Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.0% → 99.5%

Time: 18.2s

Alternatives: 19

Speedup: 6.6×

• 47.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 90.2%

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

   8. Applied rewrites 99.6%

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

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\frac{\left(8 \cdot x2\right) \cdot \left(x2 \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          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))))))
   (if (<= t_3 -2e+156)
     (/ (* (* 8.0 x2) (* x2 x1)) (fma x1 x1 1.0))
     (if (<= t_3 1e+122)
       (fma x2 -6.0 (* (- (* (fma -19.0 x1 9.0) x1) 1.0) x1))
       (* (* 6.0 (* x1 x1)) (* x1 x1))))))

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;
	double t_3 = 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)));
	double tmp;
	if (t_3 <= -2e+156) {
		tmp = ((8.0 * x2) * (x2 * x1)) / fma(x1, x1, 1.0);
	} else if (t_3 <= 1e+122) {
		tmp = fma(x2, -6.0, (((fma(-19.0, x1, 9.0) * x1) - 1.0) * x1));
	} else {
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	}
	return tmp;
}

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)
	t_3 = 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))))
	tmp = 0.0
	if (t_3 <= -2e+156)
		tmp = Float64(Float64(Float64(8.0 * x2) * Float64(x2 * x1)) / fma(x1, x1, 1.0));
	elseif (t_3 <= 1e+122)
		tmp = fma(x2, -6.0, Float64(Float64(Float64(fma(-19.0, x1, 9.0) * x1) - 1.0) * x1));
	else
		tmp = Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(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[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(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]}, If[LessEqual[t$95$3, -2e+156], N[(N[(N[(8.0 * x2), $MachinePrecision] * N[(x2 * x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+122], N[(x2 * -6.0 + N[(N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e156

   1. Initial program 99.8%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 10.2

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

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 66.9

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

   8. Applied rewrites 66.9%

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

   10. Applied rewrites 85.0%

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

   if -2e156 < (+.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)))))) < 1.00000000000000001e122

   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

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

   5. Applied rewrites 92.8%

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

   7. Applied rewrites 92.9%

      \[\leadsto \mathsf{fma}\left(x2, \color{blue}{-6}, \mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(x2 \cdot 2 - 3, -2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, 6\right), x2, \mathsf{fma}\left(x2 \cdot 2 - 3, 3, 1\right)\right) - \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 2, 2, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 3\right), x1, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(6, x2, 9\right)\right)\right)\right)\right) - 6, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   8. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(-8 \cdot \left(x1 \cdot {x2}^{2}\right), x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 90.7%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot -8, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto \mathsf{fma}\left(x2, -6, \left(x1 \cdot \left(9 + -19 \cdot x1\right) - 1\right) \cdot x1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 89.4%

       \[\leadsto \mathsf{fma}\left(x2, -6, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right) \]

   if 1.00000000000000001e122 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 48.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 80.2

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

   5. Applied rewrites 80.2%

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

   7. Applied rewrites 80.2%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 80.3%

       \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{elif}\;t\_3 \leq 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          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))))))
   (if (<= t_3 -2e+156)
     (* (* (* x2 x2) x1) 8.0)
     (if (<= t_3 1e+122)
       (fma x2 -6.0 (* (- (* (fma -19.0 x1 9.0) x1) 1.0) x1))
       (* (* 6.0 (* x1 x1)) (* x1 x1))))))

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;
	double t_3 = 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)));
	double tmp;
	if (t_3 <= -2e+156) {
		tmp = ((x2 * x2) * x1) * 8.0;
	} else if (t_3 <= 1e+122) {
		tmp = fma(x2, -6.0, (((fma(-19.0, x1, 9.0) * x1) - 1.0) * x1));
	} else {
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	}
	return tmp;
}

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)
	t_3 = 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))))
	tmp = 0.0
	if (t_3 <= -2e+156)
		tmp = Float64(Float64(Float64(x2 * x2) * x1) * 8.0);
	elseif (t_3 <= 1e+122)
		tmp = fma(x2, -6.0, Float64(Float64(Float64(fma(-19.0, x1, 9.0) * x1) - 1.0) * x1));
	else
		tmp = Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(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[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(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]}, If[LessEqual[t$95$3, -2e+156], N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision], If[LessEqual[t$95$3, 1e+122], N[(x2 * -6.0 + N[(N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e156

   1. Initial program 99.8%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 10.2

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

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 66.9

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

   8. Applied rewrites 66.9%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 66.9%

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

   if -2e156 < (+.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)))))) < 1.00000000000000001e122

   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

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

   5. Applied rewrites 92.8%

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

   7. Applied rewrites 92.9%

      \[\leadsto \mathsf{fma}\left(x2, \color{blue}{-6}, \mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(x2 \cdot 2 - 3, -2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, 6\right), x2, \mathsf{fma}\left(x2 \cdot 2 - 3, 3, 1\right)\right) - \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 2, 2, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 3\right), x1, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(6, x2, 9\right)\right)\right)\right)\right) - 6, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   8. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(-8 \cdot \left(x1 \cdot {x2}^{2}\right), x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 90.7%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot -8, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto \mathsf{fma}\left(x2, -6, \left(x1 \cdot \left(9 + -19 \cdot x1\right) - 1\right) \cdot x1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 89.4%

       \[\leadsto \mathsf{fma}\left(x2, -6, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right) \]

   if 1.00000000000000001e122 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 48.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 80.2

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

   5. Applied rewrites 80.2%

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

   7. Applied rewrites 80.2%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 80.3%

       \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 69.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{elif}\;t\_3 \leq 10^{+122}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          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))))))
   (if (<= t_3 -2e+156)
     (* (* (* x2 x2) x1) 8.0)
     (if (<= t_3 1e+122) (* -6.0 x2) (* (* 6.0 (* x1 x1)) (* x1 x1))))))

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;
	double t_3 = 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)));
	double tmp;
	if (t_3 <= -2e+156) {
		tmp = ((x2 * x2) * x1) * 8.0;
	} else if (t_3 <= 1e+122) {
		tmp = -6.0 * x2;
	} else {
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (3.0d0 * x1) * x1
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    t_3 = 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)))
    if (t_3 <= (-2d+156)) then
        tmp = ((x2 * x2) * x1) * 8.0d0
    else if (t_3 <= 1d+122) then
        tmp = (-6.0d0) * x2
    else
        tmp = (6.0d0 * (x1 * x1)) * (x1 * x1)
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = 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)));
	double tmp;
	if (t_3 <= -2e+156) {
		tmp = ((x2 * x2) * x1) * 8.0;
	} else if (t_3 <= 1e+122) {
		tmp = -6.0 * x2;
	} else {
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	}
	return tmp;
}

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
	t_3 = 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)))
	tmp = 0
	if t_3 <= -2e+156:
		tmp = ((x2 * x2) * x1) * 8.0
	elif t_3 <= 1e+122:
		tmp = -6.0 * x2
	else:
		tmp = (6.0 * (x1 * x1)) * (x1 * x1)
	return tmp

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)
	t_3 = 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))))
	tmp = 0.0
	if (t_3 <= -2e+156)
		tmp = Float64(Float64(Float64(x2 * x2) * x1) * 8.0);
	elseif (t_3 <= 1e+122)
		tmp = Float64(-6.0 * x2);
	else
		tmp = Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1));
	end
	return tmp
end

MATLAB

function tmp_2 = 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;
	t_3 = 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)));
	tmp = 0.0;
	if (t_3 <= -2e+156)
		tmp = ((x2 * x2) * x1) * 8.0;
	elseif (t_3 <= 1e+122)
		tmp = -6.0 * x2;
	else
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(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]}, If[LessEqual[t$95$3, -2e+156], N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision], If[LessEqual[t$95$3, 1e+122], N[(-6.0 * x2), $MachinePrecision], N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e156

   1. Initial program 99.8%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 10.2

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

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 66.9

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

   8. Applied rewrites 66.9%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 66.9%

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

   if -2e156 < (+.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)))))) < 1.00000000000000001e122

   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

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

      1. lower-*.f64 56.9

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

   5. Applied rewrites 56.9%

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

   if 1.00000000000000001e122 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 48.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 80.2

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

   5. Applied rewrites 80.2%

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

   7. Applied rewrites 80.2%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 80.3%

       \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 43.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+156} \lor \neg \left(t\_3 \leq 2 \cdot 10^{+254}\right):\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \end{array} \]

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))
        (t_3
         (+
          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))))))
   (if (or (<= t_3 -2e+156) (not (<= t_3 2e+254)))
     (* (* (* x2 x2) x1) 8.0)
     (* -6.0 x2))))

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;
	double t_3 = 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)));
	double tmp;
	if ((t_3 <= -2e+156) || !(t_3 <= 2e+254)) {
		tmp = ((x2 * x2) * x1) * 8.0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (3.0d0 * x1) * x1
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    t_3 = 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)))
    if ((t_3 <= (-2d+156)) .or. (.not. (t_3 <= 2d+254))) then
        tmp = ((x2 * x2) * x1) * 8.0d0
    else
        tmp = (-6.0d0) * x2
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = 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)));
	double tmp;
	if ((t_3 <= -2e+156) || !(t_3 <= 2e+254)) {
		tmp = ((x2 * x2) * x1) * 8.0;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

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
	t_3 = 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)))
	tmp = 0
	if (t_3 <= -2e+156) or not (t_3 <= 2e+254):
		tmp = ((x2 * x2) * x1) * 8.0
	else:
		tmp = -6.0 * x2
	return tmp

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)
	t_3 = 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))))
	tmp = 0.0
	if ((t_3 <= -2e+156) || !(t_3 <= 2e+254))
		tmp = Float64(Float64(Float64(x2 * x2) * x1) * 8.0);
	else
		tmp = Float64(-6.0 * x2);
	end
	return tmp
end

MATLAB

function tmp_2 = 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;
	t_3 = 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)));
	tmp = 0.0;
	if ((t_3 <= -2e+156) || ~((t_3 <= 2e+254)))
		tmp = ((x2 * x2) * x1) * 8.0;
	else
		tmp = -6.0 * x2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(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]}, If[Or[LessEqual[t$95$3, -2e+156], N[Not[LessEqual[t$95$3, 2e+254]], $MachinePrecision]], N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision], N[(-6.0 * x2), $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)))))) < -2e156 or 1.9999999999999999e254 < (+.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 52.1%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 4.2

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 29.7

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

   8. Applied rewrites 29.7%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 37.6%

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

   if -2e156 < (+.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)))))) < 1.9999999999999999e254

   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

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

      1. lower-*.f64 50.4

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

   5. Applied rewrites 50.4%

      \[\leadsto \color{blue}{-6 \cdot x2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2 \cdot 10^{+156} \lor \neg \left(x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 2 \cdot 10^{+254}\right):\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

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;
	double t_3 = 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)));
	double tmp;
	if (t_3 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	}
	return tmp;
}

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;
	double t_3 = 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)));
	double tmp;
	if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	}
	return tmp;
}

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
	t_3 = 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)))
	tmp = 0
	if t_3 <= math.inf:
		tmp = t_3
	else:
		tmp = (6.0 * (x1 * x1)) * (x1 * x1)
	return tmp

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)
	t_3 = 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))))
	tmp = 0.0
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1));
	end
	return tmp
end

MATLAB

function tmp_2 = 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;
	t_3 = 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)));
	tmp = 0.0;
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.0% accurate, 0.5× speedup

Math

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

FPCore

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

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

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)
	t_3 = 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)
	tmp = 0.0
	if (Float64(x1 + Float64(t_3 + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1)))) <= Inf)
		tmp = Float64(x1 + Float64(t_3 + Float64(3.0 * fma(Float64(Float64(Float64(x1 + Float64(3.0 - Float64(-2.0 * x2))) * x1) - 1.0), x1, Float64(-2.0 * x2)))));
	else
		tmp = Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(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[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(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]}, If[LessEqual[N[(x1 + N[(t$95$3 + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(x1 + N[(t$95$3 + N[(3.0 * N[(N[(N[(N[(x1 + N[(3.0 - N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1 + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 90.2%

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x1 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 97.5

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

   11. Applied rewrites 97.5%

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

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 96.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.15 \lor \neg \left(x1 \leq 1.35\right):\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(2 \cdot x2 - 3, 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(x1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-8, x1 \cdot x1, 8\right), x2, \mathsf{fma}\left(24, x1, 12\right) \cdot x1 - 12\right), x2, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -1.15) (not (<= x1 1.35)))
   (*
    (- 6.0 (/ (- 3.0 (/ (fma (- (* 2.0 x2) 3.0) 4.0 9.0) x1)) x1))
    (pow x1 4.0))
   (fma
    x2
    -6.0
    (fma
     (*
      x1
      (fma (fma -8.0 (* x1 x1) 8.0) x2 (- (* (fma 24.0 x1 12.0) x1) 12.0)))
     x2
     (* (- (* (fma -19.0 x1 9.0) x1) 1.0) x1)))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -1.15) || !(x1 <= 1.35)) {
		tmp = (6.0 - ((3.0 - (fma(((2.0 * x2) - 3.0), 4.0, 9.0) / x1)) / x1)) * pow(x1, 4.0);
	} else {
		tmp = fma(x2, -6.0, fma((x1 * fma(fma(-8.0, (x1 * x1), 8.0), x2, ((fma(24.0, x1, 12.0) * x1) - 12.0))), x2, (((fma(-19.0, x1, 9.0) * x1) - 1.0) * x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -1.15) || !(x1 <= 1.35))
		tmp = Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(Float64(Float64(2.0 * x2) - 3.0), 4.0, 9.0) / x1)) / x1)) * (x1 ^ 4.0));
	else
		tmp = fma(x2, -6.0, fma(Float64(x1 * fma(fma(-8.0, Float64(x1 * x1), 8.0), x2, Float64(Float64(fma(24.0, x1, 12.0) * x1) - 12.0))), x2, Float64(Float64(Float64(fma(-19.0, x1, 9.0) * x1) - 1.0) * x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -1.15], N[Not[LessEqual[x1, 1.35]], $MachinePrecision]], N[(N[(6.0 - N[(N[(3.0 - N[(N[(N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], N[(x2 * -6.0 + N[(N[(x1 * N[(N[(-8.0 * N[(x1 * x1), $MachinePrecision] + 8.0), $MachinePrecision] * x2 + N[(N[(N[(24.0 * x1 + 12.0), $MachinePrecision] * x1), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.1499999999999999 or 1.3500000000000001 < x1

   1. Initial program 47.4%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.8%

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

   if -1.1499999999999999 < x1 < 1.3500000000000001

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 74.4%

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

   7. Applied rewrites 74.5%

      \[\leadsto \mathsf{fma}\left(x2, \color{blue}{-6}, \mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(x2 \cdot 2 - 3, -2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, 6\right), x2, \mathsf{fma}\left(x2 \cdot 2 - 3, 3, 1\right)\right) - \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 2, 2, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 3\right), x1, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(6, x2, 9\right)\right)\right)\right)\right) - 6, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   8. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(-8 \cdot \left(x1 \cdot {x2}^{2}\right), x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.9%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot -8, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto \mathsf{fma}\left(x2, -6, x1 \cdot \left(x1 \cdot \left(9 + -19 \cdot x1\right) - 1\right) + x2 \cdot \left(x1 \cdot \left(x2 \cdot \left(8 + -8 \cdot {x1}^{2}\right)\right) + x1 \cdot \left(x1 \cdot \left(12 + 24 \cdot x1\right) - 12\right)\right)\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 99.8%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(x1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-8, x1 \cdot x1, 8\right), x2, \mathsf{fma}\left(24, x1, 12\right) \cdot x1 - 12\right), x2, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.15 \lor \neg \left(x1 \leq 1.35\right):\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(2 \cdot x2 - 3, 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(x1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-8, x1 \cdot x1, 8\right), x2, \mathsf{fma}\left(24, x1, 12\right) \cdot x1 - 12\right), x2, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -3.1:\\ \;\;\;\;\left(\left(6 \cdot x1 - 3\right) \cdot x1\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 1.35:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(x1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-8, x1 \cdot x1, 8\right), x2, \mathsf{fma}\left(24, x1, 12\right) \cdot x1 - 12\right), x2, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 - \frac{3}{x1}\right) \cdot {x1}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -3.1)
   (* (* (- (* 6.0 x1) 3.0) x1) (* x1 x1))
   (if (<= x1 1.35)
     (fma
      x2
      -6.0
      (fma
       (*
        x1
        (fma (fma -8.0 (* x1 x1) 8.0) x2 (- (* (fma 24.0 x1 12.0) x1) 12.0)))
       x2
       (* (- (* (fma -19.0 x1 9.0) x1) 1.0) x1)))
     (* (- 6.0 (/ 3.0 x1)) (pow x1 4.0)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -3.1) {
		tmp = (((6.0 * x1) - 3.0) * x1) * (x1 * x1);
	} else if (x1 <= 1.35) {
		tmp = fma(x2, -6.0, fma((x1 * fma(fma(-8.0, (x1 * x1), 8.0), x2, ((fma(24.0, x1, 12.0) * x1) - 12.0))), x2, (((fma(-19.0, x1, 9.0) * x1) - 1.0) * x1)));
	} else {
		tmp = (6.0 - (3.0 / x1)) * pow(x1, 4.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -3.1)
		tmp = Float64(Float64(Float64(Float64(6.0 * x1) - 3.0) * x1) * Float64(x1 * x1));
	elseif (x1 <= 1.35)
		tmp = fma(x2, -6.0, fma(Float64(x1 * fma(fma(-8.0, Float64(x1 * x1), 8.0), x2, Float64(Float64(fma(24.0, x1, 12.0) * x1) - 12.0))), x2, Float64(Float64(Float64(fma(-19.0, x1, 9.0) * x1) - 1.0) * x1)));
	else
		tmp = Float64(Float64(6.0 - Float64(3.0 / x1)) * (x1 ^ 4.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -3.1], N[(N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1.35], N[(x2 * -6.0 + N[(N[(x1 * N[(N[(-8.0 * N[(x1 * x1), $MachinePrecision] + 8.0), $MachinePrecision] * x2 + N[(N[(N[(24.0 * x1 + 12.0), $MachinePrecision] * x1), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.10000000000000009

   1. Initial program 39.3%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 86.5

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

   5. Applied rewrites 86.5%

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in x1 around 0

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

   10. Applied rewrites 86.6%

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

   if -3.10000000000000009 < x1 < 1.3500000000000001

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 74.4%

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

   7. Applied rewrites 74.5%

      \[\leadsto \mathsf{fma}\left(x2, \color{blue}{-6}, \mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(x2 \cdot 2 - 3, -2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, 6\right), x2, \mathsf{fma}\left(x2 \cdot 2 - 3, 3, 1\right)\right) - \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 2, 2, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 3\right), x1, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(6, x2, 9\right)\right)\right)\right)\right) - 6, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   8. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(-8 \cdot \left(x1 \cdot {x2}^{2}\right), x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.9%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot -8, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto \mathsf{fma}\left(x2, -6, x1 \cdot \left(x1 \cdot \left(9 + -19 \cdot x1\right) - 1\right) + x2 \cdot \left(x1 \cdot \left(x2 \cdot \left(8 + -8 \cdot {x1}^{2}\right)\right) + x1 \cdot \left(x1 \cdot \left(12 + 24 \cdot x1\right) - 12\right)\right)\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 99.8%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(x1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-8, x1 \cdot x1, 8\right), x2, \mathsf{fma}\left(24, x1, 12\right) \cdot x1 - 12\right), x2, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right)\right) \]

   if 1.3500000000000001 < x1

   1. Initial program 59.8%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 84.6

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

   5. Applied rewrites 84.6%

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

Alternative 11: 93.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -3.1:\\ \;\;\;\;\left(\left(6 \cdot x1 - 3\right) \cdot x1\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 1.35:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(x1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-8, x1 \cdot x1, 8\right), x2, \mathsf{fma}\left(24, x1, 12\right) \cdot x1 - 12\right), x2, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(6 - \frac{3}{x1}\right) \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -3.1)
   (* (* (- (* 6.0 x1) 3.0) x1) (* x1 x1))
   (if (<= x1 1.35)
     (fma
      x2
      -6.0
      (fma
       (*
        x1
        (fma (fma -8.0 (* x1 x1) 8.0) x2 (- (* (fma 24.0 x1 12.0) x1) 12.0)))
       x2
       (* (- (* (fma -19.0 x1 9.0) x1) 1.0) x1)))
     (* (* (* (* (- 6.0 (/ 3.0 x1)) x1) x1) x1) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -3.1) {
		tmp = (((6.0 * x1) - 3.0) * x1) * (x1 * x1);
	} else if (x1 <= 1.35) {
		tmp = fma(x2, -6.0, fma((x1 * fma(fma(-8.0, (x1 * x1), 8.0), x2, ((fma(24.0, x1, 12.0) * x1) - 12.0))), x2, (((fma(-19.0, x1, 9.0) * x1) - 1.0) * x1)));
	} else {
		tmp = ((((6.0 - (3.0 / x1)) * x1) * x1) * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -3.1)
		tmp = Float64(Float64(Float64(Float64(6.0 * x1) - 3.0) * x1) * Float64(x1 * x1));
	elseif (x1 <= 1.35)
		tmp = fma(x2, -6.0, fma(Float64(x1 * fma(fma(-8.0, Float64(x1 * x1), 8.0), x2, Float64(Float64(fma(24.0, x1, 12.0) * x1) - 12.0))), x2, Float64(Float64(Float64(fma(-19.0, x1, 9.0) * x1) - 1.0) * x1)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(6.0 - Float64(3.0 / x1)) * x1) * x1) * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -3.1], N[(N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1.35], N[(x2 * -6.0 + N[(N[(x1 * N[(N[(-8.0 * N[(x1 * x1), $MachinePrecision] + 8.0), $MachinePrecision] * x2 + N[(N[(N[(24.0 * x1 + 12.0), $MachinePrecision] * x1), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.10000000000000009

   1. Initial program 39.3%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 86.5

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

   5. Applied rewrites 86.5%

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in x1 around 0

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

   10. Applied rewrites 86.6%

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

   if -3.10000000000000009 < x1 < 1.3500000000000001

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 74.4%

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

   7. Applied rewrites 74.5%

      \[\leadsto \mathsf{fma}\left(x2, \color{blue}{-6}, \mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(x2 \cdot 2 - 3, -2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, 6\right), x2, \mathsf{fma}\left(x2 \cdot 2 - 3, 3, 1\right)\right) - \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 2, 2, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 3\right), x1, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(6, x2, 9\right)\right)\right)\right)\right) - 6, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   8. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(-8 \cdot \left(x1 \cdot {x2}^{2}\right), x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.9%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot -8, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto \mathsf{fma}\left(x2, -6, x1 \cdot \left(x1 \cdot \left(9 + -19 \cdot x1\right) - 1\right) + x2 \cdot \left(x1 \cdot \left(x2 \cdot \left(8 + -8 \cdot {x1}^{2}\right)\right) + x1 \cdot \left(x1 \cdot \left(12 + 24 \cdot x1\right) - 12\right)\right)\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 99.8%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(x1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-8, x1 \cdot x1, 8\right), x2, \mathsf{fma}\left(24, x1, 12\right) \cdot x1 - 12\right), x2, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right)\right) \]

   if 1.3500000000000001 < x1

   1. Initial program 59.8%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 84.5%

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

   9. Applied rewrites 84.6%

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

Alternative 12: 93.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -3.1:\\ \;\;\;\;\left(\left(6 \cdot x1 - 3\right) \cdot x1\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 1.35:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-8, x1 \cdot x1, 8\right), x2, \mathsf{fma}\left(24, x1, 12\right) \cdot x1 - 12\right) - 6, x2, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(6 - \frac{3}{x1}\right) \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -3.1)
   (* (* (- (* 6.0 x1) 3.0) x1) (* x1 x1))
   (if (<= x1 1.35)
     (fma
      (-
       (*
        x1
        (fma (fma -8.0 (* x1 x1) 8.0) x2 (- (* (fma 24.0 x1 12.0) x1) 12.0)))
       6.0)
      x2
      (* (- (* (fma -19.0 x1 9.0) x1) 1.0) x1))
     (* (* (* (* (- 6.0 (/ 3.0 x1)) x1) x1) x1) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -3.1) {
		tmp = (((6.0 * x1) - 3.0) * x1) * (x1 * x1);
	} else if (x1 <= 1.35) {
		tmp = fma(((x1 * fma(fma(-8.0, (x1 * x1), 8.0), x2, ((fma(24.0, x1, 12.0) * x1) - 12.0))) - 6.0), x2, (((fma(-19.0, x1, 9.0) * x1) - 1.0) * x1));
	} else {
		tmp = ((((6.0 - (3.0 / x1)) * x1) * x1) * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -3.1)
		tmp = Float64(Float64(Float64(Float64(6.0 * x1) - 3.0) * x1) * Float64(x1 * x1));
	elseif (x1 <= 1.35)
		tmp = fma(Float64(Float64(x1 * fma(fma(-8.0, Float64(x1 * x1), 8.0), x2, Float64(Float64(fma(24.0, x1, 12.0) * x1) - 12.0))) - 6.0), x2, Float64(Float64(Float64(fma(-19.0, x1, 9.0) * x1) - 1.0) * x1));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(6.0 - Float64(3.0 / x1)) * x1) * x1) * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -3.1], N[(N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1.35], N[(N[(N[(x1 * N[(N[(-8.0 * N[(x1 * x1), $MachinePrecision] + 8.0), $MachinePrecision] * x2 + N[(N[(N[(24.0 * x1 + 12.0), $MachinePrecision] * x1), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision] * x2 + N[(N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.10000000000000009

   1. Initial program 39.3%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 86.5

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

   5. Applied rewrites 86.5%

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in x1 around 0

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

   10. Applied rewrites 86.6%

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

   if -3.10000000000000009 < x1 < 1.3500000000000001

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(x1 \cdot \left(9 + -19 \cdot x1\right) - 1\right) + \color{blue}{x2 \cdot \left(\left(x1 \cdot \left(x2 \cdot \left(8 + -8 \cdot {x1}^{2}\right)\right) + x1 \cdot \left(x1 \cdot \left(12 + 24 \cdot x1\right) - 12\right)\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(x1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-8, x1 \cdot x1, 8\right), x2, \mathsf{fma}\left(24, x1, 12\right) \cdot x1 - 12\right) - 6, \color{blue}{x2}, \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\right) \]

   if 1.3500000000000001 < x1

   1. Initial program 59.8%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 84.5%

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

   9. Applied rewrites 84.6%

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

Alternative 13: 86.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.15 \cdot 10^{+40}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 6.5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(24, x1, 12\right), x2, -19 \cdot x1\right) + 9, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(6 - \frac{3}{x1}\right) \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.15e+40)
   (* (* 6.0 (* x1 x1)) (* x1 x1))
   (if (<= x1 6.5e+33)
     (fma
      x2
      -6.0
      (*
       (fma
        (+ (fma (fma 24.0 x1 12.0) x2 (* -19.0 x1)) 9.0)
        x1
        (- (* (* (- (* x2 2.0) 3.0) x2) 4.0) 1.0))
       x1))
     (* (* (* (* (- 6.0 (/ 3.0 x1)) x1) x1) x1) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.15e+40) {
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	} else if (x1 <= 6.5e+33) {
		tmp = fma(x2, -6.0, (fma((fma(fma(24.0, x1, 12.0), x2, (-19.0 * x1)) + 9.0), x1, (((((x2 * 2.0) - 3.0) * x2) * 4.0) - 1.0)) * x1));
	} else {
		tmp = ((((6.0 - (3.0 / x1)) * x1) * x1) * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.15e+40)
		tmp = Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1));
	elseif (x1 <= 6.5e+33)
		tmp = fma(x2, -6.0, Float64(fma(Float64(fma(fma(24.0, x1, 12.0), x2, Float64(-19.0 * x1)) + 9.0), x1, Float64(Float64(Float64(Float64(Float64(x2 * 2.0) - 3.0) * x2) * 4.0) - 1.0)) * x1));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(6.0 - Float64(3.0 / x1)) * x1) * x1) * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.15e+40], N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 6.5e+33], N[(x2 * -6.0 + N[(N[(N[(N[(N[(24.0 * x1 + 12.0), $MachinePrecision] * x2 + N[(-19.0 * x1), $MachinePrecision]), $MachinePrecision] + 9.0), $MachinePrecision] * x1 + N[(N[(N[(N[(N[(x2 * 2.0), $MachinePrecision] - 3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.1500000000000001e40

   1. Initial program 28.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 96.8

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

   5. Applied rewrites 96.8%

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.8%

       \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]

   if -2.1500000000000001e40 < x1 < 6.49999999999999993e33

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 64.8%

      \[\leadsto \mathsf{fma}\left(x2, \color{blue}{-6}, \mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(x2 \cdot 2 - 3, -2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, 6\right), x2, \mathsf{fma}\left(x2 \cdot 2 - 3, 3, 1\right)\right) - \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 2, 2, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 3\right), x1, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(6, x2, 9\right)\right)\right)\right)\right) - 6, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(9 + \left(-19 \cdot x1 + x2 \cdot \left(12 + 24 \cdot x1\right)\right), x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 83.1%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(24, x1, 12\right), x2, -19 \cdot x1\right) + 9, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   if 6.49999999999999993e33 < x1

   1. Initial program 52.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 95.4

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

   5. Applied rewrites 95.4%

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

   7. Applied rewrites 95.3%

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

   9. Applied rewrites 95.4%

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

Alternative 14: 86.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.15 \cdot 10^{+40}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 6.5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(6 - \frac{3}{x1}\right) \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.15e+40)
   (* (* 6.0 (* x1 x1)) (* x1 x1))
   (if (<= x1 6.5e+33)
     (fma
      x2
      -6.0
      (*
       (fma (fma -19.0 x1 9.0) x1 (- (* (* (- (* x2 2.0) 3.0) x2) 4.0) 1.0))
       x1))
     (* (* (* (* (- 6.0 (/ 3.0 x1)) x1) x1) x1) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.15e+40) {
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	} else if (x1 <= 6.5e+33) {
		tmp = fma(x2, -6.0, (fma(fma(-19.0, x1, 9.0), x1, (((((x2 * 2.0) - 3.0) * x2) * 4.0) - 1.0)) * x1));
	} else {
		tmp = ((((6.0 - (3.0 / x1)) * x1) * x1) * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.15e+40)
		tmp = Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1));
	elseif (x1 <= 6.5e+33)
		tmp = fma(x2, -6.0, Float64(fma(fma(-19.0, x1, 9.0), x1, Float64(Float64(Float64(Float64(Float64(x2 * 2.0) - 3.0) * x2) * 4.0) - 1.0)) * x1));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(6.0 - Float64(3.0 / x1)) * x1) * x1) * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.15e+40], N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 6.5e+33], N[(x2 * -6.0 + N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1 + N[(N[(N[(N[(N[(x2 * 2.0), $MachinePrecision] - 3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.1500000000000001e40

   1. Initial program 28.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 96.8

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

   5. Applied rewrites 96.8%

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.8%

       \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]

   if -2.1500000000000001e40 < x1 < 6.49999999999999993e33

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 64.8%

      \[\leadsto \mathsf{fma}\left(x2, \color{blue}{-6}, \mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(x2 \cdot 2 - 3, -2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, 6\right), x2, \mathsf{fma}\left(x2 \cdot 2 - 3, 3, 1\right)\right) - \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 2, 2, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 3\right), x1, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(6, x2, 9\right)\right)\right)\right)\right) - 6, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   8. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(-8 \cdot \left(x1 \cdot {x2}^{2}\right), x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 63.4%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot -8, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(9 + -19 \cdot x1, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 82.9%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(-19, x1, 9\right), x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   if 6.49999999999999993e33 < x1

   1. Initial program 52.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 95.4

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

   5. Applied rewrites 95.4%

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

   7. Applied rewrites 95.3%

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

   9. Applied rewrites 95.4%

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

Alternative 15: 86.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.15 \cdot 10^{+40}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 6.5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \left(\left(\left(2 \cdot x2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(6 - \frac{3}{x1}\right) \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.15e+40)
   (* (* 6.0 (* x1 x1)) (* x1 x1))
   (if (<= x1 6.5e+33)
     (fma x2 -6.0 (* (- (* (* (- (* 2.0 x2) 3.0) x2) 4.0) 1.0) x1))
     (* (* (* (* (- 6.0 (/ 3.0 x1)) x1) x1) x1) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.15e+40) {
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	} else if (x1 <= 6.5e+33) {
		tmp = fma(x2, -6.0, ((((((2.0 * x2) - 3.0) * x2) * 4.0) - 1.0) * x1));
	} else {
		tmp = ((((6.0 - (3.0 / x1)) * x1) * x1) * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.15e+40)
		tmp = Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1));
	elseif (x1 <= 6.5e+33)
		tmp = fma(x2, -6.0, Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x2) - 3.0) * x2) * 4.0) - 1.0) * x1));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(6.0 - Float64(3.0 / x1)) * x1) * x1) * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.15e+40], N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 6.5e+33], N[(x2 * -6.0 + N[(N[(N[(N[(N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.1500000000000001e40

   1. Initial program 28.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 96.8

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

   5. Applied rewrites 96.8%

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.8%

       \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]

   if -2.1500000000000001e40 < x1 < 6.49999999999999993e33

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 64.8%

      \[\leadsto \mathsf{fma}\left(x2, \color{blue}{-6}, \mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(x2 \cdot 2 - 3, -2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, 6\right), x2, \mathsf{fma}\left(x2 \cdot 2 - 3, 3, 1\right)\right) - \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 2, 2, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 3\right), x1, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(6, x2, 9\right)\right)\right)\right)\right) - 6, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   8. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(-8 \cdot \left(x1 \cdot {x2}^{2}\right), x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 63.4%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot -8, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 81.6%

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

   if 6.49999999999999993e33 < x1

   1. Initial program 52.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 95.4

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

   5. Applied rewrites 95.4%

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

   7. Applied rewrites 95.3%

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

   9. Applied rewrites 95.4%

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

Alternative 16: 86.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.15 \cdot 10^{+40}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 6.5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \left(\left(\left(2 \cdot x2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(6 \cdot x1 - 3\right) \cdot x1\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.15e+40)
   (* (* 6.0 (* x1 x1)) (* x1 x1))
   (if (<= x1 6.5e+33)
     (fma x2 -6.0 (* (- (* (* (- (* 2.0 x2) 3.0) x2) 4.0) 1.0) x1))
     (* (* (- (* 6.0 x1) 3.0) x1) (* x1 x1)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.15e+40) {
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	} else if (x1 <= 6.5e+33) {
		tmp = fma(x2, -6.0, ((((((2.0 * x2) - 3.0) * x2) * 4.0) - 1.0) * x1));
	} else {
		tmp = (((6.0 * x1) - 3.0) * x1) * (x1 * x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.15e+40)
		tmp = Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1));
	elseif (x1 <= 6.5e+33)
		tmp = fma(x2, -6.0, Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x2) - 3.0) * x2) * 4.0) - 1.0) * x1));
	else
		tmp = Float64(Float64(Float64(Float64(6.0 * x1) - 3.0) * x1) * Float64(x1 * x1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.15e+40], N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 6.5e+33], N[(x2 * -6.0 + N[(N[(N[(N[(N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.1500000000000001e40

   1. Initial program 28.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 96.8

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

   5. Applied rewrites 96.8%

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.8%

       \[\leadsto \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]

   if -2.1500000000000001e40 < x1 < 6.49999999999999993e33

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 64.8%

      \[\leadsto \mathsf{fma}\left(x2, \color{blue}{-6}, \mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(x2 \cdot 2 - 3, -2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, 6\right), x2, \mathsf{fma}\left(x2 \cdot 2 - 3, 3, 1\right)\right) - \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 2, 2, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 3\right), x1, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(6, x2, 9\right)\right)\right)\right)\right) - 6, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   8. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(-8 \cdot \left(x1 \cdot {x2}^{2}\right), x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 63.4%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot -8, x1, \left(\left(x2 \cdot 2 - 3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right) \]

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 81.6%

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

   if 6.49999999999999993e33 < x1

   1. Initial program 52.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 95.4

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

   5. Applied rewrites 95.4%

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in x1 around 0

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

   10. Applied rewrites 95.4%

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

Alternative 17: 56.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -4 \cdot 10^{+80}:\\ \;\;\;\;\left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\\ \mathbf{elif}\;x1 \leq -1.18 \cdot 10^{-136} \lor \neg \left(x1 \leq 5.8 \cdot 10^{-94}\right):\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -4e+80)
   (* (- (* (fma -19.0 x1 9.0) x1) 1.0) x1)
   (if (or (<= x1 -1.18e-136) (not (<= x1 5.8e-94)))
     (* (* (* x2 x2) x1) 8.0)
     (* -6.0 x2))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -4e+80) {
		tmp = ((fma(-19.0, x1, 9.0) * x1) - 1.0) * x1;
	} else if ((x1 <= -1.18e-136) || !(x1 <= 5.8e-94)) {
		tmp = ((x2 * x2) * x1) * 8.0;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -4e+80)
		tmp = Float64(Float64(Float64(fma(-19.0, x1, 9.0) * x1) - 1.0) * x1);
	elseif ((x1 <= -1.18e-136) || !(x1 <= 5.8e-94))
		tmp = Float64(Float64(Float64(x2 * x2) * x1) * 8.0);
	else
		tmp = Float64(-6.0 * x2);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -4e+80], N[(N[(N[(N[(-19.0 * x1 + 9.0), $MachinePrecision] * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision], If[Or[LessEqual[x1, -1.18e-136], N[Not[LessEqual[x1, 5.8e-94]], $MachinePrecision]], N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision], N[(-6.0 * x2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -4e80

   1. Initial program 13.2%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -19 \cdot x1\right) - 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.8%

      \[\leadsto \left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot \color{blue}{x1} \]

   if -4e80 < x1 < -1.18e-136 or 5.79999999999999991e-94 < x1

   1. Initial program 83.4%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 9.3

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

   5. Applied rewrites 9.3%

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

   6. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 33.1

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

   8. Applied rewrites 33.1%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 41.4%

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

   if -1.18e-136 < x1 < 5.79999999999999991e-94

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4 \cdot 10^{+80}:\\ \;\;\;\;\left(\mathsf{fma}\left(-19, x1, 9\right) \cdot x1 - 1\right) \cdot x1\\ \mathbf{elif}\;x1 \leq -1.18 \cdot 10^{-136} \lor \neg \left(x1 \leq 5.8 \cdot 10^{-94}\right):\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 56.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -4 \cdot 10^{+80}:\\ \;\;\;\;\left(-3 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -1.18 \cdot 10^{-136} \lor \neg \left(x1 \leq 5.8 \cdot 10^{-94}\right):\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -4e+80)
   (* (* -3.0 x1) (* x1 x1))
   (if (or (<= x1 -1.18e-136) (not (<= x1 5.8e-94)))
     (* (* (* x2 x2) x1) 8.0)
     (* -6.0 x2))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -4e+80) {
		tmp = (-3.0 * x1) * (x1 * x1);
	} else if ((x1 <= -1.18e-136) || !(x1 <= 5.8e-94)) {
		tmp = ((x2 * x2) * x1) * 8.0;
	} 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 (x1 <= (-4d+80)) then
        tmp = ((-3.0d0) * x1) * (x1 * x1)
    else if ((x1 <= (-1.18d-136)) .or. (.not. (x1 <= 5.8d-94))) then
        tmp = ((x2 * x2) * x1) * 8.0d0
    else
        tmp = (-6.0d0) * x2
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if (x1 <= -4e+80) {
		tmp = (-3.0 * x1) * (x1 * x1);
	} else if ((x1 <= -1.18e-136) || !(x1 <= 5.8e-94)) {
		tmp = ((x2 * x2) * x1) * 8.0;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if x1 <= -4e+80:
		tmp = (-3.0 * x1) * (x1 * x1)
	elif (x1 <= -1.18e-136) or not (x1 <= 5.8e-94):
		tmp = ((x2 * x2) * x1) * 8.0
	else:
		tmp = -6.0 * x2
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -4e+80)
		tmp = Float64(Float64(-3.0 * x1) * Float64(x1 * x1));
	elseif ((x1 <= -1.18e-136) || !(x1 <= 5.8e-94))
		tmp = Float64(Float64(Float64(x2 * x2) * x1) * 8.0);
	else
		tmp = Float64(-6.0 * x2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if (x1 <= -4e+80)
		tmp = (-3.0 * x1) * (x1 * x1);
	elseif ((x1 <= -1.18e-136) || ~((x1 <= 5.8e-94)))
		tmp = ((x2 * x2) * x1) * 8.0;
	else
		tmp = -6.0 * x2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -4e+80], N[(N[(-3.0 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x1, -1.18e-136], N[Not[LessEqual[x1, 5.8e-94]], $MachinePrecision]], N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision], N[(-6.0 * x2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -4e80

   1. Initial program 13.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x1 around 0

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

   10. Applied rewrites 87.8%

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

   if -4e80 < x1 < -1.18e-136 or 5.79999999999999991e-94 < x1

   1. Initial program 83.4%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 9.3

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

   5. Applied rewrites 9.3%

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

   6. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 33.1

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

   8. Applied rewrites 33.1%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 41.4%

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

   if -1.18e-136 < x1 < 5.79999999999999991e-94

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4 \cdot 10^{+80}:\\ \;\;\;\;\left(-3 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -1.18 \cdot 10^{-136} \lor \neg \left(x1 \leq 5.8 \cdot 10^{-94}\right):\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 26.0% accurate, 49.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

3. Taylor expanded in x1 around 0

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

   1. lower-*.f64 25.5

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

5. Applied rewrites 25.5%

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

Reproduce

herbie shell --seed 2024326 
(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))))))