Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 71.0% → 99.5%

Time: 45.4s

Alternatives: 23

Speedup: 4.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.0% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

   3. Simplified 99.7%

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

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

   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. Taylor expanded in x1 around 0 0.0%

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

   3. Taylor expanded in x1 around inf 98.5%

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

   4. Taylor expanded in x1 around inf 98.5%

      \[\leadsto \color{blue}{-1 \cdot x1 + 6 \cdot {x1}^{4}} \]
   5. Step-by-step derivation

      1. +-commutative 98.5%

         \[\leadsto \color{blue}{6 \cdot {x1}^{4} + -1 \cdot x1} \]

      2. neg-mul-1 98.5%

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

      3. unsub-neg 98.5%

         \[\leadsto \color{blue}{6 \cdot {x1}^{4} - x1} \]

   6. Simplified 98.5%

      \[\leadsto \color{blue}{6 \cdot {x1}^{4} - x1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x1, x2):
	t_0 = x1 * (x1 * x1)
	t_1 = x1 * (x1 * 3.0)
	t_2 = (x1 * x1) + 1.0
	t_3 = ((t_1 + (2.0 * x2)) - x1) / t_2
	t_4 = ((x1 * 2.0) * t_3) * (t_3 - 3.0)
	tmp = 0
	if (x1 + ((x1 + (((t_2 * (t_4 + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_1 * t_3)) + t_0)) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)))) <= math.inf:
		tmp = x1 + ((x1 + (t_0 + ((t_2 * (t_4 + ((x1 * x1) * ((4.0 * ((((2.0 * x2) + (3.0 * (x1 * x1))) - x1) / t_2)) - 6.0)))) + ((x1 * x1) * 9.0)))) + (3.0 * ((x2 * -2.0) - x1)))
	else:
		tmp = (6.0 * math.pow(x1, 4.0)) - x1
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 * (x1 * x1);
	t_1 = x1 * (x1 * 3.0);
	t_2 = (x1 * x1) + 1.0;
	t_3 = ((t_1 + (2.0 * x2)) - x1) / t_2;
	t_4 = ((x1 * 2.0) * t_3) * (t_3 - 3.0);
	tmp = 0.0;
	if ((x1 + ((x1 + (((t_2 * (t_4 + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_1 * t_3)) + t_0)) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)))) <= Inf)
		tmp = x1 + ((x1 + (t_0 + ((t_2 * (t_4 + ((x1 * x1) * ((4.0 * ((((2.0 * x2) + (3.0 * (x1 * x1))) - x1) / t_2)) - 6.0)))) + ((x1 * x1) * 9.0)))) + (3.0 * ((x2 * -2.0) - x1)));
	else
		tmp = (6.0 * (x1 ^ 4.0)) - x1;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   2. Taylor expanded in x1 around 0 98.6%

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

   3. Taylor expanded in x1 around 0 98.6%

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

      1. unpow2 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in x1 around inf 99.4%

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

      1. *-commutative 33.3%

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

      2. unpow2 33.3%

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

   8. Simplified 99.4%

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

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

   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. Taylor expanded in x1 around 0 0.0%

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

   3. Taylor expanded in x1 around inf 98.5%

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

   4. Taylor expanded in x1 around inf 98.5%

      \[\leadsto \color{blue}{-1 \cdot x1 + 6 \cdot {x1}^{4}} \]
   5. Step-by-step derivation

      1. +-commutative 98.5%

         \[\leadsto \color{blue}{6 \cdot {x1}^{4} + -1 \cdot x1} \]

      2. neg-mul-1 98.5%

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

      3. unsub-neg 98.5%

         \[\leadsto \color{blue}{6 \cdot {x1}^{4} - x1} \]

   6. Simplified 98.5%

      \[\leadsto \color{blue}{6 \cdot {x1}^{4} - x1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

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

Alternative 3: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot \left(x1 \cdot 9\right) - x1\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t_0 + 2 \cdot x2\right) - x1}{t_2}\\ t_4 := x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot x1\right) \cdot 9 + t_2 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(t_3 \cdot 4 - 6\right) + \left(\left(x1 \cdot 2\right) \cdot t_3\right) \cdot \left(2 \cdot \frac{x2}{x1 \cdot x1} + \left(\frac{-1}{x1} - \frac{3}{x1 \cdot x1}\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{if}\;x1 \leq -1.8 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x1 \leq -0.7:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x1 \leq 0.33:\\ \;\;\;\;x1 + \left(3 \cdot \frac{\left(t_0 - 2 \cdot x2\right) - x1}{t_2} + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (- (* x1 (* x1 9.0)) x1))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (* 3.0 (- (* x2 -2.0) x1))
           (+
            x1
            (+
             (* x1 (* x1 x1))
             (+
              (* (* x1 x1) 9.0)
              (*
               t_2
               (+
                (* (* x1 x1) (- (* t_3 4.0) 6.0))
                (*
                 (* (* x1 2.0) t_3)
                 (+
                  (* 2.0 (/ x2 (* x1 x1)))
                  (- (/ -1.0 x1) (/ 3.0 (* x1 x1))))))))))))))
   (if (<= x1 -1.8e+108)
     t_1
     (if (<= x1 -0.7)
       t_4
       (if (<= x1 0.33)
         (+
          x1
          (+
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_2))
           (+ x1 (* 4.0 (* x2 (* x1 (- (* 2.0 x2) 3.0)))))))
         (if (<= x1 2e+153) t_4 t_1))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * (x1 * 9.0)) - x1;
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + (((x1 * x1) * 9.0) + (t_2 * (((x1 * x1) * ((t_3 * 4.0) - 6.0)) + (((x1 * 2.0) * t_3) * ((2.0 * (x2 / (x1 * x1))) + ((-1.0 / x1) - (3.0 / (x1 * x1)))))))))));
	double tmp;
	if (x1 <= -1.8e+108) {
		tmp = t_1;
	} else if (x1 <= -0.7) {
		tmp = t_4;
	} else if (x1 <= 0.33) {
		tmp = x1 + ((3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	} else if (x1 <= 2e+153) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = x1 * (x1 * 3.0d0)
    t_1 = (x1 * (x1 * 9.0d0)) - x1
    t_2 = (x1 * x1) + 1.0d0
    t_3 = ((t_0 + (2.0d0 * x2)) - x1) / t_2
    t_4 = x1 + ((3.0d0 * ((x2 * (-2.0d0)) - x1)) + (x1 + ((x1 * (x1 * x1)) + (((x1 * x1) * 9.0d0) + (t_2 * (((x1 * x1) * ((t_3 * 4.0d0) - 6.0d0)) + (((x1 * 2.0d0) * t_3) * ((2.0d0 * (x2 / (x1 * x1))) + (((-1.0d0) / x1) - (3.0d0 / (x1 * x1)))))))))))
    if (x1 <= (-1.8d+108)) then
        tmp = t_1
    else if (x1 <= (-0.7d0)) then
        tmp = t_4
    else if (x1 <= 0.33d0) then
        tmp = x1 + ((3.0d0 * (((t_0 - (2.0d0 * x2)) - x1) / t_2)) + (x1 + (4.0d0 * (x2 * (x1 * ((2.0d0 * x2) - 3.0d0))))))
    else if (x1 <= 2d+153) then
        tmp = t_4
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * (x1 * 9.0)) - x1;
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + (((x1 * x1) * 9.0) + (t_2 * (((x1 * x1) * ((t_3 * 4.0) - 6.0)) + (((x1 * 2.0) * t_3) * ((2.0 * (x2 / (x1 * x1))) + ((-1.0 / x1) - (3.0 / (x1 * x1)))))))))));
	double tmp;
	if (x1 <= -1.8e+108) {
		tmp = t_1;
	} else if (x1 <= -0.7) {
		tmp = t_4;
	} else if (x1 <= 0.33) {
		tmp = x1 + ((3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	} else if (x1 <= 2e+153) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = x1 * (x1 * 3.0)
	t_1 = (x1 * (x1 * 9.0)) - x1
	t_2 = (x1 * x1) + 1.0
	t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2
	t_4 = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + (((x1 * x1) * 9.0) + (t_2 * (((x1 * x1) * ((t_3 * 4.0) - 6.0)) + (((x1 * 2.0) * t_3) * ((2.0 * (x2 / (x1 * x1))) + ((-1.0 / x1) - (3.0 / (x1 * x1)))))))))))
	tmp = 0
	if x1 <= -1.8e+108:
		tmp = t_1
	elif x1 <= -0.7:
		tmp = t_4
	elif x1 <= 0.33:
		tmp = x1 + ((3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))))
	elif x1 <= 2e+153:
		tmp = t_4
	else:
		tmp = t_1
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * Float64(x1 * 9.0)) - 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(x1 + Float64(Float64(3.0 * Float64(Float64(x2 * -2.0) - x1)) + Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(Float64(Float64(x1 * x1) * 9.0) + Float64(t_2 * Float64(Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(Float64(2.0 * Float64(x2 / Float64(x1 * x1))) + Float64(Float64(-1.0 / x1) - Float64(3.0 / Float64(x1 * x1))))))))))))
	tmp = 0.0
	if (x1 <= -1.8e+108)
		tmp = t_1;
	elseif (x1 <= -0.7)
		tmp = t_4;
	elseif (x1 <= 0.33)
		tmp = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_2)) + Float64(x1 + Float64(4.0 * Float64(x2 * Float64(x1 * Float64(Float64(2.0 * x2) - 3.0)))))));
	elseif (x1 <= 2e+153)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 * (x1 * 3.0);
	t_1 = (x1 * (x1 * 9.0)) - x1;
	t_2 = (x1 * x1) + 1.0;
	t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	t_4 = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + (((x1 * x1) * 9.0) + (t_2 * (((x1 * x1) * ((t_3 * 4.0) - 6.0)) + (((x1 * 2.0) * t_3) * ((2.0 * (x2 / (x1 * x1))) + ((-1.0 / x1) - (3.0 / (x1 * x1)))))))))));
	tmp = 0.0;
	if (x1 <= -1.8e+108)
		tmp = t_1;
	elseif (x1 <= -0.7)
		tmp = t_4;
	elseif (x1 <= 0.33)
		tmp = x1 + ((3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	elseif (x1 <= 2e+153)
		tmp = t_4;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $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[(x1 + N[(N[(3.0 * N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision] + N[(t$95$2 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(N[(2.0 * N[(x2 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / x1), $MachinePrecision] - N[(3.0 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.8e+108], t$95$1, If[LessEqual[x1, -0.7], t$95$4, If[LessEqual[x1, 0.33], N[(x1 + N[(N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(4.0 * N[(x2 * N[(x1 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2e+153], t$95$4, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.8e108 or 2e153 < x1

   1. Initial program 0.0%

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

   2. Taylor expanded in x1 around 0 0.0%

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

   3. Taylor expanded in x2 around 0 0.1%

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

      1. fma-def 0.1%

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

      2. *-commutative 0.1%

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

      3. unpow2 0.1%

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

      4. associate-*r* 0.1%

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

      5. +-rgt-identity 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.1%

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

      10. *-commutative 0.1%

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

      11. *-commutative 0.1%

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

      12. associate-*r* 0.1%

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x1 around 0 85.0%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 85.0%

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

      2. neg-mul-1 85.0%

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

      3. unsub-neg 85.0%

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

      4. *-commutative 85.0%

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

      5. unpow2 85.0%

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

      6. associate-*l* 85.0%

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

   8. Simplified 85.0%

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

   if -1.8e108 < x1 < -0.69999999999999996 or 0.330000000000000016 < x1 < 2e153

   1. Initial program 97.6%

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

   2. Taylor expanded in x1 around 0 94.6%

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

   3. Taylor expanded in x1 around inf 93.3%

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

      1. unpow2 93.3%

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

      2. associate-*r/ 93.3%

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

      3. metadata-eval 93.3%

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

      4. unpow2 93.3%

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

   5. Simplified 93.3%

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

   6. Taylor expanded in x1 around inf 93.1%

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

      1. *-commutative 93.1%

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

      2. unpow2 93.1%

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

   8. Simplified 93.1%

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

   if -0.69999999999999996 < x1 < 0.330000000000000016

   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. Taylor expanded in x1 around 0 99.3%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.8 \cdot 10^{+108}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{elif}\;x1 \leq -0.7:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot x1\right) \cdot 9 + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(2 \cdot \frac{x2}{x1 \cdot x1} + \left(\frac{-1}{x1} - \frac{3}{x1 \cdot x1}\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 0.33:\\ \;\;\;\;x1 + \left(3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot x1\right) \cdot 9 + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(2 \cdot \frac{x2}{x1 \cdot x1} + \left(\frac{-1}{x1} - \frac{3}{x1 \cdot x1}\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \end{array} \]

Alternative 4: 93.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 9\right) - x1\\ t_2 := x1 \cdot \left(x1 \cdot 3\right)\\ t_3 := \frac{\left(t_2 + 2 \cdot x2\right) - x1}{t_0}\\ t_4 := x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(t_2 \cdot t_3 + t_0 \cdot \left(\left(x1 \cdot x1\right) \cdot 6 + \left(\left(x1 \cdot 2\right) \cdot t_3\right) \cdot \left(2 \cdot \frac{x2}{x1 \cdot x1} + \left(\frac{-1}{x1} - \frac{3}{x1 \cdot x1}\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{if}\;x1 \leq -5 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x1 \leq -0.72:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x1 \leq 0.36:\\ \;\;\;\;x1 + \left(3 \cdot \frac{\left(t_2 - 2 \cdot x2\right) - x1}{t_0} + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (- (* x1 (* x1 9.0)) x1))
        (t_2 (* x1 (* x1 3.0)))
        (t_3 (/ (- (+ t_2 (* 2.0 x2)) x1) t_0))
        (t_4
         (+
          x1
          (+
           (* 3.0 (- (* x2 -2.0) x1))
           (+
            x1
            (+
             (* x1 (* x1 x1))
             (+
              (* t_2 t_3)
              (*
               t_0
               (+
                (* (* x1 x1) 6.0)
                (*
                 (* (* x1 2.0) t_3)
                 (+
                  (* 2.0 (/ x2 (* x1 x1)))
                  (- (/ -1.0 x1) (/ 3.0 (* x1 x1))))))))))))))
   (if (<= x1 -5e+106)
     t_1
     (if (<= x1 -0.72)
       t_4
       (if (<= x1 0.36)
         (+
          x1
          (+
           (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_0))
           (+ x1 (* 4.0 (* x2 (* x1 (- (* 2.0 x2) 3.0)))))))
         (if (<= x1 2e+153) t_4 t_1))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = (x1 * (x1 * 9.0)) - x1;
	double t_2 = x1 * (x1 * 3.0);
	double t_3 = ((t_2 + (2.0 * x2)) - x1) / t_0;
	double t_4 = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_3) + (t_0 * (((x1 * x1) * 6.0) + (((x1 * 2.0) * t_3) * ((2.0 * (x2 / (x1 * x1))) + ((-1.0 / x1) - (3.0 / (x1 * x1)))))))))));
	double tmp;
	if (x1 <= -5e+106) {
		tmp = t_1;
	} else if (x1 <= -0.72) {
		tmp = t_4;
	} else if (x1 <= 0.36) {
		tmp = x1 + ((3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	} else if (x1 <= 2e+153) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (x1 * x1) + 1.0d0
    t_1 = (x1 * (x1 * 9.0d0)) - x1
    t_2 = x1 * (x1 * 3.0d0)
    t_3 = ((t_2 + (2.0d0 * x2)) - x1) / t_0
    t_4 = x1 + ((3.0d0 * ((x2 * (-2.0d0)) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_3) + (t_0 * (((x1 * x1) * 6.0d0) + (((x1 * 2.0d0) * t_3) * ((2.0d0 * (x2 / (x1 * x1))) + (((-1.0d0) / x1) - (3.0d0 / (x1 * x1)))))))))))
    if (x1 <= (-5d+106)) then
        tmp = t_1
    else if (x1 <= (-0.72d0)) then
        tmp = t_4
    else if (x1 <= 0.36d0) then
        tmp = x1 + ((3.0d0 * (((t_2 - (2.0d0 * x2)) - x1) / t_0)) + (x1 + (4.0d0 * (x2 * (x1 * ((2.0d0 * x2) - 3.0d0))))))
    else if (x1 <= 2d+153) then
        tmp = t_4
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = (x1 * (x1 * 9.0)) - x1;
	double t_2 = x1 * (x1 * 3.0);
	double t_3 = ((t_2 + (2.0 * x2)) - x1) / t_0;
	double t_4 = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_3) + (t_0 * (((x1 * x1) * 6.0) + (((x1 * 2.0) * t_3) * ((2.0 * (x2 / (x1 * x1))) + ((-1.0 / x1) - (3.0 / (x1 * x1)))))))))));
	double tmp;
	if (x1 <= -5e+106) {
		tmp = t_1;
	} else if (x1 <= -0.72) {
		tmp = t_4;
	} else if (x1 <= 0.36) {
		tmp = x1 + ((3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	} else if (x1 <= 2e+153) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * x1) + 1.0
	t_1 = (x1 * (x1 * 9.0)) - x1
	t_2 = x1 * (x1 * 3.0)
	t_3 = ((t_2 + (2.0 * x2)) - x1) / t_0
	t_4 = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_3) + (t_0 * (((x1 * x1) * 6.0) + (((x1 * 2.0) * t_3) * ((2.0 * (x2 / (x1 * x1))) + ((-1.0 / x1) - (3.0 / (x1 * x1)))))))))))
	tmp = 0
	if x1 <= -5e+106:
		tmp = t_1
	elif x1 <= -0.72:
		tmp = t_4
	elif x1 <= 0.36:
		tmp = x1 + ((3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))))
	elif x1 <= 2e+153:
		tmp = t_4
	else:
		tmp = t_1
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1)
	t_2 = Float64(x1 * Float64(x1 * 3.0))
	t_3 = Float64(Float64(Float64(t_2 + Float64(2.0 * x2)) - x1) / t_0)
	t_4 = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(x2 * -2.0) - x1)) + Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(Float64(t_2 * t_3) + Float64(t_0 * Float64(Float64(Float64(x1 * x1) * 6.0) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(Float64(2.0 * Float64(x2 / Float64(x1 * x1))) + Float64(Float64(-1.0 / x1) - Float64(3.0 / Float64(x1 * x1))))))))))))
	tmp = 0.0
	if (x1 <= -5e+106)
		tmp = t_1;
	elseif (x1 <= -0.72)
		tmp = t_4;
	elseif (x1 <= 0.36)
		tmp = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_0)) + Float64(x1 + Float64(4.0 * Float64(x2 * Float64(x1 * Float64(Float64(2.0 * x2) - 3.0)))))));
	elseif (x1 <= 2e+153)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * x1) + 1.0;
	t_1 = (x1 * (x1 * 9.0)) - x1;
	t_2 = x1 * (x1 * 3.0);
	t_3 = ((t_2 + (2.0 * x2)) - x1) / t_0;
	t_4 = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_3) + (t_0 * (((x1 * x1) * 6.0) + (((x1 * 2.0) * t_3) * ((2.0 * (x2 / (x1 * x1))) + ((-1.0 / x1) - (3.0 / (x1 * x1)))))))))));
	tmp = 0.0;
	if (x1 <= -5e+106)
		tmp = t_1;
	elseif (x1 <= -0.72)
		tmp = t_4;
	elseif (x1 <= 0.36)
		tmp = x1 + ((3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	elseif (x1 <= 2e+153)
		tmp = t_4;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$2 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(3.0 * N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * t$95$3), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(N[(2.0 * N[(x2 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / x1), $MachinePrecision] - N[(3.0 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -5e+106], t$95$1, If[LessEqual[x1, -0.72], t$95$4, If[LessEqual[x1, 0.36], N[(x1 + N[(N[(3.0 * N[(N[(N[(t$95$2 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(4.0 * N[(x2 * N[(x1 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2e+153], t$95$4, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -4.9999999999999998e106 or 2e153 < x1

   1. Initial program 0.0%

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

   2. Taylor expanded in x1 around 0 0.0%

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

   3. Taylor expanded in x2 around 0 0.1%

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

      1. fma-def 0.1%

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

      2. *-commutative 0.1%

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

      3. unpow2 0.1%

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

      4. associate-*r* 0.1%

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

      5. +-rgt-identity 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.1%

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

      10. *-commutative 0.1%

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

      11. *-commutative 0.1%

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

      12. associate-*r* 0.1%

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x1 around 0 85.0%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 85.0%

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

      2. neg-mul-1 85.0%

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

      3. unsub-neg 85.0%

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

      4. *-commutative 85.0%

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

      5. unpow2 85.0%

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

      6. associate-*l* 85.0%

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

   8. Simplified 85.0%

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

   if -4.9999999999999998e106 < x1 < -0.71999999999999997 or 0.35999999999999999 < x1 < 2e153

   1. Initial program 97.6%

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

   2. Taylor expanded in x1 around 0 94.6%

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

   3. Taylor expanded in x1 around inf 93.3%

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

      1. unpow2 93.3%

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

      2. associate-*r/ 93.3%

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

      3. metadata-eval 93.3%

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

      4. unpow2 93.3%

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

   5. Simplified 93.3%

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

   6. Taylor expanded in x1 around inf 87.0%

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

      1. *-commutative 66.8%

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

      2. unpow2 66.8%

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

   8. Simplified 87.0%

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

   if -0.71999999999999997 < x1 < 0.35999999999999999

   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. Taylor expanded in x1 around 0 99.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -5 \cdot 10^{+106}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{elif}\;x1 \leq -0.72:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot 6 + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(2 \cdot \frac{x2}{x1 \cdot x1} + \left(\frac{-1}{x1} - \frac{3}{x1 \cdot x1}\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 0.36:\\ \;\;\;\;x1 + \left(3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot 6 + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(2 \cdot \frac{x2}{x1 \cdot x1} + \left(\frac{-1}{x1} - \frac{3}{x1 \cdot x1}\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \end{array} \]

Alternative 5: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{t_0}\\ \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+108} \lor \neg \left(x1 \leq 2 \cdot 10^{+153}\right):\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(t_0 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t_1\right) \cdot \left(t_1 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(2 \cdot x2 + 3 \cdot \left(x1 \cdot x1\right)\right) - x1}{t_0} - 6\right)\right) + \left(x1 \cdot x1\right) \cdot 9\right)\right)\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (/ (- (+ (* x1 (* x1 3.0)) (* 2.0 x2)) x1) t_0)))
   (if (or (<= x1 -3.6e+108) (not (<= x1 2e+153)))
     (- (* x1 (* x1 9.0)) x1)
     (+
      x1
      (+
       (+
        x1
        (+
         (* x1 (* x1 x1))
         (+
          (*
           t_0
           (+
            (* (* (* x1 2.0) t_1) (- t_1 3.0))
            (*
             (* x1 x1)
             (- (* 4.0 (/ (- (+ (* 2.0 x2) (* 3.0 (* x1 x1))) x1) t_0)) 6.0))))
          (* (* x1 x1) 9.0))))
       (* 3.0 (- (* x2 -2.0) x1)))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = (((x1 * (x1 * 3.0)) + (2.0 * x2)) - x1) / t_0;
	double tmp;
	if ((x1 <= -3.6e+108) || !(x1 <= 2e+153)) {
		tmp = (x1 * (x1 * 9.0)) - x1;
	} else {
		tmp = x1 + ((x1 + ((x1 * (x1 * x1)) + ((t_0 * ((((x1 * 2.0) * t_1) * (t_1 - 3.0)) + ((x1 * x1) * ((4.0 * ((((2.0 * x2) + (3.0 * (x1 * x1))) - x1) / t_0)) - 6.0)))) + ((x1 * x1) * 9.0)))) + (3.0 * ((x2 * -2.0) - 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) :: tmp
    t_0 = (x1 * x1) + 1.0d0
    t_1 = (((x1 * (x1 * 3.0d0)) + (2.0d0 * x2)) - x1) / t_0
    if ((x1 <= (-3.6d+108)) .or. (.not. (x1 <= 2d+153))) then
        tmp = (x1 * (x1 * 9.0d0)) - x1
    else
        tmp = x1 + ((x1 + ((x1 * (x1 * x1)) + ((t_0 * ((((x1 * 2.0d0) * t_1) * (t_1 - 3.0d0)) + ((x1 * x1) * ((4.0d0 * ((((2.0d0 * x2) + (3.0d0 * (x1 * x1))) - x1) / t_0)) - 6.0d0)))) + ((x1 * x1) * 9.0d0)))) + (3.0d0 * ((x2 * (-2.0d0)) - x1)))
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = (((x1 * (x1 * 3.0)) + (2.0 * x2)) - x1) / t_0;
	double tmp;
	if ((x1 <= -3.6e+108) || !(x1 <= 2e+153)) {
		tmp = (x1 * (x1 * 9.0)) - x1;
	} else {
		tmp = x1 + ((x1 + ((x1 * (x1 * x1)) + ((t_0 * ((((x1 * 2.0) * t_1) * (t_1 - 3.0)) + ((x1 * x1) * ((4.0 * ((((2.0 * x2) + (3.0 * (x1 * x1))) - x1) / t_0)) - 6.0)))) + ((x1 * x1) * 9.0)))) + (3.0 * ((x2 * -2.0) - x1)));
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * x1) + 1.0
	t_1 = (((x1 * (x1 * 3.0)) + (2.0 * x2)) - x1) / t_0
	tmp = 0
	if (x1 <= -3.6e+108) or not (x1 <= 2e+153):
		tmp = (x1 * (x1 * 9.0)) - x1
	else:
		tmp = x1 + ((x1 + ((x1 * (x1 * x1)) + ((t_0 * ((((x1 * 2.0) * t_1) * (t_1 - 3.0)) + ((x1 * x1) * ((4.0 * ((((2.0 * x2) + (3.0 * (x1 * x1))) - x1) / t_0)) - 6.0)))) + ((x1 * x1) * 9.0)))) + (3.0 * ((x2 * -2.0) - x1)))
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(Float64(Float64(Float64(x1 * Float64(x1 * 3.0)) + Float64(2.0 * x2)) - x1) / t_0)
	tmp = 0.0
	if ((x1 <= -3.6e+108) || !(x1 <= 2e+153))
		tmp = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1);
	else
		tmp = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(Float64(t_0 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_1) * Float64(t_1 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * Float64(Float64(Float64(Float64(2.0 * x2) + Float64(3.0 * Float64(x1 * x1))) - x1) / t_0)) - 6.0)))) + Float64(Float64(x1 * x1) * 9.0)))) + Float64(3.0 * Float64(Float64(x2 * -2.0) - x1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * x1) + 1.0;
	t_1 = (((x1 * (x1 * 3.0)) + (2.0 * x2)) - x1) / t_0;
	tmp = 0.0;
	if ((x1 <= -3.6e+108) || ~((x1 <= 2e+153)))
		tmp = (x1 * (x1 * 9.0)) - x1;
	else
		tmp = x1 + ((x1 + ((x1 * (x1 * x1)) + ((t_0 * ((((x1 * 2.0) * t_1) * (t_1 - 3.0)) + ((x1 * x1) * ((4.0 * ((((2.0 * x2) + (3.0 * (x1 * x1))) - x1) / t_0)) - 6.0)))) + ((x1 * x1) * 9.0)))) + (3.0 * ((x2 * -2.0) - x1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[Or[LessEqual[x1, -3.6e+108], N[Not[LessEqual[x1, 2e+153]], $MachinePrecision]], N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision], N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t$95$1 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * N[(N[(N[(N[(2.0 * x2), $MachinePrecision] + N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.6e108 or 2e153 < x1

   1. Initial program 0.0%

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

   2. Taylor expanded in x1 around 0 0.0%

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

   3. Taylor expanded in x2 around 0 0.1%

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

      1. fma-def 0.1%

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

      2. *-commutative 0.1%

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

      3. unpow2 0.1%

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

      4. associate-*r* 0.1%

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

      5. +-rgt-identity 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.1%

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

      10. *-commutative 0.1%

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

      11. *-commutative 0.1%

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

      12. associate-*r* 0.1%

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x1 around 0 85.0%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 85.0%

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

      2. neg-mul-1 85.0%

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

      3. unsub-neg 85.0%

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

      4. *-commutative 85.0%

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

      5. unpow2 85.0%

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

      6. associate-*l* 85.0%

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

   8. Simplified 85.0%

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

   if -3.6e108 < x1 < 2e153

   1. Initial program 98.9%

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

   2. Taylor expanded in x1 around 0 98.0%

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

   3. Taylor expanded in x1 around 0 98.1%

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

      1. unpow2 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in x1 around inf 98.9%

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

      1. *-commutative 33.1%

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

      2. unpow2 33.1%

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

   8. Simplified 98.9%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+108} \lor \neg \left(x1 \leq 2 \cdot 10^{+153}\right):\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(2 \cdot x2 + 3 \cdot \left(x1 \cdot x1\right)\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) + \left(x1 \cdot x1\right) \cdot 9\right)\right)\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right)\\ \end{array} \]

Alternative 6: 93.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \frac{\left(t_1 + 2 \cdot x2\right) - x1}{t_0}\\ \mathbf{if}\;x1 \leq -1.28 \cdot 10^{+107} \lor \neg \left(x1 \leq 2 \cdot 10^{+153}\right):\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(t_1 \cdot t_2 + t_0 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t_2\right) \cdot \left(t_2 - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (/ (- (+ t_1 (* 2.0 x2)) x1) t_0)))
   (if (or (<= x1 -1.28e+107) (not (<= x1 2e+153)))
     (- (* x1 (* x1 9.0)) x1)
     (+
      x1
      (+
       (* 3.0 (- (* x2 -2.0) x1))
       (+
        x1
        (+
         (* x1 (* x1 x1))
         (+
          (* t_1 t_2)
          (*
           t_0
           (+ (* (* (* x1 2.0) t_2) (- t_2 3.0)) (* (* x1 x1) 6.0)))))))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = ((t_1 + (2.0 * x2)) - x1) / t_0;
	double tmp;
	if ((x1 <= -1.28e+107) || !(x1 <= 2e+153)) {
		tmp = (x1 * (x1 * 9.0)) - x1;
	} else {
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_1 * t_2) + (t_0 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * 6.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x1 * x1) + 1.0d0
    t_1 = x1 * (x1 * 3.0d0)
    t_2 = ((t_1 + (2.0d0 * x2)) - x1) / t_0
    if ((x1 <= (-1.28d+107)) .or. (.not. (x1 <= 2d+153))) then
        tmp = (x1 * (x1 * 9.0d0)) - x1
    else
        tmp = x1 + ((3.0d0 * ((x2 * (-2.0d0)) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_1 * t_2) + (t_0 * ((((x1 * 2.0d0) * t_2) * (t_2 - 3.0d0)) + ((x1 * x1) * 6.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = ((t_1 + (2.0 * x2)) - x1) / t_0;
	double tmp;
	if ((x1 <= -1.28e+107) || !(x1 <= 2e+153)) {
		tmp = (x1 * (x1 * 9.0)) - x1;
	} else {
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_1 * t_2) + (t_0 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * 6.0)))))));
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * x1) + 1.0
	t_1 = x1 * (x1 * 3.0)
	t_2 = ((t_1 + (2.0 * x2)) - x1) / t_0
	tmp = 0
	if (x1 <= -1.28e+107) or not (x1 <= 2e+153):
		tmp = (x1 * (x1 * 9.0)) - x1
	else:
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_1 * t_2) + (t_0 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * 6.0)))))))
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_0)
	tmp = 0.0
	if ((x1 <= -1.28e+107) || !(x1 <= 2e+153))
		tmp = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1);
	else
		tmp = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(x2 * -2.0) - x1)) + Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(Float64(t_1 * t_2) + Float64(t_0 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * 6.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * x1) + 1.0;
	t_1 = x1 * (x1 * 3.0);
	t_2 = ((t_1 + (2.0 * x2)) - x1) / t_0;
	tmp = 0.0;
	if ((x1 <= -1.28e+107) || ~((x1 <= 2e+153)))
		tmp = (x1 * (x1 * 9.0)) - x1;
	else
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_1 * t_2) + (t_0 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * 6.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[Or[LessEqual[x1, -1.28e+107], N[Not[LessEqual[x1, 2e+153]], $MachinePrecision]], N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision], N[(x1 + N[(N[(3.0 * N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * t$95$2), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.2799999999999999e107 or 2e153 < x1

   1. Initial program 0.0%

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

   2. Taylor expanded in x1 around 0 0.0%

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

   3. Taylor expanded in x2 around 0 0.1%

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

      1. fma-def 0.1%

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

      2. *-commutative 0.1%

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

      3. unpow2 0.1%

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

      4. associate-*r* 0.1%

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

      5. +-rgt-identity 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.1%

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

      10. *-commutative 0.1%

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

      11. *-commutative 0.1%

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

      12. associate-*r* 0.1%

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x1 around 0 85.0%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 85.0%

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

      2. neg-mul-1 85.0%

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

      3. unsub-neg 85.0%

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

      4. *-commutative 85.0%

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

      5. unpow2 85.0%

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

      6. associate-*l* 85.0%

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

   8. Simplified 85.0%

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

   if -1.2799999999999999e107 < x1 < 2e153

   1. Initial program 98.9%

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

   2. Taylor expanded in x1 around 0 98.0%

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

   3. Taylor expanded in x1 around inf 96.4%

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

      1. *-commutative 57.7%

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

      2. unpow2 57.7%

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

   5. Simplified 96.4%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.28 \cdot 10^{+107} \lor \neg \left(x1 \leq 2 \cdot 10^{+153}\right):\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 9\right) - x1\\ t_2 := x1 \cdot \left(x1 \cdot 3\right)\\ t_3 := 2 \cdot x2 - 3\\ t_4 := \frac{\left(t_2 + 2 \cdot x2\right) - x1}{t_0}\\ t_5 := x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(t_2 \cdot t_4 + t_0 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(t_4 \cdot 4 - 6\right) + 4 \cdot \frac{x2 \cdot t_3}{x1}\right)\right)\right)\right)\right)\\ \mathbf{if}\;x1 \leq -7.1 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x1 \leq -0.9:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x1 \leq 1.45 \cdot 10^{-18}:\\ \;\;\;\;x1 + \left(3 \cdot \frac{\left(t_2 - 2 \cdot x2\right) - x1}{t_0} + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot t_3\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (- (* x1 (* x1 9.0)) x1))
        (t_2 (* x1 (* x1 3.0)))
        (t_3 (- (* 2.0 x2) 3.0))
        (t_4 (/ (- (+ t_2 (* 2.0 x2)) x1) t_0))
        (t_5
         (+
          x1
          (+
           (* 3.0 (- (* x2 -2.0) x1))
           (+
            x1
            (+
             (* x1 (* x1 x1))
             (+
              (* t_2 t_4)
              (*
               t_0
               (+
                (* (* x1 x1) (- (* t_4 4.0) 6.0))
                (* 4.0 (/ (* x2 t_3) x1)))))))))))
   (if (<= x1 -7.1e+108)
     t_1
     (if (<= x1 -0.9)
       t_5
       (if (<= x1 1.45e-18)
         (+
          x1
          (+
           (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_0))
           (+ x1 (* 4.0 (* x2 (* x1 t_3))))))
         (if (<= x1 2e+153) t_5 t_1))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = (x1 * (x1 * 9.0)) - x1;
	double t_2 = x1 * (x1 * 3.0);
	double t_3 = (2.0 * x2) - 3.0;
	double t_4 = ((t_2 + (2.0 * x2)) - x1) / t_0;
	double t_5 = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_4) + (t_0 * (((x1 * x1) * ((t_4 * 4.0) - 6.0)) + (4.0 * ((x2 * t_3) / x1))))))));
	double tmp;
	if (x1 <= -7.1e+108) {
		tmp = t_1;
	} else if (x1 <= -0.9) {
		tmp = t_5;
	} else if (x1 <= 1.45e-18) {
		tmp = x1 + ((3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)) + (x1 + (4.0 * (x2 * (x1 * t_3)))));
	} else if (x1 <= 2e+153) {
		tmp = t_5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (x1 * x1) + 1.0d0
    t_1 = (x1 * (x1 * 9.0d0)) - x1
    t_2 = x1 * (x1 * 3.0d0)
    t_3 = (2.0d0 * x2) - 3.0d0
    t_4 = ((t_2 + (2.0d0 * x2)) - x1) / t_0
    t_5 = x1 + ((3.0d0 * ((x2 * (-2.0d0)) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_4) + (t_0 * (((x1 * x1) * ((t_4 * 4.0d0) - 6.0d0)) + (4.0d0 * ((x2 * t_3) / x1))))))))
    if (x1 <= (-7.1d+108)) then
        tmp = t_1
    else if (x1 <= (-0.9d0)) then
        tmp = t_5
    else if (x1 <= 1.45d-18) then
        tmp = x1 + ((3.0d0 * (((t_2 - (2.0d0 * x2)) - x1) / t_0)) + (x1 + (4.0d0 * (x2 * (x1 * t_3)))))
    else if (x1 <= 2d+153) then
        tmp = t_5
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = (x1 * (x1 * 9.0)) - x1;
	double t_2 = x1 * (x1 * 3.0);
	double t_3 = (2.0 * x2) - 3.0;
	double t_4 = ((t_2 + (2.0 * x2)) - x1) / t_0;
	double t_5 = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_4) + (t_0 * (((x1 * x1) * ((t_4 * 4.0) - 6.0)) + (4.0 * ((x2 * t_3) / x1))))))));
	double tmp;
	if (x1 <= -7.1e+108) {
		tmp = t_1;
	} else if (x1 <= -0.9) {
		tmp = t_5;
	} else if (x1 <= 1.45e-18) {
		tmp = x1 + ((3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)) + (x1 + (4.0 * (x2 * (x1 * t_3)))));
	} else if (x1 <= 2e+153) {
		tmp = t_5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * x1) + 1.0
	t_1 = (x1 * (x1 * 9.0)) - x1
	t_2 = x1 * (x1 * 3.0)
	t_3 = (2.0 * x2) - 3.0
	t_4 = ((t_2 + (2.0 * x2)) - x1) / t_0
	t_5 = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_4) + (t_0 * (((x1 * x1) * ((t_4 * 4.0) - 6.0)) + (4.0 * ((x2 * t_3) / x1))))))))
	tmp = 0
	if x1 <= -7.1e+108:
		tmp = t_1
	elif x1 <= -0.9:
		tmp = t_5
	elif x1 <= 1.45e-18:
		tmp = x1 + ((3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)) + (x1 + (4.0 * (x2 * (x1 * t_3)))))
	elif x1 <= 2e+153:
		tmp = t_5
	else:
		tmp = t_1
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1)
	t_2 = Float64(x1 * Float64(x1 * 3.0))
	t_3 = Float64(Float64(2.0 * x2) - 3.0)
	t_4 = Float64(Float64(Float64(t_2 + Float64(2.0 * x2)) - x1) / t_0)
	t_5 = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(x2 * -2.0) - x1)) + Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(Float64(t_2 * t_4) + Float64(t_0 * Float64(Float64(Float64(x1 * x1) * Float64(Float64(t_4 * 4.0) - 6.0)) + Float64(4.0 * Float64(Float64(x2 * t_3) / x1)))))))))
	tmp = 0.0
	if (x1 <= -7.1e+108)
		tmp = t_1;
	elseif (x1 <= -0.9)
		tmp = t_5;
	elseif (x1 <= 1.45e-18)
		tmp = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_0)) + Float64(x1 + Float64(4.0 * Float64(x2 * Float64(x1 * t_3))))));
	elseif (x1 <= 2e+153)
		tmp = t_5;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * x1) + 1.0;
	t_1 = (x1 * (x1 * 9.0)) - x1;
	t_2 = x1 * (x1 * 3.0);
	t_3 = (2.0 * x2) - 3.0;
	t_4 = ((t_2 + (2.0 * x2)) - x1) / t_0;
	t_5 = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_4) + (t_0 * (((x1 * x1) * ((t_4 * 4.0) - 6.0)) + (4.0 * ((x2 * t_3) / x1))))))));
	tmp = 0.0;
	if (x1 <= -7.1e+108)
		tmp = t_1;
	elseif (x1 <= -0.9)
		tmp = t_5;
	elseif (x1 <= 1.45e-18)
		tmp = x1 + ((3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)) + (x1 + (4.0 * (x2 * (x1 * t_3)))));
	elseif (x1 <= 2e+153)
		tmp = t_5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$2 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(x1 + N[(N[(3.0 * N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * t$95$4), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$4 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(x2 * t$95$3), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -7.1e+108], t$95$1, If[LessEqual[x1, -0.9], t$95$5, If[LessEqual[x1, 1.45e-18], N[(x1 + N[(N[(3.0 * N[(N[(N[(t$95$2 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(4.0 * N[(x2 * N[(x1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2e+153], t$95$5, t$95$1]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -7.1000000000000003e108 or 2e153 < x1

   1. Initial program 0.0%

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

   2. Taylor expanded in x1 around 0 0.0%

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

   3. Taylor expanded in x2 around 0 0.1%

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

      1. fma-def 0.1%

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

      2. *-commutative 0.1%

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

      3. unpow2 0.1%

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

      4. associate-*r* 0.1%

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

      5. +-rgt-identity 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.1%

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

      10. *-commutative 0.1%

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

      11. *-commutative 0.1%

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

      12. associate-*r* 0.1%

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x1 around 0 85.0%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 85.0%

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

      2. neg-mul-1 85.0%

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

      3. unsub-neg 85.0%

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

      4. *-commutative 85.0%

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

      5. unpow2 85.0%

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

      6. associate-*l* 85.0%

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

   8. Simplified 85.0%

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

   if -7.1000000000000003e108 < x1 < -0.900000000000000022 or 1.45e-18 < x1 < 2e153

   1. Initial program 97.7%

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

   2. Taylor expanded in x1 around 0 94.7%

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

   3. Taylor expanded in x1 around inf 91.5%

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

      1. unpow2 91.5%

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

      2. associate-*r/ 91.5%

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

      3. metadata-eval 91.5%

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

      4. unpow2 91.5%

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

   5. Simplified 91.5%

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

   6. Taylor expanded in x1 around 0 83.6%

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

   if -0.900000000000000022 < x1 < 1.45e-18

   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. Taylor expanded in x1 around 0 99.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -7.1 \cdot 10^{+108}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{elif}\;x1 \leq -0.9:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right) + 4 \cdot \frac{x2 \cdot \left(2 \cdot x2 - 3\right)}{x1}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 1.45 \cdot 10^{-18}:\\ \;\;\;\;x1 + \left(3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right) + 4 \cdot \frac{x2 \cdot \left(2 \cdot x2 - 3\right)}{x1}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \end{array} \]

Alternative 8: 87.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot \left(x1 \cdot 9\right) - x1\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t_0 + 2 \cdot x2\right) - x1}{t_2}\\ t_4 := x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(t_0 \cdot t_3 + t_2 \cdot \left(\left(x1 \cdot x1\right) \cdot 6 + \left(\left(x1 \cdot 2\right) \cdot t_3\right) \cdot -3\right)\right)\right)\right) + 3 \cdot \left(x2 \cdot -2\right)\right)\\ \mathbf{if}\;x1 \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x1 \leq -2.05 \cdot 10^{+55}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x1 \leq 1.9 \cdot 10^{+51}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (- (* x1 (* x1 9.0)) x1))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (* x1 (* x1 x1))
             (+
              (* t_0 t_3)
              (* t_2 (+ (* (* x1 x1) 6.0) (* (* (* x1 2.0) t_3) -3.0))))))
           (* 3.0 (* x2 -2.0))))))
   (if (<= x1 -5.6e+102)
     t_1
     (if (<= x1 -2.05e+55)
       t_4
       (if (<= x1 1.9e+51)
         (+
          x1
          (+
           (* 3.0 (- (* x2 -2.0) x1))
           (+ x1 (* 4.0 (* x2 (* x1 (- (* 2.0 x2) 3.0)))))))
         (if (<= x1 2e+153) t_4 t_1))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * (x1 * 9.0)) - x1;
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + ((x1 * (x1 * x1)) + ((t_0 * t_3) + (t_2 * (((x1 * x1) * 6.0) + (((x1 * 2.0) * t_3) * -3.0)))))) + (3.0 * (x2 * -2.0)));
	double tmp;
	if (x1 <= -5.6e+102) {
		tmp = t_1;
	} else if (x1 <= -2.05e+55) {
		tmp = t_4;
	} else if (x1 <= 1.9e+51) {
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	} else if (x1 <= 2e+153) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = x1 * (x1 * 3.0d0)
    t_1 = (x1 * (x1 * 9.0d0)) - x1
    t_2 = (x1 * x1) + 1.0d0
    t_3 = ((t_0 + (2.0d0 * x2)) - x1) / t_2
    t_4 = x1 + ((x1 + ((x1 * (x1 * x1)) + ((t_0 * t_3) + (t_2 * (((x1 * x1) * 6.0d0) + (((x1 * 2.0d0) * t_3) * (-3.0d0))))))) + (3.0d0 * (x2 * (-2.0d0))))
    if (x1 <= (-5.6d+102)) then
        tmp = t_1
    else if (x1 <= (-2.05d+55)) then
        tmp = t_4
    else if (x1 <= 1.9d+51) then
        tmp = x1 + ((3.0d0 * ((x2 * (-2.0d0)) - x1)) + (x1 + (4.0d0 * (x2 * (x1 * ((2.0d0 * x2) - 3.0d0))))))
    else if (x1 <= 2d+153) then
        tmp = t_4
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * (x1 * 9.0)) - x1;
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + ((x1 * (x1 * x1)) + ((t_0 * t_3) + (t_2 * (((x1 * x1) * 6.0) + (((x1 * 2.0) * t_3) * -3.0)))))) + (3.0 * (x2 * -2.0)));
	double tmp;
	if (x1 <= -5.6e+102) {
		tmp = t_1;
	} else if (x1 <= -2.05e+55) {
		tmp = t_4;
	} else if (x1 <= 1.9e+51) {
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	} else if (x1 <= 2e+153) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = x1 * (x1 * 3.0)
	t_1 = (x1 * (x1 * 9.0)) - x1
	t_2 = (x1 * x1) + 1.0
	t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2
	t_4 = x1 + ((x1 + ((x1 * (x1 * x1)) + ((t_0 * t_3) + (t_2 * (((x1 * x1) * 6.0) + (((x1 * 2.0) * t_3) * -3.0)))))) + (3.0 * (x2 * -2.0)))
	tmp = 0
	if x1 <= -5.6e+102:
		tmp = t_1
	elif x1 <= -2.05e+55:
		tmp = t_4
	elif x1 <= 1.9e+51:
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))))
	elif x1 <= 2e+153:
		tmp = t_4
	else:
		tmp = t_1
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * Float64(x1 * 9.0)) - 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(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(Float64(t_0 * t_3) + Float64(t_2 * Float64(Float64(Float64(x1 * x1) * 6.0) + Float64(Float64(Float64(x1 * 2.0) * t_3) * -3.0)))))) + Float64(3.0 * Float64(x2 * -2.0))))
	tmp = 0.0
	if (x1 <= -5.6e+102)
		tmp = t_1;
	elseif (x1 <= -2.05e+55)
		tmp = t_4;
	elseif (x1 <= 1.9e+51)
		tmp = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(x2 * -2.0) - x1)) + Float64(x1 + Float64(4.0 * Float64(x2 * Float64(x1 * Float64(Float64(2.0 * x2) - 3.0)))))));
	elseif (x1 <= 2e+153)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 * (x1 * 3.0);
	t_1 = (x1 * (x1 * 9.0)) - x1;
	t_2 = (x1 * x1) + 1.0;
	t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	t_4 = x1 + ((x1 + ((x1 * (x1 * x1)) + ((t_0 * t_3) + (t_2 * (((x1 * x1) * 6.0) + (((x1 * 2.0) * t_3) * -3.0)))))) + (3.0 * (x2 * -2.0)));
	tmp = 0.0;
	if (x1 <= -5.6e+102)
		tmp = t_1;
	elseif (x1 <= -2.05e+55)
		tmp = t_4;
	elseif (x1 <= 1.9e+51)
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	elseif (x1 <= 2e+153)
		tmp = t_4;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $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[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * t$95$3), $MachinePrecision] + N[(t$95$2 * N[(N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(x2 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -5.6e+102], t$95$1, If[LessEqual[x1, -2.05e+55], t$95$4, If[LessEqual[x1, 1.9e+51], N[(x1 + N[(N[(3.0 * N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(4.0 * N[(x2 * N[(x1 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2e+153], t$95$4, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -5.60000000000000037e102 or 2e153 < x1

   1. Initial program 1.5%

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

   2. Taylor expanded in x1 around 0 1.5%

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

   3. Taylor expanded in x2 around 0 0.1%

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

      1. fma-def 0.1%

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

      2. *-commutative 0.1%

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

      3. unpow2 0.1%

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

      4. associate-*r* 0.1%

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

      5. +-rgt-identity 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.1%

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

      10. *-commutative 0.1%

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

      11. *-commutative 0.1%

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

      12. associate-*r* 0.1%

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x1 around 0 83.7%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 83.7%

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

      2. neg-mul-1 83.7%

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

      3. unsub-neg 83.7%

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

      4. *-commutative 83.7%

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

      5. unpow2 83.7%

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

      6. associate-*l* 83.7%

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

   8. Simplified 83.7%

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

   if -5.60000000000000037e102 < x1 < -2.04999999999999991e55 or 1.8999999999999999e51 < x1 < 2e153

   1. Initial program 96.4%

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

   2. Taylor expanded in x2 around 0 92.9%

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

      1. +-commutative 92.9%

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

      2. associate--r+ 92.9%

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

      3. associate-*r/ 92.9%

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

      4. div-sub 92.9%

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

      5. *-commutative 92.9%

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

      6. unpow2 92.9%

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

      7. associate-*r* 92.9%

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

      8. fma-neg 92.9%

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

      9. +-commutative 92.9%

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

      10. unpow2 92.9%

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

      11. fma-udef 92.9%

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

   4. Simplified 92.9%

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

   5. Taylor expanded in x1 around 0 92.9%

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

   6. Taylor expanded in x1 around 0 92.9%

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

      1. *-commutative 23.3%

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

   8. Simplified 92.9%

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

   9. Taylor expanded in x1 around inf 89.3%

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

       1. *-commutative 89.3%

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

       2. unpow2 89.3%

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

   11. Simplified 89.3%

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

   if -2.04999999999999991e55 < x1 < 1.8999999999999999e51

   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. Taylor expanded in x1 around 0 92.9%

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

   3. Taylor expanded in x1 around 0 92.9%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{elif}\;x1 \leq -2.05 \cdot 10^{+55}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot 6 + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot -3\right)\right)\right)\right) + 3 \cdot \left(x2 \cdot -2\right)\right)\\ \mathbf{elif}\;x1 \leq 1.9 \cdot 10^{+51}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot 6 + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot -3\right)\right)\right)\right) + 3 \cdot \left(x2 \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \end{array} \]

Alternative 9: 90.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 9\right) - x1\\ t_2 := x1 \cdot \left(x1 \cdot 3\right)\\ t_3 := 3 \cdot \left(x2 \cdot -2 - x1\right)\\ t_4 := \frac{\left(t_2 + 2 \cdot x2\right) - x1}{t_0}\\ t_5 := x1 + \left(t_3 + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(t_2 \cdot t_4 + t_0 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(t_4 \cdot 4 - 6\right) + -6\right)\right)\right)\right)\right)\\ \mathbf{if}\;x1 \leq -3.5 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x1 \leq -1.05 \cdot 10^{+32}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x1 \leq 6000000:\\ \;\;\;\;x1 + \left(t_3 + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (- (* x1 (* x1 9.0)) x1))
        (t_2 (* x1 (* x1 3.0)))
        (t_3 (* 3.0 (- (* x2 -2.0) x1)))
        (t_4 (/ (- (+ t_2 (* 2.0 x2)) x1) t_0))
        (t_5
         (+
          x1
          (+
           t_3
           (+
            x1
            (+
             (* x1 (* x1 x1))
             (+
              (* t_2 t_4)
              (* t_0 (+ (* (* x1 x1) (- (* t_4 4.0) 6.0)) -6.0)))))))))
   (if (<= x1 -3.5e+106)
     t_1
     (if (<= x1 -1.05e+32)
       t_5
       (if (<= x1 6000000.0)
         (+ x1 (+ t_3 (+ x1 (* 4.0 (* x2 (* x1 (- (* 2.0 x2) 3.0)))))))
         (if (<= x1 2e+153) t_5 t_1))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = (x1 * (x1 * 9.0)) - x1;
	double t_2 = x1 * (x1 * 3.0);
	double t_3 = 3.0 * ((x2 * -2.0) - x1);
	double t_4 = ((t_2 + (2.0 * x2)) - x1) / t_0;
	double t_5 = x1 + (t_3 + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_4) + (t_0 * (((x1 * x1) * ((t_4 * 4.0) - 6.0)) + -6.0))))));
	double tmp;
	if (x1 <= -3.5e+106) {
		tmp = t_1;
	} else if (x1 <= -1.05e+32) {
		tmp = t_5;
	} else if (x1 <= 6000000.0) {
		tmp = x1 + (t_3 + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	} else if (x1 <= 2e+153) {
		tmp = t_5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (x1 * x1) + 1.0d0
    t_1 = (x1 * (x1 * 9.0d0)) - x1
    t_2 = x1 * (x1 * 3.0d0)
    t_3 = 3.0d0 * ((x2 * (-2.0d0)) - x1)
    t_4 = ((t_2 + (2.0d0 * x2)) - x1) / t_0
    t_5 = x1 + (t_3 + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_4) + (t_0 * (((x1 * x1) * ((t_4 * 4.0d0) - 6.0d0)) + (-6.0d0)))))))
    if (x1 <= (-3.5d+106)) then
        tmp = t_1
    else if (x1 <= (-1.05d+32)) then
        tmp = t_5
    else if (x1 <= 6000000.0d0) then
        tmp = x1 + (t_3 + (x1 + (4.0d0 * (x2 * (x1 * ((2.0d0 * x2) - 3.0d0))))))
    else if (x1 <= 2d+153) then
        tmp = t_5
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = (x1 * (x1 * 9.0)) - x1;
	double t_2 = x1 * (x1 * 3.0);
	double t_3 = 3.0 * ((x2 * -2.0) - x1);
	double t_4 = ((t_2 + (2.0 * x2)) - x1) / t_0;
	double t_5 = x1 + (t_3 + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_4) + (t_0 * (((x1 * x1) * ((t_4 * 4.0) - 6.0)) + -6.0))))));
	double tmp;
	if (x1 <= -3.5e+106) {
		tmp = t_1;
	} else if (x1 <= -1.05e+32) {
		tmp = t_5;
	} else if (x1 <= 6000000.0) {
		tmp = x1 + (t_3 + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	} else if (x1 <= 2e+153) {
		tmp = t_5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * x1) + 1.0
	t_1 = (x1 * (x1 * 9.0)) - x1
	t_2 = x1 * (x1 * 3.0)
	t_3 = 3.0 * ((x2 * -2.0) - x1)
	t_4 = ((t_2 + (2.0 * x2)) - x1) / t_0
	t_5 = x1 + (t_3 + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_4) + (t_0 * (((x1 * x1) * ((t_4 * 4.0) - 6.0)) + -6.0))))))
	tmp = 0
	if x1 <= -3.5e+106:
		tmp = t_1
	elif x1 <= -1.05e+32:
		tmp = t_5
	elif x1 <= 6000000.0:
		tmp = x1 + (t_3 + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))))
	elif x1 <= 2e+153:
		tmp = t_5
	else:
		tmp = t_1
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1)
	t_2 = Float64(x1 * Float64(x1 * 3.0))
	t_3 = Float64(3.0 * Float64(Float64(x2 * -2.0) - x1))
	t_4 = Float64(Float64(Float64(t_2 + Float64(2.0 * x2)) - x1) / t_0)
	t_5 = Float64(x1 + Float64(t_3 + Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(Float64(t_2 * t_4) + Float64(t_0 * Float64(Float64(Float64(x1 * x1) * Float64(Float64(t_4 * 4.0) - 6.0)) + -6.0)))))))
	tmp = 0.0
	if (x1 <= -3.5e+106)
		tmp = t_1;
	elseif (x1 <= -1.05e+32)
		tmp = t_5;
	elseif (x1 <= 6000000.0)
		tmp = Float64(x1 + Float64(t_3 + Float64(x1 + Float64(4.0 * Float64(x2 * Float64(x1 * Float64(Float64(2.0 * x2) - 3.0)))))));
	elseif (x1 <= 2e+153)
		tmp = t_5;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * x1) + 1.0;
	t_1 = (x1 * (x1 * 9.0)) - x1;
	t_2 = x1 * (x1 * 3.0);
	t_3 = 3.0 * ((x2 * -2.0) - x1);
	t_4 = ((t_2 + (2.0 * x2)) - x1) / t_0;
	t_5 = x1 + (t_3 + (x1 + ((x1 * (x1 * x1)) + ((t_2 * t_4) + (t_0 * (((x1 * x1) * ((t_4 * 4.0) - 6.0)) + -6.0))))));
	tmp = 0.0;
	if (x1 <= -3.5e+106)
		tmp = t_1;
	elseif (x1 <= -1.05e+32)
		tmp = t_5;
	elseif (x1 <= 6000000.0)
		tmp = x1 + (t_3 + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	elseif (x1 <= 2e+153)
		tmp = t_5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$2 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 * N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(x1 + N[(t$95$3 + N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * t$95$4), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$4 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -3.5e+106], t$95$1, If[LessEqual[x1, -1.05e+32], t$95$5, If[LessEqual[x1, 6000000.0], N[(x1 + N[(t$95$3 + N[(x1 + N[(4.0 * N[(x2 * N[(x1 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2e+153], t$95$5, t$95$1]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.49999999999999981e106 or 2e153 < x1

   1. Initial program 0.0%

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

   2. Taylor expanded in x1 around 0 0.0%

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

   3. Taylor expanded in x2 around 0 0.1%

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

      1. fma-def 0.1%

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

      2. *-commutative 0.1%

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

      3. unpow2 0.1%

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

      4. associate-*r* 0.1%

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

      5. +-rgt-identity 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.1%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.1%

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

      10. *-commutative 0.1%

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

      11. *-commutative 0.1%

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

      12. associate-*r* 0.1%

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x1 around 0 85.0%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 85.0%

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

      2. neg-mul-1 85.0%

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

      3. unsub-neg 85.0%

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

      4. *-commutative 85.0%

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

      5. unpow2 85.0%

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

      6. associate-*l* 85.0%

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

   8. Simplified 85.0%

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

   if -3.49999999999999981e106 < x1 < -1.05e32 or 6e6 < x1 < 2e153

   1. Initial program 97.4%

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

   2. Taylor expanded in x1 around 0 97.4%

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

   3. Taylor expanded in x1 around inf 97.4%

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

      1. unpow2 97.4%

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

      2. associate-*r/ 97.4%

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

      3. metadata-eval 97.4%

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

      4. unpow2 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x1 around inf 85.6%

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

   if -1.05e32 < x1 < 6e6

   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. Taylor expanded in x1 around 0 96.8%

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

   3. Taylor expanded in x1 around 0 96.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.5 \cdot 10^{+106}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{elif}\;x1 \leq -1.05 \cdot 10^{+32}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right) + -6\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 6000000:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right) + -6\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \end{array} \]

Alternative 10: 79.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -3 \cdot 10^{+107} \lor \neg \left(x1 \leq 2.3 \cdot 10^{+148}\right):\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -3e+107) (not (<= x1 2.3e+148)))
   (- (* x1 (* x1 9.0)) x1)
   (+
    x1
    (+
     (* 3.0 (/ (- (- (* x1 (* x1 3.0)) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))
     (+ x1 (* 4.0 (* x2 (* x1 (- (* 2.0 x2) 3.0)))))))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -3e+107) || !(x1 <= 2.3e+148)) {
		tmp = (x1 * (x1 * 9.0)) - x1;
	} else {
		tmp = x1 + ((3.0 * ((((x1 * (x1 * 3.0)) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((x1 <= (-3d+107)) .or. (.not. (x1 <= 2.3d+148))) then
        tmp = (x1 * (x1 * 9.0d0)) - x1
    else
        tmp = x1 + ((3.0d0 * ((((x1 * (x1 * 3.0d0)) - (2.0d0 * x2)) - x1) / ((x1 * x1) + 1.0d0))) + (x1 + (4.0d0 * (x2 * (x1 * ((2.0d0 * x2) - 3.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -3e+107) || !(x1 <= 2.3e+148)) {
		tmp = (x1 * (x1 * 9.0)) - x1;
	} else {
		tmp = x1 + ((3.0 * ((((x1 * (x1 * 3.0)) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x1 <= -3e+107) or not (x1 <= 2.3e+148):
		tmp = (x1 * (x1 * 9.0)) - x1
	else:
		tmp = x1 + ((3.0 * ((((x1 * (x1 * 3.0)) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))))
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -3e+107) || !(x1 <= 2.3e+148))
		tmp = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1);
	else
		tmp = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(Float64(Float64(x1 * Float64(x1 * 3.0)) - Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0))) + Float64(x1 + Float64(4.0 * Float64(x2 * Float64(x1 * Float64(Float64(2.0 * x2) - 3.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x1 <= -3e+107) || ~((x1 <= 2.3e+148)))
		tmp = (x1 * (x1 * 9.0)) - x1;
	else
		tmp = x1 + ((3.0 * ((((x1 * (x1 * 3.0)) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -3e+107], N[Not[LessEqual[x1, 2.3e+148]], $MachinePrecision]], N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision], N[(x1 + N[(N[(3.0 * N[(N[(N[(N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(4.0 * N[(x2 * N[(x1 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.00000000000000023e107 or 2.3000000000000001e148 < x1

   1. Initial program 2.9%

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

   2. Taylor expanded in x1 around 0 0.2%

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

   3. Taylor expanded in x2 around 0 0.2%

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

      1. fma-def 0.2%

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

      2. *-commutative 0.2%

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

      3. unpow2 0.2%

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

      4. associate-*r* 0.2%

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

      5. +-rgt-identity 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.2%

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

      10. *-commutative 0.2%

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

      11. *-commutative 0.2%

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

      12. associate-*r* 0.2%

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

   5. Simplified 0.2%

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

   6. Taylor expanded in x1 around 0 82.8%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 82.8%

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

      2. neg-mul-1 82.8%

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

      3. unsub-neg 82.8%

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

      4. *-commutative 82.8%

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

      5. unpow2 82.8%

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

      6. associate-*l* 82.8%

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

   8. Simplified 82.8%

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

   if -3.00000000000000023e107 < x1 < 2.3000000000000001e148

   1. Initial program 98.9%

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

   2. Taylor expanded in x1 around 0 83.1%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3 \cdot 10^{+107} \lor \neg \left(x1 \leq 2.3 \cdot 10^{+148}\right):\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 11: 74.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 9\right) - x1\\ t_1 := x1 + \left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + x2 \cdot -6\right)\\ \mathbf{if}\;x1 \leq -3.2 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x1 \leq -1.86 \cdot 10^{-202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x1 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (* x1 (* x1 9.0)) x1))
        (t_1
         (+
          x1
          (+ (* x1 (- (* 4.0 (* x2 (- (* 2.0 x2) 3.0))) 2.0)) (* x2 -6.0)))))
   (if (<= x1 -3.2e+107)
     t_0
     (if (<= x1 -1.86e-202)
       t_1
       (if (<= x1 -5e-309)
         (- (* x2 -6.0) x1)
         (if (<= x1 2.3e+148) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * (x1 * 9.0)) - x1;
	double t_1 = x1 + ((x1 * ((4.0 * (x2 * ((2.0 * x2) - 3.0))) - 2.0)) + (x2 * -6.0));
	double tmp;
	if (x1 <= -3.2e+107) {
		tmp = t_0;
	} else if (x1 <= -1.86e-202) {
		tmp = t_1;
	} else if (x1 <= -5e-309) {
		tmp = (x2 * -6.0) - x1;
	} else if (x1 <= 2.3e+148) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x1 * (x1 * 9.0d0)) - x1
    t_1 = x1 + ((x1 * ((4.0d0 * (x2 * ((2.0d0 * x2) - 3.0d0))) - 2.0d0)) + (x2 * (-6.0d0)))
    if (x1 <= (-3.2d+107)) then
        tmp = t_0
    else if (x1 <= (-1.86d-202)) then
        tmp = t_1
    else if (x1 <= (-5d-309)) then
        tmp = (x2 * (-6.0d0)) - x1
    else if (x1 <= 2.3d+148) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * (x1 * 9.0)) - x1;
	double t_1 = x1 + ((x1 * ((4.0 * (x2 * ((2.0 * x2) - 3.0))) - 2.0)) + (x2 * -6.0));
	double tmp;
	if (x1 <= -3.2e+107) {
		tmp = t_0;
	} else if (x1 <= -1.86e-202) {
		tmp = t_1;
	} else if (x1 <= -5e-309) {
		tmp = (x2 * -6.0) - x1;
	} else if (x1 <= 2.3e+148) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * (x1 * 9.0)) - x1
	t_1 = x1 + ((x1 * ((4.0 * (x2 * ((2.0 * x2) - 3.0))) - 2.0)) + (x2 * -6.0))
	tmp = 0
	if x1 <= -3.2e+107:
		tmp = t_0
	elif x1 <= -1.86e-202:
		tmp = t_1
	elif x1 <= -5e-309:
		tmp = (x2 * -6.0) - x1
	elif x1 <= 2.3e+148:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1)
	t_1 = Float64(x1 + Float64(Float64(x1 * Float64(Float64(4.0 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0))) - 2.0)) + Float64(x2 * -6.0)))
	tmp = 0.0
	if (x1 <= -3.2e+107)
		tmp = t_0;
	elseif (x1 <= -1.86e-202)
		tmp = t_1;
	elseif (x1 <= -5e-309)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	elseif (x1 <= 2.3e+148)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * (x1 * 9.0)) - x1;
	t_1 = x1 + ((x1 * ((4.0 * (x2 * ((2.0 * x2) - 3.0))) - 2.0)) + (x2 * -6.0));
	tmp = 0.0;
	if (x1 <= -3.2e+107)
		tmp = t_0;
	elseif (x1 <= -1.86e-202)
		tmp = t_1;
	elseif (x1 <= -5e-309)
		tmp = (x2 * -6.0) - x1;
	elseif (x1 <= 2.3e+148)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$1 = N[(x1 + N[(N[(x1 * N[(N[(4.0 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -3.2e+107], t$95$0, If[LessEqual[x1, -1.86e-202], t$95$1, If[LessEqual[x1, -5e-309], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], If[LessEqual[x1, 2.3e+148], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.20000000000000029e107 or 2.3000000000000001e148 < x1

   1. Initial program 2.9%

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

   2. Taylor expanded in x1 around 0 0.2%

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

   3. Taylor expanded in x2 around 0 0.2%

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

      1. fma-def 0.2%

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

      2. *-commutative 0.2%

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

      3. unpow2 0.2%

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

      4. associate-*r* 0.2%

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

      5. +-rgt-identity 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.2%

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

      10. *-commutative 0.2%

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

      11. *-commutative 0.2%

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

      12. associate-*r* 0.2%

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

   5. Simplified 0.2%

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

   6. Taylor expanded in x1 around 0 82.8%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 82.8%

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

      2. neg-mul-1 82.8%

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

      3. unsub-neg 82.8%

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

      4. *-commutative 82.8%

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

      5. unpow2 82.8%

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

      6. associate-*l* 82.8%

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

   8. Simplified 82.8%

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

   if -3.20000000000000029e107 < x1 < -1.85999999999999998e-202 or -4.9999999999999995e-309 < x1 < 2.3000000000000001e148

   1. Initial program 98.7%

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

   2. Taylor expanded in x1 around 0 80.5%

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

   3. Taylor expanded in x1 around 0 75.6%

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

   if -1.85999999999999998e-202 < x1 < -4.9999999999999995e-309

   1. Initial program 99.7%

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

   2. Taylor expanded in x1 around 0 99.7%

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

   3. Taylor expanded in x1 around inf 96.2%

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

   4. Taylor expanded in x1 around 0 96.3%

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

      1. +-commutative 96.3%

         \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]

      2. fma-def 96.2%

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

      3. neg-mul-1 96.2%

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

      4. fma-neg 96.3%

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

   6. Simplified 96.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.2 \cdot 10^{+107}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{elif}\;x1 \leq -1.86 \cdot 10^{-202}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + x2 \cdot -6\right)\\ \mathbf{elif}\;x1 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \end{array} \]

Alternative 12: 74.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot x2 - 3\\ t_1 := x1 \cdot \left(x1 \cdot 9\right) - x1\\ t_2 := x1 + \left(x1 \cdot \left(4 \cdot \left(x2 \cdot t_0\right) - 2\right) + x2 \cdot -6\right)\\ \mathbf{if}\;x1 \leq -2.6 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x1 \leq -4.2 \cdot 10^{-208}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x1 \leq 3 \cdot 10^{-159}:\\ \;\;\;\;x1 + \left(\left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot t_0\right)\right)\right) + 3 \cdot \left(x2 \cdot -2\right)\right)\\ \mathbf{elif}\;x1 \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (* 2.0 x2) 3.0))
        (t_1 (- (* x1 (* x1 9.0)) x1))
        (t_2 (+ x1 (+ (* x1 (- (* 4.0 (* x2 t_0)) 2.0)) (* x2 -6.0)))))
   (if (<= x1 -2.6e+107)
     t_1
     (if (<= x1 -4.2e-208)
       t_2
       (if (<= x1 3e-159)
         (+ x1 (+ (+ x1 (* 4.0 (* x2 (* x1 t_0)))) (* 3.0 (* x2 -2.0))))
         (if (<= x1 2.3e+148) t_2 t_1))))))

C

double code(double x1, double x2) {
	double t_0 = (2.0 * x2) - 3.0;
	double t_1 = (x1 * (x1 * 9.0)) - x1;
	double t_2 = x1 + ((x1 * ((4.0 * (x2 * t_0)) - 2.0)) + (x2 * -6.0));
	double tmp;
	if (x1 <= -2.6e+107) {
		tmp = t_1;
	} else if (x1 <= -4.2e-208) {
		tmp = t_2;
	} else if (x1 <= 3e-159) {
		tmp = x1 + ((x1 + (4.0 * (x2 * (x1 * t_0)))) + (3.0 * (x2 * -2.0)));
	} else if (x1 <= 2.3e+148) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (2.0d0 * x2) - 3.0d0
    t_1 = (x1 * (x1 * 9.0d0)) - x1
    t_2 = x1 + ((x1 * ((4.0d0 * (x2 * t_0)) - 2.0d0)) + (x2 * (-6.0d0)))
    if (x1 <= (-2.6d+107)) then
        tmp = t_1
    else if (x1 <= (-4.2d-208)) then
        tmp = t_2
    else if (x1 <= 3d-159) then
        tmp = x1 + ((x1 + (4.0d0 * (x2 * (x1 * t_0)))) + (3.0d0 * (x2 * (-2.0d0))))
    else if (x1 <= 2.3d+148) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (2.0 * x2) - 3.0;
	double t_1 = (x1 * (x1 * 9.0)) - x1;
	double t_2 = x1 + ((x1 * ((4.0 * (x2 * t_0)) - 2.0)) + (x2 * -6.0));
	double tmp;
	if (x1 <= -2.6e+107) {
		tmp = t_1;
	} else if (x1 <= -4.2e-208) {
		tmp = t_2;
	} else if (x1 <= 3e-159) {
		tmp = x1 + ((x1 + (4.0 * (x2 * (x1 * t_0)))) + (3.0 * (x2 * -2.0)));
	} else if (x1 <= 2.3e+148) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (2.0 * x2) - 3.0
	t_1 = (x1 * (x1 * 9.0)) - x1
	t_2 = x1 + ((x1 * ((4.0 * (x2 * t_0)) - 2.0)) + (x2 * -6.0))
	tmp = 0
	if x1 <= -2.6e+107:
		tmp = t_1
	elif x1 <= -4.2e-208:
		tmp = t_2
	elif x1 <= 3e-159:
		tmp = x1 + ((x1 + (4.0 * (x2 * (x1 * t_0)))) + (3.0 * (x2 * -2.0)))
	elif x1 <= 2.3e+148:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(2.0 * x2) - 3.0)
	t_1 = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1)
	t_2 = Float64(x1 + Float64(Float64(x1 * Float64(Float64(4.0 * Float64(x2 * t_0)) - 2.0)) + Float64(x2 * -6.0)))
	tmp = 0.0
	if (x1 <= -2.6e+107)
		tmp = t_1;
	elseif (x1 <= -4.2e-208)
		tmp = t_2;
	elseif (x1 <= 3e-159)
		tmp = Float64(x1 + Float64(Float64(x1 + Float64(4.0 * Float64(x2 * Float64(x1 * t_0)))) + Float64(3.0 * Float64(x2 * -2.0))));
	elseif (x1 <= 2.3e+148)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (2.0 * x2) - 3.0;
	t_1 = (x1 * (x1 * 9.0)) - x1;
	t_2 = x1 + ((x1 * ((4.0 * (x2 * t_0)) - 2.0)) + (x2 * -6.0));
	tmp = 0.0;
	if (x1 <= -2.6e+107)
		tmp = t_1;
	elseif (x1 <= -4.2e-208)
		tmp = t_2;
	elseif (x1 <= 3e-159)
		tmp = x1 + ((x1 + (4.0 * (x2 * (x1 * t_0)))) + (3.0 * (x2 * -2.0)));
	elseif (x1 <= 2.3e+148)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$2 = N[(x1 + N[(N[(x1 * N[(N[(4.0 * N[(x2 * t$95$0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -2.6e+107], t$95$1, If[LessEqual[x1, -4.2e-208], t$95$2, If[LessEqual[x1, 3e-159], N[(x1 + N[(N[(x1 + N[(4.0 * N[(x2 * N[(x1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(x2 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.3e+148], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.6000000000000001e107 or 2.3000000000000001e148 < x1

   1. Initial program 2.9%

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

   2. Taylor expanded in x1 around 0 0.2%

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

   3. Taylor expanded in x2 around 0 0.2%

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

      1. fma-def 0.2%

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

      2. *-commutative 0.2%

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

      3. unpow2 0.2%

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

      4. associate-*r* 0.2%

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

      5. +-rgt-identity 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.2%

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

      10. *-commutative 0.2%

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

      11. *-commutative 0.2%

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

      12. associate-*r* 0.2%

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

   5. Simplified 0.2%

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

   6. Taylor expanded in x1 around 0 82.8%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 82.8%

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

      2. neg-mul-1 82.8%

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

      3. unsub-neg 82.8%

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

      4. *-commutative 82.8%

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

      5. unpow2 82.8%

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

      6. associate-*l* 82.8%

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

   8. Simplified 82.8%

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

   if -2.6000000000000001e107 < x1 < -4.20000000000000024e-208 or 3.00000000000000009e-159 < x1 < 2.3000000000000001e148

   1. Initial program 98.5%

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

   2. Taylor expanded in x1 around 0 75.5%

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

   3. Taylor expanded in x1 around 0 72.9%

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

   if -4.20000000000000024e-208 < x1 < 3.00000000000000009e-159

   1. Initial program 99.6%

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

   2. Taylor expanded in x1 around 0 99.6%

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

   3. Taylor expanded in x1 around 0 92.8%

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

      1. *-commutative 92.8%

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

   5. Simplified 92.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.6 \cdot 10^{+107}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{elif}\;x1 \leq -4.2 \cdot 10^{-208}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + x2 \cdot -6\right)\\ \mathbf{elif}\;x1 \leq 3 \cdot 10^{-159}:\\ \;\;\;\;x1 + \left(\left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right) + 3 \cdot \left(x2 \cdot -2\right)\right)\\ \mathbf{elif}\;x1 \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \end{array} \]

Alternative 13: 79.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -3.9 \cdot 10^{+108} \lor \neg \left(x1 \leq 2.3 \cdot 10^{+148}\right):\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -3.9e+108) (not (<= x1 2.3e+148)))
   (- (* x1 (* x1 9.0)) x1)
   (+
    x1
    (+
     (* 3.0 (- (* x2 -2.0) x1))
     (+ x1 (* 4.0 (* x2 (* x1 (- (* 2.0 x2) 3.0)))))))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -3.9e+108) || !(x1 <= 2.3e+148)) {
		tmp = (x1 * (x1 * 9.0)) - x1;
	} else {
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((x1 <= (-3.9d+108)) .or. (.not. (x1 <= 2.3d+148))) then
        tmp = (x1 * (x1 * 9.0d0)) - x1
    else
        tmp = x1 + ((3.0d0 * ((x2 * (-2.0d0)) - x1)) + (x1 + (4.0d0 * (x2 * (x1 * ((2.0d0 * x2) - 3.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -3.9e+108) || !(x1 <= 2.3e+148)) {
		tmp = (x1 * (x1 * 9.0)) - x1;
	} else {
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x1 <= -3.9e+108) or not (x1 <= 2.3e+148):
		tmp = (x1 * (x1 * 9.0)) - x1
	else:
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))))
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -3.9e+108) || !(x1 <= 2.3e+148))
		tmp = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1);
	else
		tmp = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(x2 * -2.0) - x1)) + Float64(x1 + Float64(4.0 * Float64(x2 * Float64(x1 * Float64(Float64(2.0 * x2) - 3.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x1 <= -3.9e+108) || ~((x1 <= 2.3e+148)))
		tmp = (x1 * (x1 * 9.0)) - x1;
	else
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + (4.0 * (x2 * (x1 * ((2.0 * x2) - 3.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -3.9e+108], N[Not[LessEqual[x1, 2.3e+148]], $MachinePrecision]], N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision], N[(x1 + N[(N[(3.0 * N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(4.0 * N[(x2 * N[(x1 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.89999999999999985e108 or 2.3000000000000001e148 < x1

   1. Initial program 2.9%

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

   2. Taylor expanded in x1 around 0 0.2%

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

   3. Taylor expanded in x2 around 0 0.2%

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

      1. fma-def 0.2%

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

      2. *-commutative 0.2%

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

      3. unpow2 0.2%

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

      4. associate-*r* 0.2%

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

      5. +-rgt-identity 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.2%

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

      10. *-commutative 0.2%

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

      11. *-commutative 0.2%

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

      12. associate-*r* 0.2%

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

   5. Simplified 0.2%

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

   6. Taylor expanded in x1 around 0 82.8%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 82.8%

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

      2. neg-mul-1 82.8%

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

      3. unsub-neg 82.8%

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

      4. *-commutative 82.8%

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

      5. unpow2 82.8%

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

      6. associate-*l* 82.8%

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

   8. Simplified 82.8%

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

   if -3.89999999999999985e108 < x1 < 2.3000000000000001e148

   1. Initial program 98.9%

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

   2. Taylor expanded in x1 around 0 83.1%

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

   3. Taylor expanded in x1 around 0 83.1%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.9 \cdot 10^{+108} \lor \neg \left(x1 \leq 2.3 \cdot 10^{+148}\right):\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + 4 \cdot \left(x2 \cdot \left(x1 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 14: 66.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{if}\;x1 \leq -1.06 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x1 \leq -1.85 \cdot 10^{-37}:\\ \;\;\;\;x1 + \left(x2 \cdot x2\right) \cdot \left(x1 \cdot 8\right)\\ \mathbf{elif}\;x1 \leq 0.7:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x1 + x1 \cdot \left(1 + 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (* x1 (* x1 9.0)) x1)))
   (if (<= x1 -1.06e+108)
     t_0
     (if (<= x1 -1.85e-37)
       (+ x1 (* (* x2 x2) (* x1 8.0)))
       (if (<= x1 0.7)
         (- (* x2 -6.0) x1)
         (if (<= x1 2.3e+148)
           (+ x1 (* x1 (+ 1.0 (* 4.0 (* x2 (- (* 2.0 x2) 3.0))))))
           t_0))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * (x1 * 9.0)) - x1;
	double tmp;
	if (x1 <= -1.06e+108) {
		tmp = t_0;
	} else if (x1 <= -1.85e-37) {
		tmp = x1 + ((x2 * x2) * (x1 * 8.0));
	} else if (x1 <= 0.7) {
		tmp = (x2 * -6.0) - x1;
	} else if (x1 <= 2.3e+148) {
		tmp = x1 + (x1 * (1.0 + (4.0 * (x2 * ((2.0 * x2) - 3.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x1 * (x1 * 9.0d0)) - x1
    if (x1 <= (-1.06d+108)) then
        tmp = t_0
    else if (x1 <= (-1.85d-37)) then
        tmp = x1 + ((x2 * x2) * (x1 * 8.0d0))
    else if (x1 <= 0.7d0) then
        tmp = (x2 * (-6.0d0)) - x1
    else if (x1 <= 2.3d+148) then
        tmp = x1 + (x1 * (1.0d0 + (4.0d0 * (x2 * ((2.0d0 * x2) - 3.0d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * (x1 * 9.0)) - x1;
	double tmp;
	if (x1 <= -1.06e+108) {
		tmp = t_0;
	} else if (x1 <= -1.85e-37) {
		tmp = x1 + ((x2 * x2) * (x1 * 8.0));
	} else if (x1 <= 0.7) {
		tmp = (x2 * -6.0) - x1;
	} else if (x1 <= 2.3e+148) {
		tmp = x1 + (x1 * (1.0 + (4.0 * (x2 * ((2.0 * x2) - 3.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * (x1 * 9.0)) - x1
	tmp = 0
	if x1 <= -1.06e+108:
		tmp = t_0
	elif x1 <= -1.85e-37:
		tmp = x1 + ((x2 * x2) * (x1 * 8.0))
	elif x1 <= 0.7:
		tmp = (x2 * -6.0) - x1
	elif x1 <= 2.3e+148:
		tmp = x1 + (x1 * (1.0 + (4.0 * (x2 * ((2.0 * x2) - 3.0)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1)
	tmp = 0.0
	if (x1 <= -1.06e+108)
		tmp = t_0;
	elseif (x1 <= -1.85e-37)
		tmp = Float64(x1 + Float64(Float64(x2 * x2) * Float64(x1 * 8.0)));
	elseif (x1 <= 0.7)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	elseif (x1 <= 2.3e+148)
		tmp = Float64(x1 + Float64(x1 * Float64(1.0 + Float64(4.0 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * (x1 * 9.0)) - x1;
	tmp = 0.0;
	if (x1 <= -1.06e+108)
		tmp = t_0;
	elseif (x1 <= -1.85e-37)
		tmp = x1 + ((x2 * x2) * (x1 * 8.0));
	elseif (x1 <= 0.7)
		tmp = (x2 * -6.0) - x1;
	elseif (x1 <= 2.3e+148)
		tmp = x1 + (x1 * (1.0 + (4.0 * (x2 * ((2.0 * x2) - 3.0)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, If[LessEqual[x1, -1.06e+108], t$95$0, If[LessEqual[x1, -1.85e-37], N[(x1 + N[(N[(x2 * x2), $MachinePrecision] * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 0.7], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], If[LessEqual[x1, 2.3e+148], N[(x1 + N[(x1 * N[(1.0 + N[(4.0 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.06e108 or 2.3000000000000001e148 < x1

   1. Initial program 2.9%

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

   2. Taylor expanded in x1 around 0 0.2%

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

   3. Taylor expanded in x2 around 0 0.2%

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

      1. fma-def 0.2%

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

      2. *-commutative 0.2%

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

      3. unpow2 0.2%

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

      4. associate-*r* 0.2%

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

      5. +-rgt-identity 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.2%

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

      10. *-commutative 0.2%

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

      11. *-commutative 0.2%

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

      12. associate-*r* 0.2%

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

   5. Simplified 0.2%

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

   6. Taylor expanded in x1 around 0 82.8%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 82.8%

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

      2. neg-mul-1 82.8%

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

      3. unsub-neg 82.8%

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

      4. *-commutative 82.8%

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

      5. unpow2 82.8%

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

      6. associate-*l* 82.8%

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

   8. Simplified 82.8%

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

   if -1.06e108 < x1 < -1.85e-37

   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. Taylor expanded in x1 around 0 60.1%

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

   3. Taylor expanded in x2 around inf 49.4%

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

      1. *-commutative 49.4%

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

      2. associate-*l* 49.4%

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

      3. unpow2 49.4%

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

   5. Simplified 49.4%

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

   if -1.85e-37 < x1 < 0.69999999999999996

   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. Taylor expanded in x1 around 0 98.7%

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

   3. Taylor expanded in x1 around inf 81.4%

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

   4. Taylor expanded in x1 around 0 82.0%

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

      1. +-commutative 82.0%

         \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]

      2. fma-def 82.1%

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

      3. neg-mul-1 82.1%

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

      4. fma-neg 82.0%

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

   6. Simplified 82.0%

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

   if 0.69999999999999996 < x1 < 2.3000000000000001e148

   1. Initial program 95.8%

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

   2. Taylor expanded in x1 around 0 32.0%

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

   3. Taylor expanded in x1 around inf 32.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.06 \cdot 10^{+108}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{elif}\;x1 \leq -1.85 \cdot 10^{-37}:\\ \;\;\;\;x1 + \left(x2 \cdot x2\right) \cdot \left(x1 \cdot 8\right)\\ \mathbf{elif}\;x1 \leq 0.7:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x1 + x1 \cdot \left(1 + 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \end{array} \]

Alternative 15: 66.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{if}\;x1 \leq -4 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x1 \leq -2.1 \cdot 10^{-37}:\\ \;\;\;\;\left(x1 + \left(9 + \left(x2 \cdot x2\right) \cdot \left(x1 \cdot 8\right)\right)\right) - \frac{3}{x1}\\ \mathbf{elif}\;x1 \leq 0.7:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x1 + x1 \cdot \left(1 + 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (* x1 (* x1 9.0)) x1)))
   (if (<= x1 -4e+107)
     t_0
     (if (<= x1 -2.1e-37)
       (- (+ x1 (+ 9.0 (* (* x2 x2) (* x1 8.0)))) (/ 3.0 x1))
       (if (<= x1 0.7)
         (- (* x2 -6.0) x1)
         (if (<= x1 2.3e+148)
           (+ x1 (* x1 (+ 1.0 (* 4.0 (* x2 (- (* 2.0 x2) 3.0))))))
           t_0))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * (x1 * 9.0)) - x1;
	double tmp;
	if (x1 <= -4e+107) {
		tmp = t_0;
	} else if (x1 <= -2.1e-37) {
		tmp = (x1 + (9.0 + ((x2 * x2) * (x1 * 8.0)))) - (3.0 / x1);
	} else if (x1 <= 0.7) {
		tmp = (x2 * -6.0) - x1;
	} else if (x1 <= 2.3e+148) {
		tmp = x1 + (x1 * (1.0 + (4.0 * (x2 * ((2.0 * x2) - 3.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x1 * (x1 * 9.0d0)) - x1
    if (x1 <= (-4d+107)) then
        tmp = t_0
    else if (x1 <= (-2.1d-37)) then
        tmp = (x1 + (9.0d0 + ((x2 * x2) * (x1 * 8.0d0)))) - (3.0d0 / x1)
    else if (x1 <= 0.7d0) then
        tmp = (x2 * (-6.0d0)) - x1
    else if (x1 <= 2.3d+148) then
        tmp = x1 + (x1 * (1.0d0 + (4.0d0 * (x2 * ((2.0d0 * x2) - 3.0d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * (x1 * 9.0)) - x1;
	double tmp;
	if (x1 <= -4e+107) {
		tmp = t_0;
	} else if (x1 <= -2.1e-37) {
		tmp = (x1 + (9.0 + ((x2 * x2) * (x1 * 8.0)))) - (3.0 / x1);
	} else if (x1 <= 0.7) {
		tmp = (x2 * -6.0) - x1;
	} else if (x1 <= 2.3e+148) {
		tmp = x1 + (x1 * (1.0 + (4.0 * (x2 * ((2.0 * x2) - 3.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * (x1 * 9.0)) - x1
	tmp = 0
	if x1 <= -4e+107:
		tmp = t_0
	elif x1 <= -2.1e-37:
		tmp = (x1 + (9.0 + ((x2 * x2) * (x1 * 8.0)))) - (3.0 / x1)
	elif x1 <= 0.7:
		tmp = (x2 * -6.0) - x1
	elif x1 <= 2.3e+148:
		tmp = x1 + (x1 * (1.0 + (4.0 * (x2 * ((2.0 * x2) - 3.0)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1)
	tmp = 0.0
	if (x1 <= -4e+107)
		tmp = t_0;
	elseif (x1 <= -2.1e-37)
		tmp = Float64(Float64(x1 + Float64(9.0 + Float64(Float64(x2 * x2) * Float64(x1 * 8.0)))) - Float64(3.0 / x1));
	elseif (x1 <= 0.7)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	elseif (x1 <= 2.3e+148)
		tmp = Float64(x1 + Float64(x1 * Float64(1.0 + Float64(4.0 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * (x1 * 9.0)) - x1;
	tmp = 0.0;
	if (x1 <= -4e+107)
		tmp = t_0;
	elseif (x1 <= -2.1e-37)
		tmp = (x1 + (9.0 + ((x2 * x2) * (x1 * 8.0)))) - (3.0 / x1);
	elseif (x1 <= 0.7)
		tmp = (x2 * -6.0) - x1;
	elseif (x1 <= 2.3e+148)
		tmp = x1 + (x1 * (1.0 + (4.0 * (x2 * ((2.0 * x2) - 3.0)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, If[LessEqual[x1, -4e+107], t$95$0, If[LessEqual[x1, -2.1e-37], N[(N[(x1 + N[(9.0 + N[(N[(x2 * x2), $MachinePrecision] * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 0.7], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], If[LessEqual[x1, 2.3e+148], N[(x1 + N[(x1 * N[(1.0 + N[(4.0 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -3.9999999999999999e107 or 2.3000000000000001e148 < x1

   1. Initial program 2.9%

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

   2. Taylor expanded in x1 around 0 0.2%

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

   3. Taylor expanded in x2 around 0 0.2%

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

      1. fma-def 0.2%

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

      2. *-commutative 0.2%

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

      3. unpow2 0.2%

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

      4. associate-*r* 0.2%

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

      5. +-rgt-identity 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.2%

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

      10. *-commutative 0.2%

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

      11. *-commutative 0.2%

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

      12. associate-*r* 0.2%

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

   5. Simplified 0.2%

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

   6. Taylor expanded in x1 around 0 82.8%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 82.8%

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

      2. neg-mul-1 82.8%

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

      3. unsub-neg 82.8%

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

      4. *-commutative 82.8%

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

      5. unpow2 82.8%

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

      6. associate-*l* 82.8%

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

   8. Simplified 82.8%

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

   if -3.9999999999999999e107 < x1 < -2.1000000000000001e-37

   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. Taylor expanded in x1 around 0 60.1%

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

   3. Taylor expanded in x1 around inf 50.4%

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

   4. Taylor expanded in x2 around inf 50.4%

      \[\leadsto x1 + \left(\left(9 + \color{blue}{8 \cdot \left({x2}^{2} \cdot x1\right)}\right) - 3 \cdot \frac{1}{x1}\right) \]
   5. Step-by-step derivation

      1. *-commutative 50.4%

         \[\leadsto x1 + \left(\left(9 + \color{blue}{\left({x2}^{2} \cdot x1\right) \cdot 8}\right) - 3 \cdot \frac{1}{x1}\right) \]

      2. associate-*l* 50.4%

         \[\leadsto x1 + \left(\left(9 + \color{blue}{{x2}^{2} \cdot \left(x1 \cdot 8\right)}\right) - 3 \cdot \frac{1}{x1}\right) \]

      3. unpow2 50.4%

         \[\leadsto x1 + \left(\left(9 + \color{blue}{\left(x2 \cdot x2\right)} \cdot \left(x1 \cdot 8\right)\right) - 3 \cdot \frac{1}{x1}\right) \]

   6. Simplified 50.4%

      \[\leadsto x1 + \left(\left(9 + \color{blue}{\left(x2 \cdot x2\right) \cdot \left(x1 \cdot 8\right)}\right) - 3 \cdot \frac{1}{x1}\right) \]
   7. Step-by-step derivation

      1. associate-+r- 50.4%

         \[\leadsto \color{blue}{\left(x1 + \left(9 + \left(x2 \cdot x2\right) \cdot \left(x1 \cdot 8\right)\right)\right) - 3 \cdot \frac{1}{x1}} \]

      2. +-commutative 50.4%

         \[\leadsto \left(x1 + \color{blue}{\left(\left(x2 \cdot x2\right) \cdot \left(x1 \cdot 8\right) + 9\right)}\right) - 3 \cdot \frac{1}{x1} \]

      3. un-div-inv 50.4%

         \[\leadsto \left(x1 + \left(\left(x2 \cdot x2\right) \cdot \left(x1 \cdot 8\right) + 9\right)\right) - \color{blue}{\frac{3}{x1}} \]

   8. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{\left(x1 + \left(\left(x2 \cdot x2\right) \cdot \left(x1 \cdot 8\right) + 9\right)\right) - \frac{3}{x1}} \]

   if -2.1000000000000001e-37 < x1 < 0.69999999999999996

   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. Taylor expanded in x1 around 0 98.7%

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

   3. Taylor expanded in x1 around inf 81.4%

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

   4. Taylor expanded in x1 around 0 82.0%

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

      1. +-commutative 82.0%

         \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]

      2. fma-def 82.1%

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

      3. neg-mul-1 82.1%

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

      4. fma-neg 82.0%

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

   6. Simplified 82.0%

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

   if 0.69999999999999996 < x1 < 2.3000000000000001e148

   1. Initial program 95.8%

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

   2. Taylor expanded in x1 around 0 32.0%

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

   3. Taylor expanded in x1 around inf 32.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4 \cdot 10^{+107}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{elif}\;x1 \leq -2.1 \cdot 10^{-37}:\\ \;\;\;\;\left(x1 + \left(9 + \left(x2 \cdot x2\right) \cdot \left(x1 \cdot 8\right)\right)\right) - \frac{3}{x1}\\ \mathbf{elif}\;x1 \leq 0.7:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x1 + x1 \cdot \left(1 + 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \end{array} \]

Alternative 16: 66.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 9\right) - x1\\ t_1 := x1 \cdot \left(1 + 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\\ \mathbf{if}\;x1 \leq -8.4 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x1 \leq -1.85 \cdot 10^{-37}:\\ \;\;\;\;x1 + \left(9 + t_1\right)\\ \mathbf{elif}\;x1 \leq 0.7:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (* x1 (* x1 9.0)) x1))
        (t_1 (* x1 (+ 1.0 (* 4.0 (* x2 (- (* 2.0 x2) 3.0)))))))
   (if (<= x1 -8.4e+108)
     t_0
     (if (<= x1 -1.85e-37)
       (+ x1 (+ 9.0 t_1))
       (if (<= x1 0.7)
         (- (* x2 -6.0) x1)
         (if (<= x1 2.3e+148) (+ x1 t_1) t_0))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * (x1 * 9.0)) - x1;
	double t_1 = x1 * (1.0 + (4.0 * (x2 * ((2.0 * x2) - 3.0))));
	double tmp;
	if (x1 <= -8.4e+108) {
		tmp = t_0;
	} else if (x1 <= -1.85e-37) {
		tmp = x1 + (9.0 + t_1);
	} else if (x1 <= 0.7) {
		tmp = (x2 * -6.0) - x1;
	} else if (x1 <= 2.3e+148) {
		tmp = x1 + t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x1 * (x1 * 9.0d0)) - x1
    t_1 = x1 * (1.0d0 + (4.0d0 * (x2 * ((2.0d0 * x2) - 3.0d0))))
    if (x1 <= (-8.4d+108)) then
        tmp = t_0
    else if (x1 <= (-1.85d-37)) then
        tmp = x1 + (9.0d0 + t_1)
    else if (x1 <= 0.7d0) then
        tmp = (x2 * (-6.0d0)) - x1
    else if (x1 <= 2.3d+148) then
        tmp = x1 + t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * (x1 * 9.0)) - x1;
	double t_1 = x1 * (1.0 + (4.0 * (x2 * ((2.0 * x2) - 3.0))));
	double tmp;
	if (x1 <= -8.4e+108) {
		tmp = t_0;
	} else if (x1 <= -1.85e-37) {
		tmp = x1 + (9.0 + t_1);
	} else if (x1 <= 0.7) {
		tmp = (x2 * -6.0) - x1;
	} else if (x1 <= 2.3e+148) {
		tmp = x1 + t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * (x1 * 9.0)) - x1
	t_1 = x1 * (1.0 + (4.0 * (x2 * ((2.0 * x2) - 3.0))))
	tmp = 0
	if x1 <= -8.4e+108:
		tmp = t_0
	elif x1 <= -1.85e-37:
		tmp = x1 + (9.0 + t_1)
	elif x1 <= 0.7:
		tmp = (x2 * -6.0) - x1
	elif x1 <= 2.3e+148:
		tmp = x1 + t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1)
	t_1 = Float64(x1 * Float64(1.0 + Float64(4.0 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0)))))
	tmp = 0.0
	if (x1 <= -8.4e+108)
		tmp = t_0;
	elseif (x1 <= -1.85e-37)
		tmp = Float64(x1 + Float64(9.0 + t_1));
	elseif (x1 <= 0.7)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	elseif (x1 <= 2.3e+148)
		tmp = Float64(x1 + t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * (x1 * 9.0)) - x1;
	t_1 = x1 * (1.0 + (4.0 * (x2 * ((2.0 * x2) - 3.0))));
	tmp = 0.0;
	if (x1 <= -8.4e+108)
		tmp = t_0;
	elseif (x1 <= -1.85e-37)
		tmp = x1 + (9.0 + t_1);
	elseif (x1 <= 0.7)
		tmp = (x2 * -6.0) - x1;
	elseif (x1 <= 2.3e+148)
		tmp = x1 + t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(1.0 + N[(4.0 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -8.4e+108], t$95$0, If[LessEqual[x1, -1.85e-37], N[(x1 + N[(9.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 0.7], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], If[LessEqual[x1, 2.3e+148], N[(x1 + t$95$1), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -8.40000000000000039e108 or 2.3000000000000001e148 < x1

   1. Initial program 2.9%

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

   2. Taylor expanded in x1 around 0 0.2%

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

   3. Taylor expanded in x2 around 0 0.2%

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

      1. fma-def 0.2%

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

      2. *-commutative 0.2%

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

      3. unpow2 0.2%

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

      4. associate-*r* 0.2%

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

      5. +-rgt-identity 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.2%

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

      10. *-commutative 0.2%

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

      11. *-commutative 0.2%

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

      12. associate-*r* 0.2%

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

   5. Simplified 0.2%

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

   6. Taylor expanded in x1 around 0 82.8%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 82.8%

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

      2. neg-mul-1 82.8%

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

      3. unsub-neg 82.8%

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

      4. *-commutative 82.8%

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

      5. unpow2 82.8%

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

      6. associate-*l* 82.8%

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

   8. Simplified 82.8%

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

   if -8.40000000000000039e108 < x1 < -1.85e-37

   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. Taylor expanded in x1 around 0 60.1%

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

   3. Taylor expanded in x1 around inf 50.5%

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

   if -1.85e-37 < x1 < 0.69999999999999996

   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. Taylor expanded in x1 around 0 98.7%

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

   3. Taylor expanded in x1 around inf 81.4%

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

   4. Taylor expanded in x1 around 0 82.0%

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

      1. +-commutative 82.0%

         \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]

      2. fma-def 82.1%

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

      3. neg-mul-1 82.1%

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

      4. fma-neg 82.0%

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

   6. Simplified 82.0%

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

   if 0.69999999999999996 < x1 < 2.3000000000000001e148

   1. Initial program 95.8%

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

   2. Taylor expanded in x1 around 0 32.0%

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

   3. Taylor expanded in x1 around inf 32.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -8.4 \cdot 10^{+108}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{elif}\;x1 \leq -1.85 \cdot 10^{-37}:\\ \;\;\;\;x1 + \left(9 + x1 \cdot \left(1 + 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 0.7:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x1 + x1 \cdot \left(1 + 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \end{array} \]

Alternative 17: 66.5% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 9\right) - x1\\ t_1 := x1 + \left(x2 \cdot x2\right) \cdot \left(x1 \cdot 8\right)\\ \mathbf{if}\;x1 \leq -7.1 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x1 \leq -2.6 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x1 \leq 0.7:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (* x1 (* x1 9.0)) x1)) (t_1 (+ x1 (* (* x2 x2) (* x1 8.0)))))
   (if (<= x1 -7.1e+108)
     t_0
     (if (<= x1 -2.6e-37)
       t_1
       (if (<= x1 0.7) (- (* x2 -6.0) x1) (if (<= x1 2.3e+148) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * (x1 * 9.0)) - x1;
	double t_1 = x1 + ((x2 * x2) * (x1 * 8.0));
	double tmp;
	if (x1 <= -7.1e+108) {
		tmp = t_0;
	} else if (x1 <= -2.6e-37) {
		tmp = t_1;
	} else if (x1 <= 0.7) {
		tmp = (x2 * -6.0) - x1;
	} else if (x1 <= 2.3e+148) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x1 * (x1 * 9.0d0)) - x1
    t_1 = x1 + ((x2 * x2) * (x1 * 8.0d0))
    if (x1 <= (-7.1d+108)) then
        tmp = t_0
    else if (x1 <= (-2.6d-37)) then
        tmp = t_1
    else if (x1 <= 0.7d0) then
        tmp = (x2 * (-6.0d0)) - x1
    else if (x1 <= 2.3d+148) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * (x1 * 9.0)) - x1;
	double t_1 = x1 + ((x2 * x2) * (x1 * 8.0));
	double tmp;
	if (x1 <= -7.1e+108) {
		tmp = t_0;
	} else if (x1 <= -2.6e-37) {
		tmp = t_1;
	} else if (x1 <= 0.7) {
		tmp = (x2 * -6.0) - x1;
	} else if (x1 <= 2.3e+148) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * (x1 * 9.0)) - x1
	t_1 = x1 + ((x2 * x2) * (x1 * 8.0))
	tmp = 0
	if x1 <= -7.1e+108:
		tmp = t_0
	elif x1 <= -2.6e-37:
		tmp = t_1
	elif x1 <= 0.7:
		tmp = (x2 * -6.0) - x1
	elif x1 <= 2.3e+148:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1)
	t_1 = Float64(x1 + Float64(Float64(x2 * x2) * Float64(x1 * 8.0)))
	tmp = 0.0
	if (x1 <= -7.1e+108)
		tmp = t_0;
	elseif (x1 <= -2.6e-37)
		tmp = t_1;
	elseif (x1 <= 0.7)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	elseif (x1 <= 2.3e+148)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * (x1 * 9.0)) - x1;
	t_1 = x1 + ((x2 * x2) * (x1 * 8.0));
	tmp = 0.0;
	if (x1 <= -7.1e+108)
		tmp = t_0;
	elseif (x1 <= -2.6e-37)
		tmp = t_1;
	elseif (x1 <= 0.7)
		tmp = (x2 * -6.0) - x1;
	elseif (x1 <= 2.3e+148)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$1 = N[(x1 + N[(N[(x2 * x2), $MachinePrecision] * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -7.1e+108], t$95$0, If[LessEqual[x1, -2.6e-37], t$95$1, If[LessEqual[x1, 0.7], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], If[LessEqual[x1, 2.3e+148], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -7.1000000000000003e108 or 2.3000000000000001e148 < x1

   1. Initial program 2.9%

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

   2. Taylor expanded in x1 around 0 0.2%

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

   3. Taylor expanded in x2 around 0 0.2%

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

      1. fma-def 0.2%

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

      2. *-commutative 0.2%

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

      3. unpow2 0.2%

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

      4. associate-*r* 0.2%

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

      5. +-rgt-identity 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 0.2%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 0.2%

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

      10. *-commutative 0.2%

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

      11. *-commutative 0.2%

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

      12. associate-*r* 0.2%

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

   5. Simplified 0.2%

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

   6. Taylor expanded in x1 around 0 82.8%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 82.8%

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

      2. neg-mul-1 82.8%

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

      3. unsub-neg 82.8%

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

      4. *-commutative 82.8%

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

      5. unpow2 82.8%

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

      6. associate-*l* 82.8%

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

   8. Simplified 82.8%

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

   if -7.1000000000000003e108 < x1 < -2.5999999999999998e-37 or 0.69999999999999996 < x1 < 2.3000000000000001e148

   1. Initial program 97.9%

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

   2. Taylor expanded in x1 around 0 47.8%

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

   3. Taylor expanded in x2 around inf 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-*l* 41.7%

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

      3. unpow2 41.7%

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

   5. Simplified 41.7%

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

   if -2.5999999999999998e-37 < x1 < 0.69999999999999996

   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. Taylor expanded in x1 around 0 98.7%

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

   3. Taylor expanded in x1 around inf 81.4%

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

   4. Taylor expanded in x1 around 0 82.0%

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

      1. +-commutative 82.0%

         \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]

      2. fma-def 82.1%

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

      3. neg-mul-1 82.1%

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

      4. fma-neg 82.0%

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

   6. Simplified 82.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -7.1 \cdot 10^{+108}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{elif}\;x1 \leq -2.6 \cdot 10^{-37}:\\ \;\;\;\;x1 + \left(x2 \cdot x2\right) \cdot \left(x1 \cdot 8\right)\\ \mathbf{elif}\;x1 \leq 0.7:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x1 + \left(x2 \cdot x2\right) \cdot \left(x1 \cdot 8\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \end{array} \]

Alternative 18: 62.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.4 \cdot 10^{-31} \lor \neg \left(x1 \leq 3.4\right):\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{else}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -2.4e-31) (not (<= x1 3.4)))
   (- (* x1 (* x1 9.0)) x1)
   (- (* x2 -6.0) x1)))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -2.4e-31) || !(x1 <= 3.4)) {
		tmp = (x1 * (x1 * 9.0)) - x1;
	} else {
		tmp = (x2 * -6.0) - x1;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((x1 <= (-2.4d-31)) .or. (.not. (x1 <= 3.4d0))) then
        tmp = (x1 * (x1 * 9.0d0)) - x1
    else
        tmp = (x2 * (-6.0d0)) - x1
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -2.4e-31) || !(x1 <= 3.4)) {
		tmp = (x1 * (x1 * 9.0)) - x1;
	} else {
		tmp = (x2 * -6.0) - x1;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x1 <= -2.4e-31) or not (x1 <= 3.4):
		tmp = (x1 * (x1 * 9.0)) - x1
	else:
		tmp = (x2 * -6.0) - x1
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -2.4e-31) || !(x1 <= 3.4))
		tmp = Float64(Float64(x1 * Float64(x1 * 9.0)) - x1);
	else
		tmp = Float64(Float64(x2 * -6.0) - x1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x1 <= -2.4e-31) || ~((x1 <= 3.4)))
		tmp = (x1 * (x1 * 9.0)) - x1;
	else
		tmp = (x2 * -6.0) - x1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -2.4e-31], N[Not[LessEqual[x1, 3.4]], $MachinePrecision]], N[(N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.4e-31 or 3.39999999999999991 < x1

   1. Initial program 44.1%

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

   2. Taylor expanded in x1 around 0 19.8%

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

   3. Taylor expanded in x2 around 0 3.9%

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

      1. fma-def 3.9%

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

      2. *-commutative 3.9%

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

      3. unpow2 3.9%

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

      4. associate-*r* 3.9%

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

      5. +-rgt-identity 3.9%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 3.9%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 3.9%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 3.9%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 3.9%

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

      10. *-commutative 3.9%

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

      11. *-commutative 3.9%

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

      12. associate-*r* 3.9%

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

   5. Simplified 3.9%

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

   6. Taylor expanded in x1 around 0 51.2%

      \[\leadsto \color{blue}{-1 \cdot x1 + 9 \cdot {x1}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 51.2%

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

      2. neg-mul-1 51.2%

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

      3. unsub-neg 51.2%

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

      4. *-commutative 51.2%

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

      5. unpow2 51.2%

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

      6. associate-*l* 51.2%

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

   8. Simplified 51.2%

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

   if -2.4e-31 < x1 < 3.39999999999999991

   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. Taylor expanded in x1 around 0 98.7%

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

   3. Taylor expanded in x1 around inf 79.3%

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

   4. Taylor expanded in x1 around 0 79.8%

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

      1. +-commutative 79.8%

         \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]

      2. fma-def 79.9%

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

      3. neg-mul-1 79.9%

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

      4. fma-neg 79.8%

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

   6. Simplified 79.8%

      \[\leadsto \color{blue}{-6 \cdot x2 - x1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.4 \cdot 10^{-31} \lor \neg \left(x1 \leq 3.4\right):\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right) - x1\\ \mathbf{else}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \end{array} \]

Alternative 19: 31.2% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \leq -1.1 \cdot 10^{-97} \lor \neg \left(x2 \leq 1.65 \cdot 10^{-195}\right):\\ \;\;\;\;x1 + x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x2 -1.1e-97) (not (<= x2 1.65e-195)))
   (+ x1 (* x2 -6.0))
   (+ x1 (* x1 -2.0))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x2 <= -1.1e-97) || !(x2 <= 1.65e-195)) {
		tmp = x1 + (x2 * -6.0);
	} else {
		tmp = x1 + (x1 * -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((x2 <= (-1.1d-97)) .or. (.not. (x2 <= 1.65d-195))) then
        tmp = x1 + (x2 * (-6.0d0))
    else
        tmp = x1 + (x1 * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((x2 <= -1.1e-97) || !(x2 <= 1.65e-195)) {
		tmp = x1 + (x2 * -6.0);
	} else {
		tmp = x1 + (x1 * -2.0);
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x2 <= -1.1e-97) or not (x2 <= 1.65e-195):
		tmp = x1 + (x2 * -6.0)
	else:
		tmp = x1 + (x1 * -2.0)
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x2 <= -1.1e-97) || !(x2 <= 1.65e-195))
		tmp = Float64(x1 + Float64(x2 * -6.0));
	else
		tmp = Float64(x1 + Float64(x1 * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x2 <= -1.1e-97) || ~((x2 <= 1.65e-195)))
		tmp = x1 + (x2 * -6.0);
	else
		tmp = x1 + (x1 * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x2, -1.1e-97], N[Not[LessEqual[x2, 1.65e-195]], $MachinePrecision]], N[(x1 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x2 < -1.0999999999999999e-97 or 1.65e-195 < x2

   1. Initial program 72.6%

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

   2. Taylor expanded in x1 around 0 62.2%

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

   3. Taylor expanded in x1 around 0 36.8%

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

      1. *-commutative 36.8%

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

   5. Simplified 36.8%

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

   if -1.0999999999999999e-97 < x2 < 1.65e-195

   1. Initial program 74.0%

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

   2. Taylor expanded in x1 around 0 56.9%

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

   3. Taylor expanded in x2 around 0 41.9%

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

      1. fma-def 42.6%

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

      2. *-commutative 42.6%

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

      3. unpow2 42.6%

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

      4. associate-*r* 42.6%

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

      5. +-rgt-identity 42.6%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

      6. +-commutative 42.6%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

      7. unpow2 42.6%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

      8. fma-udef 42.6%

         \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

      9. +-rgt-identity 42.6%

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

      10. *-commutative 42.6%

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

      11. *-commutative 42.6%

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

      12. associate-*r* 42.6%

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

   5. Simplified 42.6%

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

   6. Taylor expanded in x1 around 0 43.4%

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

      1. *-commutative 43.4%

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

   8. Simplified 43.4%

      \[\leadsto x1 + \color{blue}{x1 \cdot -2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x2 \leq -1.1 \cdot 10^{-97} \lor \neg \left(x2 \leq 1.65 \cdot 10^{-195}\right):\\ \;\;\;\;x1 + x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot -2\\ \end{array} \]

Alternative 20: 14.0% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ x1 + x1 \cdot -2 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (+ x1 (* x1 -2.0)))

C

double code(double x1, double x2) {
	return x1 + (x1 * -2.0);
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = x1 + (x1 * (-2.0d0))
end function

Java

public static double code(double x1, double x2) {
	return x1 + (x1 * -2.0);
}

Python

def code(x1, x2):
	return x1 + (x1 * -2.0)

Julia

function code(x1, x2)
	return Float64(x1 + Float64(x1 * -2.0))
end

MATLAB

function tmp = code(x1, x2)
	tmp = x1 + (x1 * -2.0);
end

Wolfram

code[x1_, x2_] := N[(x1 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.0%

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

2. Taylor expanded in x1 around 0 60.7%

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

3. Taylor expanded in x2 around 0 14.3%

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

   1. fma-def 14.6%

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

   2. *-commutative 14.6%

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

   3. unpow2 14.6%

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

   4. associate-*r* 14.6%

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

   5. +-rgt-identity 14.6%

      \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

   6. +-commutative 14.6%

      \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

   7. unpow2 14.6%

      \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

   8. fma-udef 14.6%

      \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

   9. +-rgt-identity 14.6%

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

   10. *-commutative 14.6%

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

   11. *-commutative 14.6%

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

   12. associate-*r* 14.6%

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

5. Simplified 14.6%

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

6. Taylor expanded in x1 around 0 15.1%

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

   1. *-commutative 15.1%

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

8. Simplified 15.1%

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

9. Final simplification 15.1%

   \[\leadsto x1 + x1 \cdot -2 \]

Alternative 21: 38.2% accurate, 25.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.0%

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

2. Taylor expanded in x1 around 0 72.4%

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

3. Taylor expanded in x1 around inf 81.1%

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

4. Taylor expanded in x1 around 0 44.2%

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

   1. +-commutative 44.2%

      \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]

   2. fma-def 44.2%

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

   3. neg-mul-1 44.2%

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

   4. fma-neg 44.2%

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

6. Simplified 44.2%

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

7. Final simplification 44.2%

   \[\leadsto x2 \cdot -6 - x1 \]

Alternative 22: 3.3% accurate, 42.3× speedup

Math

\[\begin{array}{l} \\ x1 \cdot 2 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (* x1 2.0))

C

double code(double x1, double x2) {
	return x1 * 2.0;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = x1 * 2.0d0
end function

Java

public static double code(double x1, double x2) {
	return x1 * 2.0;
}

Python

def code(x1, x2):
	return x1 * 2.0

Julia

function code(x1, x2)
	return Float64(x1 * 2.0)
end

MATLAB

function tmp = code(x1, x2)
	tmp = x1 * 2.0;
end

Wolfram

code[x1_, x2_] := N[(x1 * 2.0), $MachinePrecision]

Derivation

1. Initial program 73.0%

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

2. Taylor expanded in x1 around 0 60.7%

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

3. Taylor expanded in x2 around 0 14.3%

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

   1. fma-def 14.6%

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

   2. *-commutative 14.6%

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

   3. unpow2 14.6%

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

   4. associate-*r* 14.6%

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

   5. +-rgt-identity 14.6%

      \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\color{blue}{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right)} - x1}{1 + {x1}^{2}}, x1\right) \]

   6. +-commutative 14.6%

      \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{{x1}^{2} + 1}}, x1\right) \]

   7. unpow2 14.6%

      \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{x1 \cdot x1} + 1}, x1\right) \]

   8. fma-udef 14.6%

      \[\leadsto x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 0\right) - x1}{\color{blue}{\mathsf{fma}\left(x1, x1, 1\right)}}, x1\right) \]

   9. +-rgt-identity 14.6%

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

   10. *-commutative 14.6%

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

   11. *-commutative 14.6%

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

   12. associate-*r* 14.6%

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

5. Simplified 14.6%

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

6. Taylor expanded in x1 around inf 3.4%

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

   1. *-commutative 3.4%

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

8. Simplified 3.4%

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

9. Final simplification 3.4%

   \[\leadsto x1 \cdot 2 \]

Alternative 23: 3.2% accurate, 127.0× speedup

Math

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

FPCore

(FPCore (x1 x2) :precision binary64 x1)

C

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

Fortran

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

Java

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

Python

def code(x1, x2):
	return x1

Julia

function code(x1, x2)
	return x1
end

MATLAB

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

Wolfram

code[x1_, x2_] := x1

Derivation

1. Initial program 73.0%

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

2. Taylor expanded in x1 around 0 60.7%

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

3. Taylor expanded in x1 around 0 31.0%

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

   1. *-commutative 31.0%

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

5. Simplified 31.0%

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

6. Taylor expanded in x1 around inf 3.4%

   \[\leadsto \color{blue}{x1} \]

7. Final simplification 3.4%

   \[\leadsto x1 \]

Reproduce

herbie shell --seed 2023171 
(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))))))