ \mathbf{if}\;t\_6 \leq -1 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot 8 + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* (* (* x2 x1) 8.0) x2) x1))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (- (+ (* x2 2.0) t_0) x1))
        (t_4 (- (* x1 x1) -1.0))
        (t_5 (/ t_3 t_4))
        (t_6
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_3 t_2) t_0)
              (*
               t_2
               (-
                (* (- 3.0 t_5) (* t_5 (* 2.0 x1)))
                (* (- (* 4.0 t_5) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) t_4) 3.0)))))
   (if (<= t_6 -1e+207)
     t_1
     (if (<= t_6 5e+259)
       (fma (fma -12.0 x1 -6.0) x2 (- x1))
       (if (<= t_6 INFINITY) t_1 (+ (* (* (* x1 x1) x2) 8.0) x1))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (((x2 * x1) * 8.0) * x2) + x1;
	double t_2 = -1.0 - (x1 * x1);
	double t_3 = ((x2 * 2.0) + t_0) - x1;
	double t_4 = (x1 * x1) - -1.0;
	double t_5 = t_3 / t_4;
	double t_6 = x1 - ((((((t_3 / t_2) * t_0) - (t_2 * (((3.0 - t_5) * (t_5 * (2.0 * x1))) - (((4.0 * t_5) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_4) * 3.0));
	double tmp;
	if (t_6 <= -1e+207) {
		tmp = t_1;
	} else if (t_6 <= 5e+259) {
		tmp = fma(fma(-12.0, x1, -6.0), x2, -x1);
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (((x1 * x1) * x2) * 8.0) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(Float64(Float64(x2 * x1) * 8.0) * x2) + x1)
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_4 = Float64(Float64(x1 * x1) - -1.0)
	t_5 = Float64(t_3 / t_4)
	t_6 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_3 / t_2) * t_0) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_5) * Float64(t_5 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_5) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_4) * 3.0)))
	tmp = 0.0
	if (t_6 <= -1e+207)
		tmp = t_1;
	elseif (t_6 <= 5e+259)
		tmp = fma(fma(-12.0, x1, -6.0), x2, Float64(-x1));
	elseif (t_6 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(x1 * x1) * x2) * 8.0) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$3 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$5), $MachinePrecision] * N[(t$95$5 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$5), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -1e+207], t$95$1, If[LessEqual[t$95$6, 5e+259], N[(N[(-12.0 * x1 + -6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], If[LessEqual[t$95$6, Infinity], t$95$1, N[(N[(N[(N[(x1 * x1), $MachinePrecision] * x2), $MachinePrecision] * 8.0), $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)))))) < -1e207 or 5.00000000000000033e259 < (+.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 99.9%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

         \[\leadsto x1 + \color{blue}{\frac{8 \cdot x1}{1 + {x1}^{2}} \cdot {x2}^{2}} \]

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 60.0

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 63.2%

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

   if -1e207 < (+.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)))))) < 5.00000000000000033e259

   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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 75.7

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

   9. Applied rewrites 75.7%

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

   10. Taylor expanded in x2 around 0

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

   12. Applied rewrites 78.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 47.7%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq -1 \cdot 10^{+207}:\\ \;\;\;\;\left(\left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq \infty:\\ \;\;\;\;\left(\left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot 8 + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_4 := x1 \cdot x1 - -1\\ t_5 := \frac{t\_3}{t\_4}\\ t_6 := x1 - \left(\left(\left(\left(\frac{t\_3}{t\_2} \cdot t\_0 - t\_2 \cdot \left(\left(3 - t\_5\right) \cdot \left(t\_5 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_5 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_4} \cdot 3\right)\\ \mathbf{if}\;t\_6 \leq -5 \cdot 10^{+272}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;t\_1 + x1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot 8 + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (* (* (* x2 x2) x1) 8.0))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (- (+ (* x2 2.0) t_0) x1))
        (t_4 (- (* x1 x1) -1.0))
        (t_5 (/ t_3 t_4))
        (t_6
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_3 t_2) t_0)
              (*
               t_2
               (-
                (* (- 3.0 t_5) (* t_5 (* 2.0 x1)))
                (* (- (* 4.0 t_5) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) t_4) 3.0)))))
   (if (<= t_6 -5e+272)
     t_1
     (if (<= t_6 5e+259)
       (fma (fma -12.0 x1 -6.0) x2 (- x1))
       (if (<= t_6 INFINITY) (+ t_1 x1) (+ (* (* (* x1 x1) x2) 8.0) x1))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = ((x2 * x2) * x1) * 8.0;
	double t_2 = -1.0 - (x1 * x1);
	double t_3 = ((x2 * 2.0) + t_0) - x1;
	double t_4 = (x1 * x1) - -1.0;
	double t_5 = t_3 / t_4;
	double t_6 = x1 - ((((((t_3 / t_2) * t_0) - (t_2 * (((3.0 - t_5) * (t_5 * (2.0 * x1))) - (((4.0 * t_5) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_4) * 3.0));
	double tmp;
	if (t_6 <= -5e+272) {
		tmp = t_1;
	} else if (t_6 <= 5e+259) {
		tmp = fma(fma(-12.0, x1, -6.0), x2, -x1);
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = t_1 + x1;
	} else {
		tmp = (((x1 * x1) * x2) * 8.0) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(Float64(x2 * x2) * x1) * 8.0)
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_4 = Float64(Float64(x1 * x1) - -1.0)
	t_5 = Float64(t_3 / t_4)
	t_6 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_3 / t_2) * t_0) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_5) * Float64(t_5 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_5) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_4) * 3.0)))
	tmp = 0.0
	if (t_6 <= -5e+272)
		tmp = t_1;
	elseif (t_6 <= 5e+259)
		tmp = fma(fma(-12.0, x1, -6.0), x2, Float64(-x1));
	elseif (t_6 <= Inf)
		tmp = Float64(t_1 + x1);
	else
		tmp = Float64(Float64(Float64(Float64(x1 * x1) * x2) * 8.0) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$3 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$5), $MachinePrecision] * N[(t$95$5 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$5), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -5e+272], t$95$1, If[LessEqual[t$95$6, 5e+259], N[(N[(-12.0 * x1 + -6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(t$95$1 + x1), $MachinePrecision], N[(N[(N[(N[(x1 * x1), $MachinePrecision] * x2), $MachinePrecision] * 8.0), $MachinePrecision] + x1), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -4.99999999999999973e272

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 81.7

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

   9. Applied rewrites 81.7%

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

   10. Taylor expanded in x2 around inf

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

   12. Applied rewrites 81.7%

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

   if -4.99999999999999973e272 < (+.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)))))) < 5.00000000000000033e259

   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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 69.7

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

   9. Applied rewrites 69.7%

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

   10. Taylor expanded in x2 around 0

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

   12. Applied rewrites 72.7%

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

   if 5.00000000000000033e259 < (+.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 99.9%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

         \[\leadsto x1 + \color{blue}{\frac{8 \cdot x1}{1 + {x1}^{2}} \cdot {x2}^{2}} \]

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 45.8

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 43.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 47.7%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq -5 \cdot 10^{+272}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq \infty:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 + x1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot 8 + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_4 := x1 \cdot x1 - -1\\ t_5 := \frac{t\_3}{t\_4}\\ t_6 := x1 - \left(\left(\left(\left(\frac{t\_3}{t\_2} \cdot t\_0 - t\_2 \cdot \left(\left(3 - t\_5\right) \cdot \left(t\_5 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_5 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_4} \cdot 3\right)\\ \mathbf{if}\;t\_6 \leq -5 \cdot 10^{+272}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (* (* (* x2 x2) x1) 8.0))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (- (+ (* x2 2.0) t_0) x1))
        (t_4 (- (* x1 x1) -1.0))
        (t_5 (/ t_3 t_4))
        (t_6
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_3 t_2) t_0)
              (*
               t_2
               (-
                (* (- 3.0 t_5) (* t_5 (* 2.0 x1)))
                (* (- (* 4.0 t_5) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) t_4) 3.0)))))
   (if (<= t_6 -5e+272)
     t_1
     (if (<= t_6 5e+259) (fma (fma -12.0 x1 -6.0) x2 (- x1)) t_1))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = ((x2 * x2) * x1) * 8.0;
	double t_2 = -1.0 - (x1 * x1);
	double t_3 = ((x2 * 2.0) + t_0) - x1;
	double t_4 = (x1 * x1) - -1.0;
	double t_5 = t_3 / t_4;
	double t_6 = x1 - ((((((t_3 / t_2) * t_0) - (t_2 * (((3.0 - t_5) * (t_5 * (2.0 * x1))) - (((4.0 * t_5) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_4) * 3.0));
	double tmp;
	if (t_6 <= -5e+272) {
		tmp = t_1;
	} else if (t_6 <= 5e+259) {
		tmp = fma(fma(-12.0, x1, -6.0), x2, -x1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(Float64(x2 * x2) * x1) * 8.0)
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_4 = Float64(Float64(x1 * x1) - -1.0)
	t_5 = Float64(t_3 / t_4)
	t_6 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_3 / t_2) * t_0) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_5) * Float64(t_5 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_5) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_4) * 3.0)))
	tmp = 0.0
	if (t_6 <= -5e+272)
		tmp = t_1;
	elseif (t_6 <= 5e+259)
		tmp = fma(fma(-12.0, x1, -6.0), x2, Float64(-x1));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$3 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$5), $MachinePrecision] * N[(t$95$5 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$5), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -5e+272], t$95$1, If[LessEqual[t$95$6, 5e+259], N[(N[(-12.0 * x1 + -6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -4.99999999999999973e272 or 5.00000000000000033e259 < (+.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 37.4%

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

   4. Applied rewrites 46.5%

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

   6. Applied rewrites 37.3%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 31.9

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

   9. Applied rewrites 31.9%

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

   10. Taylor expanded in x2 around inf

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

   12. Applied rewrites 31.3%

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

   if -4.99999999999999973e272 < (+.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)))))) < 5.00000000000000033e259

   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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 69.7

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

   9. Applied rewrites 69.7%

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

   10. Taylor expanded in x2 around 0

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

   12. Applied rewrites 72.7%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq -5 \cdot 10^{+272}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.5%

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

   8. Applied rewrites 99.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 6: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.7%

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

Alternative 7: 95.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\\ \mathbf{if}\;x1 \leq -1.56:\\ \;\;\;\;\left(\left(6 - \frac{3 - \frac{t\_0}{x1}}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, -8, 8\right), x2, \mathsf{fma}\left(\mathsf{fma}\left(24, x1, 12\right), x1, -12\right)\right), -6\right), x2, \mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, t\_0\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (fma 2.0 x2 -3.0) 4.0 9.0)))
   (if (<= x1 -1.56)
     (+ (* (* (- 6.0 (/ (- 3.0 (/ t_0 x1)) x1)) (* x1 x1)) (* x1 x1)) x1)
     (if (<= x1 1.4)
       (+
        (fma
         (fma
          x1
          (fma (fma (* x1 x1) -8.0 8.0) x2 (fma (fma 24.0 x1 12.0) x1 -12.0))
          -6.0)
         x2
         (* (fma (fma -19.0 x1 9.0) x1 -2.0) x1))
        x1)
       (+ (* (fma (fma 6.0 x1 -3.0) x1 t_0) (* x1 x1)) x1)))))

C

double code(double x1, double x2) {
	double t_0 = fma(fma(2.0, x2, -3.0), 4.0, 9.0);
	double tmp;
	if (x1 <= -1.56) {
		tmp = (((6.0 - ((3.0 - (t_0 / x1)) / x1)) * (x1 * x1)) * (x1 * x1)) + x1;
	} else if (x1 <= 1.4) {
		tmp = fma(fma(x1, fma(fma((x1 * x1), -8.0, 8.0), x2, fma(fma(24.0, x1, 12.0), x1, -12.0)), -6.0), x2, (fma(fma(-19.0, x1, 9.0), x1, -2.0) * x1)) + x1;
	} else {
		tmp = (fma(fma(6.0, x1, -3.0), x1, t_0) * (x1 * x1)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(fma(2.0, x2, -3.0), 4.0, 9.0)
	tmp = 0.0
	if (x1 <= -1.56)
		tmp = Float64(Float64(Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(t_0 / x1)) / x1)) * Float64(x1 * x1)) * Float64(x1 * x1)) + x1);
	elseif (x1 <= 1.4)
		tmp = Float64(fma(fma(x1, fma(fma(Float64(x1 * x1), -8.0, 8.0), x2, fma(fma(24.0, x1, 12.0), x1, -12.0)), -6.0), x2, Float64(fma(fma(-19.0, x1, 9.0), x1, -2.0) * x1)) + x1);
	else
		tmp = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, t_0) * Float64(x1 * x1)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]}, If[LessEqual[x1, -1.56], N[(N[(N[(N[(6.0 - N[(N[(3.0 - N[(t$95$0 / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1.4], N[(N[(N[(x1 * N[(N[(N[(x1 * x1), $MachinePrecision] * -8.0 + 8.0), $MachinePrecision] * x2 + N[(N[(24.0 * x1 + 12.0), $MachinePrecision] * x1 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] * x2 + N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + t$95$0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.5600000000000001

   1. Initial program 31.3%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.8%

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

   7. Applied rewrites 96.9%

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

   if -1.5600000000000001 < x1 < 1.3999999999999999

   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

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(x1 \cdot \left(x1 \cdot \left(9 + -19 \cdot x1\right) - 2\right) + \color{blue}{x2 \cdot \left(\left(x1 \cdot \left(x2 \cdot \left(8 + -8 \cdot {x1}^{2}\right)\right) + x1 \cdot \left(x1 \cdot \left(12 + 24 \cdot x1\right) - 12\right)\right) - 6\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.0%

      \[\leadsto x1 + \mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, -8, 8\right), x2, \mathsf{fma}\left(\mathsf{fma}\left(24, x1, 12\right), x1, -12\right)\right), -6\right), \color{blue}{x2}, \mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, -2\right) \cdot x1\right) \]

   if 1.3999999999999999 < x1

   1. Initial program 40.3%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.56:\\ \;\;\;\;\left(\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, -8, 8\right), x2, \mathsf{fma}\left(\mathsf{fma}\left(24, x1, 12\right), x1, -12\right)\right), -6\right), x2, \mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -1.56:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, -8, 8\right), x2, \mathsf{fma}\left(\mathsf{fma}\left(24, x1, 12\right), x1, -12\right)\right), -6\right), x2, \mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (fma (fma 6.0 x1 -3.0) x1 (fma (fma 2.0 x2 -3.0) 4.0 9.0))
           (* x1 x1))
          x1)))
   (if (<= x1 -1.56)
     t_0
     (if (<= x1 1.4)
       (+
        (fma
         (fma
          x1
          (fma (fma (* x1 x1) -8.0 8.0) x2 (fma (fma 24.0 x1 12.0) x1 -12.0))
          -6.0)
         x2
         (* (fma (fma -19.0 x1 9.0) x1 -2.0) x1))
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (fma(fma(6.0, x1, -3.0), x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)) * (x1 * x1)) + x1;
	double tmp;
	if (x1 <= -1.56) {
		tmp = t_0;
	} else if (x1 <= 1.4) {
		tmp = fma(fma(x1, fma(fma((x1 * x1), -8.0, 8.0), x2, fma(fma(24.0, x1, 12.0), x1, -12.0)), -6.0), x2, (fma(fma(-19.0, x1, 9.0), x1, -2.0) * x1)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)) * Float64(x1 * x1)) + x1)
	tmp = 0.0
	if (x1 <= -1.56)
		tmp = t_0;
	elseif (x1 <= 1.4)
		tmp = Float64(fma(fma(x1, fma(fma(Float64(x1 * x1), -8.0, 8.0), x2, fma(fma(24.0, x1, 12.0), x1, -12.0)), -6.0), x2, Float64(fma(fma(-19.0, x1, 9.0), x1, -2.0) * x1)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -1.56], t$95$0, If[LessEqual[x1, 1.4], N[(N[(N[(x1 * N[(N[(N[(x1 * x1), $MachinePrecision] * -8.0 + 8.0), $MachinePrecision] * x2 + N[(N[(24.0 * x1 + 12.0), $MachinePrecision] * x1 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] * x2 + N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.5600000000000001 or 1.3999999999999999 < x1

   1. Initial program 35.3%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.8%

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

   if -1.5600000000000001 < x1 < 1.3999999999999999

   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

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(x1 \cdot \left(x1 \cdot \left(9 + -19 \cdot x1\right) - 2\right) + \color{blue}{x2 \cdot \left(\left(x1 \cdot \left(x2 \cdot \left(8 + -8 \cdot {x1}^{2}\right)\right) + x1 \cdot \left(x1 \cdot \left(12 + 24 \cdot x1\right) - 12\right)\right) - 6\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.0%

      \[\leadsto x1 + \mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, -8, 8\right), x2, \mathsf{fma}\left(\mathsf{fma}\left(24, x1, 12\right), x1, -12\right)\right), -6\right), \color{blue}{x2}, \mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, -2\right) \cdot x1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.56:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, -8, 8\right), x2, \mathsf{fma}\left(\mathsf{fma}\left(24, x1, 12\right), x1, -12\right)\right), -6\right), x2, \mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -2700000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 350000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(-12, x1, -6\right)\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (fma (fma 6.0 x1 -3.0) x1 (fma (fma 2.0 x2 -3.0) 4.0 9.0))
           (* x1 x1))
          x1)))
   (if (<= x1 -2700000.0)
     t_0
     (if (<= x1 350000000.0)
       (fma (fma (* x2 x1) 8.0 (fma -12.0 x1 -6.0)) x2 (- x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (fma(fma(6.0, x1, -3.0), x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)) * (x1 * x1)) + x1;
	double tmp;
	if (x1 <= -2700000.0) {
		tmp = t_0;
	} else if (x1 <= 350000000.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(-12.0, x1, -6.0)), x2, -x1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)) * Float64(x1 * x1)) + x1)
	tmp = 0.0
	if (x1 <= -2700000.0)
		tmp = t_0;
	elseif (x1 <= 350000000.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(-12.0, x1, -6.0)), x2, Float64(-x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2700000.0], t$95$0, If[LessEqual[x1, 350000000.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(-12.0 * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.7e6 or 3.5e8 < x1

   1. Initial program 34.3%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 98.2%

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

   if -2.7e6 < x1 < 3.5e8

   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 99.8%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 79.4

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

   9. Applied rewrites 79.4%

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

   10. Taylor expanded in x2 around 0

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

   12. Applied rewrites 97.2%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2700000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 350000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(-12, x1, -6\right)\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, -2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq -6.8 \cdot 10^{-39}:\\ \;\;\;\;\left(\left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\\ \mathbf{elif}\;x1 \leq 4.4 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{elif}\;x1 \leq 7.8 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.2e+75)
   (+ (* (fma (fma -19.0 x1 9.0) x1 -2.0) x1) x1)
   (if (<= x1 -6.8e-39)
     (+ (* (* (* x2 x1) 8.0) x2) x1)
     (if (<= x1 4.4e-27)
       (fma (fma -12.0 x1 -6.0) x2 (- x1))
       (if (<= x1 7.8e+101)
         (fma (* (* x2 x2) 8.0) x1 (* -6.0 x2))
         (+ (fma (* x1 x1) x1 (* -6.0 x2)) x1))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.2e+75) {
		tmp = (fma(fma(-19.0, x1, 9.0), x1, -2.0) * x1) + x1;
	} else if (x1 <= -6.8e-39) {
		tmp = (((x2 * x1) * 8.0) * x2) + x1;
	} else if (x1 <= 4.4e-27) {
		tmp = fma(fma(-12.0, x1, -6.0), x2, -x1);
	} else if (x1 <= 7.8e+101) {
		tmp = fma(((x2 * x2) * 8.0), x1, (-6.0 * x2));
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.2e+75)
		tmp = Float64(Float64(fma(fma(-19.0, x1, 9.0), x1, -2.0) * x1) + x1);
	elseif (x1 <= -6.8e-39)
		tmp = Float64(Float64(Float64(Float64(x2 * x1) * 8.0) * x2) + x1);
	elseif (x1 <= 4.4e-27)
		tmp = fma(fma(-12.0, x1, -6.0), x2, Float64(-x1));
	elseif (x1 <= 7.8e+101)
		tmp = fma(Float64(Float64(x2 * x2) * 8.0), x1, Float64(-6.0 * x2));
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(-6.0 * x2)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.2e+75], N[(N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -6.8e-39], N[(N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 4.4e-27], N[(N[(-12.0 * x1 + -6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], If[LessEqual[x1, 7.8e+101], N[(N[(N[(x2 * x2), $MachinePrecision] * 8.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -1.2e75

   1. Initial program 11.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -19 \cdot x1\right) - 2\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.8%

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

   if -1.2e75 < x1 < -6.7999999999999998e-39

   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 inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

         \[\leadsto x1 + \color{blue}{\frac{8 \cdot x1}{1 + {x1}^{2}} \cdot {x2}^{2}} \]

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 40.8

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

   5. Applied rewrites 40.8%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 40.6%

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

   if -6.7999999999999998e-39 < x1 < 4.39999999999999974e-27

   1. Initial program 99.4%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 79.7

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

   9. Applied rewrites 79.7%

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

   10. Taylor expanded in x2 around 0

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

   12. Applied rewrites 77.9%

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

   if 4.39999999999999974e-27 < x1 < 7.8e101

   1. Initial program 99.6%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 38.9

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

   9. Applied rewrites 38.9%

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

   10. Taylor expanded in x2 around inf

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

   12. Applied rewrites 35.3%

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

   if 7.8e101 < x1

   1. Initial program 20.9%

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, -2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq -6.8 \cdot 10^{-39}:\\ \;\;\;\;\left(\left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\\ \mathbf{elif}\;x1 \leq 4.4 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{elif}\;x1 \leq 7.8 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, -2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq -6.8 \cdot 10^{-39}:\\ \;\;\;\;\left(\left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\\ \mathbf{elif}\;x1 \leq 4.9 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{elif}\;x1 \leq 7.8 \cdot 10^{+101}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.2e+75)
   (+ (* (fma (fma -19.0 x1 9.0) x1 -2.0) x1) x1)
   (if (<= x1 -6.8e-39)
     (+ (* (* (* x2 x1) 8.0) x2) x1)
     (if (<= x1 4.9e-23)
       (fma (fma -12.0 x1 -6.0) x2 (- x1))
       (if (<= x1 7.8e+101)
         (+ (* (* (* x2 x2) x1) 8.0) x1)
         (+ (fma (* x1 x1) x1 (* -6.0 x2)) x1))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.2e+75) {
		tmp = (fma(fma(-19.0, x1, 9.0), x1, -2.0) * x1) + x1;
	} else if (x1 <= -6.8e-39) {
		tmp = (((x2 * x1) * 8.0) * x2) + x1;
	} else if (x1 <= 4.9e-23) {
		tmp = fma(fma(-12.0, x1, -6.0), x2, -x1);
	} else if (x1 <= 7.8e+101) {
		tmp = (((x2 * x2) * x1) * 8.0) + x1;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.2e+75)
		tmp = Float64(Float64(fma(fma(-19.0, x1, 9.0), x1, -2.0) * x1) + x1);
	elseif (x1 <= -6.8e-39)
		tmp = Float64(Float64(Float64(Float64(x2 * x1) * 8.0) * x2) + x1);
	elseif (x1 <= 4.9e-23)
		tmp = fma(fma(-12.0, x1, -6.0), x2, Float64(-x1));
	elseif (x1 <= 7.8e+101)
		tmp = Float64(Float64(Float64(Float64(x2 * x2) * x1) * 8.0) + x1);
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(-6.0 * x2)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.2e+75], N[(N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -6.8e-39], N[(N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 4.9e-23], N[(N[(-12.0 * x1 + -6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], If[LessEqual[x1, 7.8e+101], N[(N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -1.2e75

   1. Initial program 11.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -19 \cdot x1\right) - 2\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.8%

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

   if -1.2e75 < x1 < -6.7999999999999998e-39

   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 inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

         \[\leadsto x1 + \color{blue}{\frac{8 \cdot x1}{1 + {x1}^{2}} \cdot {x2}^{2}} \]

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 40.8

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

   5. Applied rewrites 40.8%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 40.6%

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

   if -6.7999999999999998e-39 < x1 < 4.8999999999999998e-23

   1. Initial program 99.4%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 79.7

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

   9. Applied rewrites 79.7%

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

   10. Taylor expanded in x2 around 0

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

   12. Applied rewrites 77.9%

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

   if 4.8999999999999998e-23 < x1 < 7.8e101

   1. Initial program 99.6%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

         \[\leadsto x1 + \color{blue}{\frac{8 \cdot x1}{1 + {x1}^{2}} \cdot {x2}^{2}} \]

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 34.4%

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

   if 7.8e101 < x1

   1. Initial program 20.9%

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, -2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq -6.8 \cdot 10^{-39}:\\ \;\;\;\;\left(\left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\\ \mathbf{elif}\;x1 \leq 4.9 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{elif}\;x1 \leq 7.8 \cdot 10^{+101}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -245000000000:\\ \;\;\;\;\mathsf{fma}\left(-3, x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 7.8 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(-12, x1, -6\right)\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -245000000000.0)
   (+ (* (fma -3.0 x1 (fma (fma 2.0 x2 -3.0) 4.0 9.0)) (* x1 x1)) x1)
   (if (<= x1 7.8e+101)
     (fma (fma (* x2 x1) 8.0 (fma -12.0 x1 -6.0)) x2 (- x1))
     (+ (fma (* x1 x1) x1 (* -6.0 x2)) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -245000000000.0) {
		tmp = (fma(-3.0, x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)) * (x1 * x1)) + x1;
	} else if (x1 <= 7.8e+101) {
		tmp = fma(fma((x2 * x1), 8.0, fma(-12.0, x1, -6.0)), x2, -x1);
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -245000000000.0)
		tmp = Float64(Float64(fma(-3.0, x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)) * Float64(x1 * x1)) + x1);
	elseif (x1 <= 7.8e+101)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(-12.0, x1, -6.0)), x2, Float64(-x1));
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(-6.0 * x2)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -245000000000.0], N[(N[(N[(-3.0 * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 7.8e+101], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(-12.0 * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.45e11

   1. Initial program 31.3%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 75.0%

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

   if -2.45e11 < x1 < 7.8e101

   1. Initial program 99.4%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 74.3

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

   9. Applied rewrites 74.3%

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

   10. Taylor expanded in x2 around 0

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

   12. Applied rewrites 90.7%

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

   if 7.8e101 < x1

   1. Initial program 20.9%

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -245000000000:\\ \;\;\;\;\mathsf{fma}\left(-3, x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 7.8 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(-12, x1, -6\right)\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, -2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 7.8 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(-12, x1, -6\right)\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.2e+75)
   (+ (* (fma (fma -19.0 x1 9.0) x1 -2.0) x1) x1)
   (if (<= x1 7.8e+101)
     (fma (fma (* x2 x1) 8.0 (fma -12.0 x1 -6.0)) x2 (- x1))
     (+ (fma (* x1 x1) x1 (* -6.0 x2)) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.2e+75) {
		tmp = (fma(fma(-19.0, x1, 9.0), x1, -2.0) * x1) + x1;
	} else if (x1 <= 7.8e+101) {
		tmp = fma(fma((x2 * x1), 8.0, fma(-12.0, x1, -6.0)), x2, -x1);
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.2e+75)
		tmp = Float64(Float64(fma(fma(-19.0, x1, 9.0), x1, -2.0) * x1) + x1);
	elseif (x1 <= 7.8e+101)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(-12.0, x1, -6.0)), x2, Float64(-x1));
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(-6.0 * x2)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.2e+75], N[(N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 7.8e+101], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(-12.0 * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.2e75

   1. Initial program 11.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -19 \cdot x1\right) - 2\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.8%

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

   if -1.2e75 < x1 < 7.8e101

   1. Initial program 99.4%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 69.1

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

   9. Applied rewrites 69.1%

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

   10. Taylor expanded in x2 around 0

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

   12. Applied rewrites 83.9%

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

   if 7.8e101 < x1

   1. Initial program 20.9%

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, -2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 7.8 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(-12, x1, -6\right)\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -4 \cdot 10^{-192}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \cdot 2 \leq 2 \cdot 10^{-85}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2 + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* x2 2.0) -4e-192)
   (* -6.0 x2)
   (if (<= (* x2 2.0) 2e-85) (- x1) (+ (* -6.0 x2) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x2 * 2.0) <= -4e-192) {
		tmp = -6.0 * x2;
	} else if ((x2 * 2.0) <= 2e-85) {
		tmp = -x1;
	} else {
		tmp = (-6.0 * x2) + x1;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((x2 * 2.0d0) <= (-4d-192)) then
        tmp = (-6.0d0) * x2
    else if ((x2 * 2.0d0) <= 2d-85) then
        tmp = -x1
    else
        tmp = ((-6.0d0) * x2) + x1
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((x2 * 2.0) <= -4e-192) {
		tmp = -6.0 * x2;
	} else if ((x2 * 2.0) <= 2e-85) {
		tmp = -x1;
	} else {
		tmp = (-6.0 * x2) + x1;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x2 * 2.0) <= -4e-192:
		tmp = -6.0 * x2
	elif (x2 * 2.0) <= 2e-85:
		tmp = -x1
	else:
		tmp = (-6.0 * x2) + x1
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(x2 * 2.0) <= -4e-192)
		tmp = Float64(-6.0 * x2);
	elseif (Float64(x2 * 2.0) <= 2e-85)
		tmp = Float64(-x1);
	else
		tmp = Float64(Float64(-6.0 * x2) + x1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x2 * 2.0) <= -4e-192)
		tmp = -6.0 * x2;
	elseif ((x2 * 2.0) <= 2e-85)
		tmp = -x1;
	else
		tmp = (-6.0 * x2) + x1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(x2 * 2.0), $MachinePrecision], -4e-192], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[N[(x2 * 2.0), $MachinePrecision], 2e-85], (-x1), N[(N[(-6.0 * x2), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -4.0000000000000004e-192

   1. Initial program 73.1%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 30.8

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.3

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

   8. Applied rewrites 31.3%

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

   if -4.0000000000000004e-192 < (*.f64 #s(literal 2 binary64) x2) < 2e-85

   1. Initial program 62.1%

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

   4. Applied rewrites 65.6%

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

   6. Applied rewrites 62.4%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 49.0

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

   9. Applied rewrites 49.0%

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

   10. Taylor expanded in x2 around 0

       \[\leadsto -1 \cdot \color{blue}{x1} \]
   11. Step-by-step derivation

   12. Applied rewrites 38.4%

       \[\leadsto -x1 \]

   if 2e-85 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 64.2%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 28.2

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

   5. Applied rewrites 28.2%

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -4 \cdot 10^{-192}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \cdot 2 \leq 2 \cdot 10^{-85}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2 + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.2% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -4 \cdot 10^{-192}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \cdot 2 \leq 2 \cdot 10^{-85}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* x2 2.0) -4e-192)
   (* -6.0 x2)
   (if (<= (* x2 2.0) 2e-85) (- x1) (* -6.0 x2))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x2 * 2.0) <= -4e-192) {
		tmp = -6.0 * x2;
	} else if ((x2 * 2.0) <= 2e-85) {
		tmp = -x1;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((x2 * 2.0d0) <= (-4d-192)) then
        tmp = (-6.0d0) * x2
    else if ((x2 * 2.0d0) <= 2d-85) then
        tmp = -x1
    else
        tmp = (-6.0d0) * x2
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((x2 * 2.0) <= -4e-192) {
		tmp = -6.0 * x2;
	} else if ((x2 * 2.0) <= 2e-85) {
		tmp = -x1;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x2 * 2.0) <= -4e-192:
		tmp = -6.0 * x2
	elif (x2 * 2.0) <= 2e-85:
		tmp = -x1
	else:
		tmp = -6.0 * x2
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(x2 * 2.0) <= -4e-192)
		tmp = Float64(-6.0 * x2);
	elseif (Float64(x2 * 2.0) <= 2e-85)
		tmp = Float64(-x1);
	else
		tmp = Float64(-6.0 * x2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x2 * 2.0) <= -4e-192)
		tmp = -6.0 * x2;
	elseif ((x2 * 2.0) <= 2e-85)
		tmp = -x1;
	else
		tmp = -6.0 * x2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(x2 * 2.0), $MachinePrecision], -4e-192], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[N[(x2 * 2.0), $MachinePrecision], 2e-85], (-x1), N[(-6.0 * x2), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -4.0000000000000004e-192 or 2e-85 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 69.6%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 29.8

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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 29.7

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

   8. Applied rewrites 29.7%

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

   if -4.0000000000000004e-192 < (*.f64 #s(literal 2 binary64) x2) < 2e-85

   1. Initial program 62.1%

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

   4. Applied rewrites 65.6%

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

   6. Applied rewrites 62.4%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 49.0

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

   9. Applied rewrites 49.0%

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

   10. Taylor expanded in x2 around 0

       \[\leadsto -1 \cdot \color{blue}{x1} \]
   11. Step-by-step derivation

   12. Applied rewrites 38.4%

       \[\leadsto -x1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -4 \cdot 10^{-192}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \cdot 2 \leq 2 \cdot 10^{-85}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq 4.9 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 4.9e-23)
   (fma (fma -12.0 x1 -6.0) x2 (- x1))
   (+ (* (* (* x2 x2) x1) 8.0) x1)))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= 4.9e-23) {
		tmp = fma(fma(-12.0, x1, -6.0), x2, -x1);
	} else {
		tmp = (((x2 * x2) * x1) * 8.0) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= 4.9e-23)
		tmp = fma(fma(-12.0, x1, -6.0), x2, Float64(-x1));
	else
		tmp = Float64(Float64(Float64(Float64(x2 * x2) * x1) * 8.0) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, 4.9e-23], N[(N[(-12.0 * x1 + -6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], N[(N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision] + x1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < 4.8999999999999998e-23

   1. Initial program 74.8%

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

   4. Applied rewrites 81.2%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 53.7

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

   9. Applied rewrites 53.7%

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

   10. Taylor expanded in x2 around 0

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

   12. Applied rewrites 53.4%

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

   if 4.8999999999999998e-23 < x1

   1. Initial program 45.0%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

         \[\leadsto x1 + \color{blue}{\frac{8 \cdot x1}{1 + {x1}^{2}} \cdot {x2}^{2}} \]

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 18.1

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

   5. Applied rewrites 18.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 40.7%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq 4.9 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.1% accurate, 19.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right) \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (fma (fma -12.0 x1 -6.0) x2 (- x1)))

C

double code(double x1, double x2) {
	return fma(fma(-12.0, x1, -6.0), x2, -x1);
}

Julia

function code(x1, x2)
	return fma(fma(-12.0, x1, -6.0), x2, Float64(-x1))
end

Wolfram

code[x1_, x2_] := N[(N[(-12.0 * x1 + -6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision]

Derivation

1. Initial program 67.6%

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

4. Applied rewrites 72.5%

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

6. Applied rewrites 67.6%

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

7. Taylor expanded in x1 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 50.4

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

9. Applied rewrites 50.4%

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

10. Taylor expanded in x2 around 0

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

12. Applied rewrites 42.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), \color{blue}{x2}, -x1\right) \]
13. Add Preprocessing

Alternative 18: 38.5% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-1, x1, -6 \cdot x2\right) \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (fma -1.0 x1 (* -6.0 x2)))

C

double code(double x1, double x2) {
	return fma(-1.0, x1, (-6.0 * x2));
}

Julia

function code(x1, x2)
	return fma(-1.0, x1, Float64(-6.0 * x2))
end

Wolfram

code[x1_, x2_] := N[(-1.0 * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.6%

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

4. Applied rewrites 72.5%

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

6. Applied rewrites 67.6%

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

7. Taylor expanded in x1 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 50.4

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

9. Applied rewrites 50.4%

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

10. Taylor expanded in x2 around 0

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

12. Applied rewrites 37.1%

    \[\leadsto \mathsf{fma}\left(-1, x1, -6 \cdot x2\right) \]
13. Add Preprocessing

Alternative 19: 13.5% accurate, 99.3× speedup

Math

\[\begin{array}{l} \\ -x1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x1, x2):
	return -x1

Julia

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

MATLAB

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

Wolfram

code[x1_, x2_] := (-x1)

Derivation

1. Initial program 67.6%

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

4. Applied rewrites 72.5%

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

6. Applied rewrites 67.6%

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

7. Taylor expanded in x1 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 50.4

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

9. Applied rewrites 50.4%

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

10. Taylor expanded in x2 around 0

    \[\leadsto -1 \cdot \color{blue}{x1} \]
11. Step-by-step derivation

12. Applied rewrites 13.7%

    \[\leadsto -x1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024266 
(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))))))