Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.1% → 97.6%

Time: 20.0s

Alternatives: 21

Speedup: 5.6×

• 47.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.1% 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: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := \frac{t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_3 := \frac{\mathsf{fma}\left(x2, 2, t\_1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;x1 \leq -1.2 \cdot 10^{+60}:\\ \;\;\;\;\left(6 \cdot x1\right) \cdot {x1}^{3} + x1\\ \mathbf{elif}\;x1 \leq 0.84:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, t\_1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4, t\_3, -6\right), x1 \cdot x1, \left(t\_3 \cdot \left(2 \cdot x1\right)\right) \cdot \left(t\_3 - 3\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(6 \cdot x1, x2, 1\right) \cdot x1\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot t\_0\right) \cdot \left(t\_2 - 3\right), 2, \mathsf{fma}\left(t\_2, 4, -6\right) \cdot \left(x1 \cdot x1\right)\right) \cdot x1, x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* x1 x1) 3.0 (fma 2.0 x2 (- x1))))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (/ t_0 (fma x1 x1 1.0)))
        (t_3 (/ (- (fma x2 2.0 t_1) x1) (fma x1 x1 1.0))))
   (if (<= x1 -1.2e+60)
     (+ (* (* 6.0 x1) (pow x1 3.0)) x1)
     (if (<= x1 0.84)
       (+
        (fma
         (/ (- (fma -2.0 x2 t_1) x1) (fma x1 x1 1.0))
         3.0
         (fma
          (fma (fma 4.0 t_3 -6.0) (* x1 x1) (* (* t_3 (* 2.0 x1)) (- t_3 3.0)))
          (fma x1 x1 1.0)
          (* (fma (* 6.0 x1) x2 1.0) x1)))
        x1)
       (if (<= x1 5e+153)
         (+
          (fma
           (* x1 x1)
           x1
           (fma
            (*
             (fma
              (* (* (/ x1 (fma x1 x1 1.0)) t_0) (- t_2 3.0))
              2.0
              (* (fma t_2 4.0 -6.0) (* x1 x1)))
             x1)
            x1
            (* -6.0 x2)))
          x1)
         (+ (* 9.0 (* x1 x1)) x1))))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * x1), 3.0, fma(2.0, x2, -x1));
	double t_1 = (3.0 * x1) * x1;
	double t_2 = t_0 / fma(x1, x1, 1.0);
	double t_3 = (fma(x2, 2.0, t_1) - x1) / fma(x1, x1, 1.0);
	double tmp;
	if (x1 <= -1.2e+60) {
		tmp = ((6.0 * x1) * pow(x1, 3.0)) + x1;
	} else if (x1 <= 0.84) {
		tmp = fma(((fma(-2.0, x2, t_1) - x1) / fma(x1, x1, 1.0)), 3.0, fma(fma(fma(4.0, t_3, -6.0), (x1 * x1), ((t_3 * (2.0 * x1)) * (t_3 - 3.0))), fma(x1, x1, 1.0), (fma((6.0 * x1), x2, 1.0) * x1))) + x1;
	} else if (x1 <= 5e+153) {
		tmp = fma((x1 * x1), x1, fma((fma((((x1 / fma(x1, x1, 1.0)) * t_0) * (t_2 - 3.0)), 2.0, (fma(t_2, 4.0, -6.0) * (x1 * x1))) * x1), x1, (-6.0 * x2))) + x1;
	} else {
		tmp = (9.0 * (x1 * x1)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * x1), 3.0, fma(2.0, x2, Float64(-x1)))
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(t_0 / fma(x1, x1, 1.0))
	t_3 = Float64(Float64(fma(x2, 2.0, t_1) - x1) / fma(x1, x1, 1.0))
	tmp = 0.0
	if (x1 <= -1.2e+60)
		tmp = Float64(Float64(Float64(6.0 * x1) * (x1 ^ 3.0)) + x1);
	elseif (x1 <= 0.84)
		tmp = Float64(fma(Float64(Float64(fma(-2.0, x2, t_1) - x1) / fma(x1, x1, 1.0)), 3.0, fma(fma(fma(4.0, t_3, -6.0), Float64(x1 * x1), Float64(Float64(t_3 * Float64(2.0 * x1)) * Float64(t_3 - 3.0))), fma(x1, x1, 1.0), Float64(fma(Float64(6.0 * x1), x2, 1.0) * x1))) + x1);
	elseif (x1 <= 5e+153)
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(Float64(fma(Float64(Float64(Float64(x1 / fma(x1, x1, 1.0)) * t_0) * Float64(t_2 - 3.0)), 2.0, Float64(fma(t_2, 4.0, -6.0) * Float64(x1 * x1))) * x1), x1, Float64(-6.0 * x2))) + x1);
	else
		tmp = Float64(Float64(9.0 * Float64(x1 * x1)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * 3.0 + N[(2.0 * x2 + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x2 * 2.0 + t$95$1), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.2e+60], N[(N[(N[(6.0 * x1), $MachinePrecision] * N[Power[x1, 3.0], $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 0.84], N[(N[(N[(N[(N[(-2.0 * x2 + t$95$1), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(N[(4.0 * t$95$3 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$3 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(N[(6.0 * x1), $MachinePrecision] * x2 + 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 5e+153], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(N[(N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(t$95$2 * 4.0 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.2e60

   1. Initial program 19.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 98.2

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.2%

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

   if -1.2e60 < x1 < 0.839999999999999969

   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. Add Preprocessing

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

   if 0.839999999999999969 < x1 < 5.00000000000000018e153

   1. Initial program 90.5%

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.4

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

   9. Applied rewrites 97.4%

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

   if 5.00000000000000018e153 < x1

   1. Initial program 3.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 100.0%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.2 \cdot 10^{+60}:\\ \;\;\;\;\left(6 \cdot x1\right) \cdot {x1}^{3} + x1\\ \mathbf{elif}\;x1 \leq 0.84:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4, \frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, -6\right), x1 \cdot x1, \left(\frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(2 \cdot x1\right)\right) \cdot \left(\frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(6 \cdot x1, x2, 1\right) \cdot x1\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)\right) \cdot \left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right), 2, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right) \cdot \left(x1 \cdot x1\right)\right) \cdot x1, x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x1 \cdot x1\\ t_1 := \left(\left(8 \cdot x1\right) \cdot x2\right) \cdot x2\\ t_2 := \left(3 \cdot x1\right) \cdot x1\\ t_3 := \frac{\left(x2 \cdot 2 + t\_2\right) - x1}{t\_0}\\ t_4 := \left(\frac{\left(t\_2 - x2 \cdot 2\right) - x1}{t\_0} \cdot 3 + \left(\left(\left(x1 \cdot x1\right) \cdot x1 + \left(t\_3 \cdot t\_2 + \left(\left(4 \cdot t\_3 - 6\right) \cdot \left(x1 \cdot x1\right) + \left(t\_3 - 3\right) \cdot \left(t\_3 \cdot \left(2 \cdot x1\right)\right)\right) \cdot t\_0\right)\right) + x1\right)\right) + x1\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(9, x1, -2\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x1 x1)))
        (t_1 (* (* (* 8.0 x1) x2) x2))
        (t_2 (* (* 3.0 x1) x1))
        (t_3 (/ (- (+ (* x2 2.0) t_2) x1) t_0))
        (t_4
         (+
          (+
           (* (/ (- (- t_2 (* x2 2.0)) x1) t_0) 3.0)
           (+
            (+
             (* (* x1 x1) x1)
             (+
              (* t_3 t_2)
              (*
               (+
                (* (- (* 4.0 t_3) 6.0) (* x1 x1))
                (* (- t_3 3.0) (* t_3 (* 2.0 x1))))
               t_0)))
            x1))
          x1)))
   (if (<= t_4 -2e+239)
     t_1
     (if (<= t_4 1e+96)
       (+ (fma (fma 9.0 x1 -2.0) x1 (* -6.0 x2)) x1)
       (if (<= t_4 INFINITY) t_1 (+ (* 9.0 (* x1 x1)) x1))))))

C

double code(double x1, double x2) {
	double t_0 = 1.0 + (x1 * x1);
	double t_1 = ((8.0 * x1) * x2) * x2;
	double t_2 = (3.0 * x1) * x1;
	double t_3 = (((x2 * 2.0) + t_2) - x1) / t_0;
	double t_4 = (((((t_2 - (x2 * 2.0)) - x1) / t_0) * 3.0) + ((((x1 * x1) * x1) + ((t_3 * t_2) + (((((4.0 * t_3) - 6.0) * (x1 * x1)) + ((t_3 - 3.0) * (t_3 * (2.0 * x1)))) * t_0))) + x1)) + x1;
	double tmp;
	if (t_4 <= -2e+239) {
		tmp = t_1;
	} else if (t_4 <= 1e+96) {
		tmp = fma(fma(9.0, x1, -2.0), x1, (-6.0 * x2)) + x1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (9.0 * (x1 * x1)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(1.0 + Float64(x1 * x1))
	t_1 = Float64(Float64(Float64(8.0 * x1) * x2) * x2)
	t_2 = Float64(Float64(3.0 * x1) * x1)
	t_3 = Float64(Float64(Float64(Float64(x2 * 2.0) + t_2) - x1) / t_0)
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(t_2 - Float64(x2 * 2.0)) - x1) / t_0) * 3.0) + Float64(Float64(Float64(Float64(x1 * x1) * x1) + Float64(Float64(t_3 * t_2) + Float64(Float64(Float64(Float64(Float64(4.0 * t_3) - 6.0) * Float64(x1 * x1)) + Float64(Float64(t_3 - 3.0) * Float64(t_3 * Float64(2.0 * x1)))) * t_0))) + x1)) + x1)
	tmp = 0.0
	if (t_4 <= -2e+239)
		tmp = t_1;
	elseif (t_4 <= 1e+96)
		tmp = Float64(fma(fma(9.0, x1, -2.0), x1, Float64(-6.0 * x2)) + x1);
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(9.0 * Float64(x1 * x1)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(8.0 * x1), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(t$95$2 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision] + N[(N[(t$95$3 * t$95$2), $MachinePrecision] + N[(N[(N[(N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - 3.0), $MachinePrecision] * N[(t$95$3 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+239], t$95$1, If[LessEqual[t$95$4, 1e+96], N[(N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.99999999999999998e239 or 1.00000000000000005e96 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 9.8

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 47.6

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

   8. Applied rewrites 47.6%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 52.1%

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

   if -1.99999999999999998e239 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.00000000000000005e96

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 86.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 86.2%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 86.2%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} \cdot 3 + \left(\left(\left(x1 \cdot x1\right) \cdot x1 + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) + \left(\left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 3\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(2 \cdot x1\right)\right)\right) \cdot \left(1 + x1 \cdot x1\right)\right)\right) + x1\right)\right) + x1 \leq -2 \cdot 10^{+239}:\\ \;\;\;\;\left(\left(8 \cdot x1\right) \cdot x2\right) \cdot x2\\ \mathbf{elif}\;\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} \cdot 3 + \left(\left(\left(x1 \cdot x1\right) \cdot x1 + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) + \left(\left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 3\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(2 \cdot x1\right)\right)\right) \cdot \left(1 + x1 \cdot x1\right)\right)\right) + x1\right)\right) + x1 \leq 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(9, x1, -2\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} \cdot 3 + \left(\left(\left(x1 \cdot x1\right) \cdot x1 + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) + \left(\left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 3\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(2 \cdot x1\right)\right)\right) \cdot \left(1 + x1 \cdot x1\right)\right)\right) + x1\right)\right) + x1 \leq \infty:\\ \;\;\;\;\left(\left(8 \cdot x1\right) \cdot x2\right) \cdot x2\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x1 \cdot x1\\ t_1 := \left(\left(8 \cdot x1\right) \cdot x2\right) \cdot x2\\ t_2 := \left(3 \cdot x1\right) \cdot x1\\ t_3 := \frac{\left(x2 \cdot 2 + t\_2\right) - x1}{t\_0}\\ t_4 := \left(\frac{\left(t\_2 - x2 \cdot 2\right) - x1}{t\_0} \cdot 3 + \left(\left(\left(x1 \cdot x1\right) \cdot x1 + \left(t\_3 \cdot t\_2 + \left(\left(4 \cdot t\_3 - 6\right) \cdot \left(x1 \cdot x1\right) + \left(t\_3 - 3\right) \cdot \left(t\_3 \cdot \left(2 \cdot x1\right)\right)\right) \cdot t\_0\right)\right) + x1\right)\right) + x1\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{+96}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x1 x1)))
        (t_1 (* (* (* 8.0 x1) x2) x2))
        (t_2 (* (* 3.0 x1) x1))
        (t_3 (/ (- (+ (* x2 2.0) t_2) x1) t_0))
        (t_4
         (+
          (+
           (* (/ (- (- t_2 (* x2 2.0)) x1) t_0) 3.0)
           (+
            (+
             (* (* x1 x1) x1)
             (+
              (* t_3 t_2)
              (*
               (+
                (* (- (* 4.0 t_3) 6.0) (* x1 x1))
                (* (- t_3 3.0) (* t_3 (* 2.0 x1))))
               t_0)))
            x1))
          x1)))
   (if (<= t_4 -2e+239)
     t_1
     (if (<= t_4 1e+96)
       (* -6.0 x2)
       (if (<= t_4 INFINITY) t_1 (+ (* 9.0 (* x1 x1)) x1))))))

C

double code(double x1, double x2) {
	double t_0 = 1.0 + (x1 * x1);
	double t_1 = ((8.0 * x1) * x2) * x2;
	double t_2 = (3.0 * x1) * x1;
	double t_3 = (((x2 * 2.0) + t_2) - x1) / t_0;
	double t_4 = (((((t_2 - (x2 * 2.0)) - x1) / t_0) * 3.0) + ((((x1 * x1) * x1) + ((t_3 * t_2) + (((((4.0 * t_3) - 6.0) * (x1 * x1)) + ((t_3 - 3.0) * (t_3 * (2.0 * x1)))) * t_0))) + x1)) + x1;
	double tmp;
	if (t_4 <= -2e+239) {
		tmp = t_1;
	} else if (t_4 <= 1e+96) {
		tmp = -6.0 * x2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (9.0 * (x1 * x1)) + x1;
	}
	return tmp;
}

Java

public static double code(double x1, double x2) {
	double t_0 = 1.0 + (x1 * x1);
	double t_1 = ((8.0 * x1) * x2) * x2;
	double t_2 = (3.0 * x1) * x1;
	double t_3 = (((x2 * 2.0) + t_2) - x1) / t_0;
	double t_4 = (((((t_2 - (x2 * 2.0)) - x1) / t_0) * 3.0) + ((((x1 * x1) * x1) + ((t_3 * t_2) + (((((4.0 * t_3) - 6.0) * (x1 * x1)) + ((t_3 - 3.0) * (t_3 * (2.0 * x1)))) * t_0))) + x1)) + x1;
	double tmp;
	if (t_4 <= -2e+239) {
		tmp = t_1;
	} else if (t_4 <= 1e+96) {
		tmp = -6.0 * x2;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = (9.0 * (x1 * x1)) + x1;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = 1.0 + (x1 * x1)
	t_1 = ((8.0 * x1) * x2) * x2
	t_2 = (3.0 * x1) * x1
	t_3 = (((x2 * 2.0) + t_2) - x1) / t_0
	t_4 = (((((t_2 - (x2 * 2.0)) - x1) / t_0) * 3.0) + ((((x1 * x1) * x1) + ((t_3 * t_2) + (((((4.0 * t_3) - 6.0) * (x1 * x1)) + ((t_3 - 3.0) * (t_3 * (2.0 * x1)))) * t_0))) + x1)) + x1
	tmp = 0
	if t_4 <= -2e+239:
		tmp = t_1
	elif t_4 <= 1e+96:
		tmp = -6.0 * x2
	elif t_4 <= math.inf:
		tmp = t_1
	else:
		tmp = (9.0 * (x1 * x1)) + x1
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(1.0 + Float64(x1 * x1))
	t_1 = Float64(Float64(Float64(8.0 * x1) * x2) * x2)
	t_2 = Float64(Float64(3.0 * x1) * x1)
	t_3 = Float64(Float64(Float64(Float64(x2 * 2.0) + t_2) - x1) / t_0)
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(t_2 - Float64(x2 * 2.0)) - x1) / t_0) * 3.0) + Float64(Float64(Float64(Float64(x1 * x1) * x1) + Float64(Float64(t_3 * t_2) + Float64(Float64(Float64(Float64(Float64(4.0 * t_3) - 6.0) * Float64(x1 * x1)) + Float64(Float64(t_3 - 3.0) * Float64(t_3 * Float64(2.0 * x1)))) * t_0))) + x1)) + x1)
	tmp = 0.0
	if (t_4 <= -2e+239)
		tmp = t_1;
	elseif (t_4 <= 1e+96)
		tmp = Float64(-6.0 * x2);
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(9.0 * Float64(x1 * x1)) + x1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = 1.0 + (x1 * x1);
	t_1 = ((8.0 * x1) * x2) * x2;
	t_2 = (3.0 * x1) * x1;
	t_3 = (((x2 * 2.0) + t_2) - x1) / t_0;
	t_4 = (((((t_2 - (x2 * 2.0)) - x1) / t_0) * 3.0) + ((((x1 * x1) * x1) + ((t_3 * t_2) + (((((4.0 * t_3) - 6.0) * (x1 * x1)) + ((t_3 - 3.0) * (t_3 * (2.0 * x1)))) * t_0))) + x1)) + x1;
	tmp = 0.0;
	if (t_4 <= -2e+239)
		tmp = t_1;
	elseif (t_4 <= 1e+96)
		tmp = -6.0 * x2;
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = (9.0 * (x1 * x1)) + x1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(8.0 * x1), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(t$95$2 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision] + N[(N[(t$95$3 * t$95$2), $MachinePrecision] + N[(N[(N[(N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - 3.0), $MachinePrecision] * N[(t$95$3 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+239], t$95$1, If[LessEqual[t$95$4, 1e+96], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.99999999999999998e239 or 1.00000000000000005e96 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 9.8

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 47.6

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

   8. Applied rewrites 47.6%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 52.1%

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

   if -1.99999999999999998e239 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.00000000000000005e96

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 55.1

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

   8. Applied rewrites 55.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 86.2%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 86.2%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} \cdot 3 + \left(\left(\left(x1 \cdot x1\right) \cdot x1 + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) + \left(\left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 3\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(2 \cdot x1\right)\right)\right) \cdot \left(1 + x1 \cdot x1\right)\right)\right) + x1\right)\right) + x1 \leq -2 \cdot 10^{+239}:\\ \;\;\;\;\left(\left(8 \cdot x1\right) \cdot x2\right) \cdot x2\\ \mathbf{elif}\;\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} \cdot 3 + \left(\left(\left(x1 \cdot x1\right) \cdot x1 + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) + \left(\left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 3\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(2 \cdot x1\right)\right)\right) \cdot \left(1 + x1 \cdot x1\right)\right)\right) + x1\right)\right) + x1 \leq 10^{+96}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} \cdot 3 + \left(\left(\left(x1 \cdot x1\right) \cdot x1 + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) + \left(\left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 3\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(2 \cdot x1\right)\right)\right) \cdot \left(1 + x1 \cdot x1\right)\right)\right) + x1\right)\right) + x1 \leq \infty:\\ \;\;\;\;\left(\left(8 \cdot x1\right) \cdot x2\right) \cdot x2\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.3%

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 95.1

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 95.1%

      \[\leadsto x1 + \left({x1}^{3} \cdot x1\right) \cdot 6 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

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

Alternative 5: 81.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x1 \cdot x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := \frac{\left(x2 \cdot 2 + t\_1\right) - x1}{t\_0}\\ t_3 := \left(\frac{\left(t\_1 - x2 \cdot 2\right) - x1}{t\_0} \cdot 3 + \left(\left(\left(x1 \cdot x1\right) \cdot x1 + \left(t\_2 \cdot t\_1 + \left(\left(4 \cdot t\_2 - 6\right) \cdot \left(x1 \cdot x1\right) + \left(t\_2 - 3\right) \cdot \left(t\_2 \cdot \left(2 \cdot x1\right)\right)\right) \cdot t\_0\right)\right) + x1\right)\right) + x1\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+239}:\\ \;\;\;\;\left(\left(8 \cdot x1\right) \cdot x2\right) \cdot x2\\ \mathbf{elif}\;t\_3 \leq 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(9, x1, -2\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x1 x1)))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (/ (- (+ (* x2 2.0) t_1) x1) t_0))
        (t_3
         (+
          (+
           (* (/ (- (- t_1 (* x2 2.0)) x1) t_0) 3.0)
           (+
            (+
             (* (* x1 x1) x1)
             (+
              (* t_2 t_1)
              (*
               (+
                (* (- (* 4.0 t_2) 6.0) (* x1 x1))
                (* (- t_2 3.0) (* t_2 (* 2.0 x1))))
               t_0)))
            x1))
          x1)))
   (if (<= t_3 -2e+239)
     (* (* (* 8.0 x1) x2) x2)
     (if (<= t_3 1e+41)
       (+ (fma (fma 9.0 x1 -2.0) x1 (* -6.0 x2)) x1)
       (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1)))))

C

double code(double x1, double x2) {
	double t_0 = 1.0 + (x1 * x1);
	double t_1 = (3.0 * x1) * x1;
	double t_2 = (((x2 * 2.0) + t_1) - x1) / t_0;
	double t_3 = (((((t_1 - (x2 * 2.0)) - x1) / t_0) * 3.0) + ((((x1 * x1) * x1) + ((t_2 * t_1) + (((((4.0 * t_2) - 6.0) * (x1 * x1)) + ((t_2 - 3.0) * (t_2 * (2.0 * x1)))) * t_0))) + x1)) + x1;
	double tmp;
	if (t_3 <= -2e+239) {
		tmp = ((8.0 * x1) * x2) * x2;
	} else if (t_3 <= 1e+41) {
		tmp = fma(fma(9.0, x1, -2.0), x1, (-6.0 * x2)) + x1;
	} else {
		tmp = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(1.0 + Float64(x1 * x1))
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(Float64(Float64(Float64(x2 * 2.0) + t_1) - x1) / t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(Float64(t_1 - Float64(x2 * 2.0)) - x1) / t_0) * 3.0) + Float64(Float64(Float64(Float64(x1 * x1) * x1) + Float64(Float64(t_2 * t_1) + Float64(Float64(Float64(Float64(Float64(4.0 * t_2) - 6.0) * Float64(x1 * x1)) + Float64(Float64(t_2 - 3.0) * Float64(t_2 * Float64(2.0 * x1)))) * t_0))) + x1)) + x1)
	tmp = 0.0
	if (t_3 <= -2e+239)
		tmp = Float64(Float64(Float64(8.0 * x1) * x2) * x2);
	elseif (t_3 <= 1e+41)
		tmp = Float64(fma(fma(9.0, x1, -2.0), x1, Float64(-6.0 * x2)) + x1);
	else
		tmp = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$1), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(N[(t$95$1 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision] + N[(N[(t$95$2 * t$95$1), $MachinePrecision] + N[(N[(N[(N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - 3.0), $MachinePrecision] * N[(t$95$2 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+239], N[(N[(N[(8.0 * x1), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision], If[LessEqual[t$95$3, 1e+41], N[(N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.99999999999999998e239

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 8.9

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

   5. Applied rewrites 8.9%

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

   6. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 90.4

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

   8. Applied rewrites 90.4%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 90.4%

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

   if -1.99999999999999998e239 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.00000000000000001e41

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 91.6%

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

   if 1.00000000000000001e41 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 44.8%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 82.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} \cdot 3 + \left(\left(\left(x1 \cdot x1\right) \cdot x1 + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) + \left(\left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 3\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(2 \cdot x1\right)\right)\right) \cdot \left(1 + x1 \cdot x1\right)\right)\right) + x1\right)\right) + x1 \leq -2 \cdot 10^{+239}:\\ \;\;\;\;\left(\left(8 \cdot x1\right) \cdot x2\right) \cdot x2\\ \mathbf{elif}\;\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} \cdot 3 + \left(\left(\left(x1 \cdot x1\right) \cdot x1 + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) + \left(\left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} - 3\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{1 + x1 \cdot x1} \cdot \left(2 \cdot x1\right)\right)\right) \cdot \left(1 + x1 \cdot x1\right)\right)\right) + x1\right)\right) + x1 \leq 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(9, x1, -2\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.3%

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 95.1

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 95.1%

      \[\leadsto x1 + \left({x1}^{3} \cdot x1\right) \cdot 6 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

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

Alternative 7: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.3%

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 95.1

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 95.1%

      \[\leadsto x1 + \left({x1}^{3} \cdot x1\right) \cdot 6 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

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

Alternative 8: 95.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)\\ t_1 := \frac{t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;x1 \leq -27000:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot t\_0\right) \cdot \left(t\_1 - 3\right), 2, \mathsf{fma}\left(t\_1, 4, -6\right) \cdot \left(x1 \cdot x1\right)\right) \cdot x1, x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* x1 x1) 3.0 (fma 2.0 x2 (- x1))))
        (t_1 (/ t_0 (fma x1 x1 1.0))))
   (if (<= x1 -27000.0)
     (* (pow x1 4.0) (- 6.0 (/ 3.0 x1)))
     (if (<= x1 0.55)
       (+
        (fma
         (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
         x2
         (* (fma 9.0 x1 -2.0) x1))
        x1)
       (if (<= x1 5e+153)
         (+
          (fma
           (* x1 x1)
           x1
           (fma
            (*
             (fma
              (* (* (/ x1 (fma x1 x1 1.0)) t_0) (- t_1 3.0))
              2.0
              (* (fma t_1 4.0 -6.0) (* x1 x1)))
             x1)
            x1
            (* -6.0 x2)))
          x1)
         (+ (* 9.0 (* x1 x1)) x1))))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * x1), 3.0, fma(2.0, x2, -x1));
	double t_1 = t_0 / fma(x1, x1, 1.0);
	double tmp;
	if (x1 <= -27000.0) {
		tmp = pow(x1, 4.0) * (6.0 - (3.0 / x1));
	} else if (x1 <= 0.55) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -2.0) * x1)) + x1;
	} else if (x1 <= 5e+153) {
		tmp = fma((x1 * x1), x1, fma((fma((((x1 / fma(x1, x1, 1.0)) * t_0) * (t_1 - 3.0)), 2.0, (fma(t_1, 4.0, -6.0) * (x1 * x1))) * x1), x1, (-6.0 * x2))) + x1;
	} else {
		tmp = (9.0 * (x1 * x1)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * x1), 3.0, fma(2.0, x2, Float64(-x1)))
	t_1 = Float64(t_0 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (x1 <= -27000.0)
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 - Float64(3.0 / x1)));
	elseif (x1 <= 0.55)
		tmp = Float64(fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -2.0) * x1)) + x1);
	elseif (x1 <= 5e+153)
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(Float64(fma(Float64(Float64(Float64(x1 / fma(x1, x1, 1.0)) * t_0) * Float64(t_1 - 3.0)), 2.0, Float64(fma(t_1, 4.0, -6.0) * Float64(x1 * x1))) * x1), x1, Float64(-6.0 * x2))) + x1);
	else
		tmp = Float64(Float64(9.0 * Float64(x1 * x1)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * 3.0 + N[(2.0 * x2 + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -27000.0], N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 0.55], N[(N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 5e+153], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(N[(N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(t$95$1 - 3.0), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(t$95$1 * 4.0 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -27000

   1. Initial program 32.1%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 94.0

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

   8. Applied rewrites 94.0%

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

   if -27000 < x1 < 0.55000000000000004

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 98.5%

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

   if 0.55000000000000004 < x1 < 5.00000000000000018e153

   1. Initial program 90.5%

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.4

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

   9. Applied rewrites 97.4%

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

   if 5.00000000000000018e153 < x1

   1. Initial program 3.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 100.0%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -27000:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)\right) \cdot \left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right), 2, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right) \cdot \left(x1 \cdot x1\right)\right) \cdot x1, x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -27000:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 215000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(6 - \frac{4}{x1}\right) \cdot {x1}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -27000.0)
   (* (pow x1 4.0) (- 6.0 (/ 3.0 x1)))
   (if (<= x1 215000.0)
     (+
      (fma
       (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
       x2
       (* (fma 9.0 x1 -2.0) x1))
      x1)
     (fma (fma x1 x1 1.0) x1 (* (- 6.0 (/ 4.0 x1)) (pow x1 4.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -27000.0) {
		tmp = pow(x1, 4.0) * (6.0 - (3.0 / x1));
	} else if (x1 <= 215000.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -2.0) * x1)) + x1;
	} else {
		tmp = fma(fma(x1, x1, 1.0), x1, ((6.0 - (4.0 / x1)) * pow(x1, 4.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -27000.0)
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 - Float64(3.0 / x1)));
	elseif (x1 <= 215000.0)
		tmp = Float64(fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -2.0) * x1)) + x1);
	else
		tmp = fma(fma(x1, x1, 1.0), x1, Float64(Float64(6.0 - Float64(4.0 / x1)) * (x1 ^ 4.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -27000.0], N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 215000.0], N[(N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(6.0 - N[(4.0 / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -27000

   1. Initial program 32.1%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 94.0

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

   8. Applied rewrites 94.0%

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

   if -27000 < x1 < 215000

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 97.7%

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

   if 215000 < x1

   1. Initial program 45.1%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 92.3

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

   7. Applied rewrites 92.3%

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

   9. Applied rewrites 92.3%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -27000:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 215000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(6 - \frac{4}{x1}\right) \cdot {x1}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot \left(6 - \frac{3}{x1}\right)\\ \mathbf{if}\;x1 \leq -27000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 215000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (pow x1 4.0) (- 6.0 (/ 3.0 x1)))))
   (if (<= x1 -27000.0)
     t_0
     (if (<= x1 215000.0)
       (+
        (fma
         (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
         x2
         (* (fma 9.0 x1 -2.0) x1))
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = pow(x1, 4.0) * (6.0 - (3.0 / x1));
	double tmp;
	if (x1 <= -27000.0) {
		tmp = t_0;
	} else if (x1 <= 215000.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -2.0) * x1)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64((x1 ^ 4.0) * Float64(6.0 - Float64(3.0 / x1)))
	tmp = 0.0
	if (x1 <= -27000.0)
		tmp = t_0;
	elseif (x1 <= 215000.0)
		tmp = Float64(fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -2.0) * x1)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -27000.0], t$95$0, If[LessEqual[x1, 215000.0], N[(N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -27000 or 215000 < x1

   1. Initial program 38.4%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 93.2

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

   8. Applied rewrites 93.2%

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

   if -27000 < x1 < 215000

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 97.7%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -27000:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 215000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot 6\\ \mathbf{if}\;x1 \leq -35000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 820000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (pow x1 4.0) 6.0)))
   (if (<= x1 -35000.0)
     t_0
     (if (<= x1 820000000.0)
       (+
        (fma
         (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
         x2
         (* (fma 9.0 x1 -2.0) x1))
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = pow(x1, 4.0) * 6.0;
	double tmp;
	if (x1 <= -35000.0) {
		tmp = t_0;
	} else if (x1 <= 820000000.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -2.0) * x1)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64((x1 ^ 4.0) * 6.0)
	tmp = 0.0
	if (x1 <= -35000.0)
		tmp = t_0;
	elseif (x1 <= 820000000.0)
		tmp = Float64(fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -2.0) * x1)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[x1, -35000.0], t$95$0, If[LessEqual[x1, 820000000.0], N[(N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -35000 or 8.2e8 < x1

   1. Initial program 38.4%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 92.6

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

   8. Applied rewrites 92.6%

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

   if -35000 < x1 < 8.2e8

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 97.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -35000:\\ \;\;\;\;{x1}^{4} \cdot 6\\ \mathbf{elif}\;x1 \leq 820000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -35000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 820000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1)))
   (if (<= x1 -35000.0)
     t_0
     (if (<= x1 820000000.0)
       (+
        (fma
         (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
         x2
         (* (fma 9.0 x1 -2.0) x1))
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	double tmp;
	if (x1 <= -35000.0) {
		tmp = t_0;
	} else if (x1 <= 820000000.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -2.0) * x1)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	tmp = 0.0
	if (x1 <= -35000.0)
		tmp = t_0;
	elseif (x1 <= 820000000.0)
		tmp = Float64(fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -2.0) * x1)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -35000.0], t$95$0, If[LessEqual[x1, 820000000.0], N[(N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -35000 or 8.2e8 < x1

   1. Initial program 38.4%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 92.6

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

   5. Applied rewrites 92.6%

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

   7. Applied rewrites 92.6%

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

   if -35000 < x1 < 8.2e8

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 97.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -35000:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 820000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 88.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{if}\;x1 \leq -33000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -6.2 \cdot 10^{-224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq 5.8 \cdot 10^{-273}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right), x2, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 750000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1))
        (t_1 (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))))
   (if (<= x1 -33000.0)
     t_0
     (if (<= x1 -6.2e-224)
       t_1
       (if (<= x1 5.8e-273)
         (+
          (fma (fma (fma 12.0 x1 -12.0) x1 -6.0) x2 (* (fma 9.0 x1 -2.0) x1))
          x1)
         (if (<= x1 750000000.0) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	double t_1 = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, (-6.0 * x2));
	double tmp;
	if (x1 <= -33000.0) {
		tmp = t_0;
	} else if (x1 <= -6.2e-224) {
		tmp = t_1;
	} else if (x1 <= 5.8e-273) {
		tmp = fma(fma(fma(12.0, x1, -12.0), x1, -6.0), x2, (fma(9.0, x1, -2.0) * x1)) + x1;
	} else if (x1 <= 750000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	t_1 = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2))
	tmp = 0.0
	if (x1 <= -33000.0)
		tmp = t_0;
	elseif (x1 <= -6.2e-224)
		tmp = t_1;
	elseif (x1 <= 5.8e-273)
		tmp = Float64(fma(fma(fma(12.0, x1, -12.0), x1, -6.0), x2, Float64(fma(9.0, x1, -2.0) * x1)) + x1);
	elseif (x1 <= 750000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -33000.0], t$95$0, If[LessEqual[x1, -6.2e-224], t$95$1, If[LessEqual[x1, 5.8e-273], N[(N[(N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 750000000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -33000 or 7.5e8 < x1

   1. Initial program 38.4%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 92.6

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

   5. Applied rewrites 92.6%

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

   7. Applied rewrites 92.6%

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

   if -33000 < x1 < -6.20000000000000017e-224 or 5.79999999999999973e-273 < x1 < 7.5e8

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 34.8

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

   5. Applied rewrites 34.8%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.7

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

   8. Applied rewrites 89.7%

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

   if -6.20000000000000017e-224 < x1 < 5.79999999999999973e-273

   1. Initial program 96.6%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 88.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -33000:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq -6.2 \cdot 10^{-224}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 5.8 \cdot 10^{-273}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right), x2, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 750000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 88.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{if}\;x1 \leq -33000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -6.2 \cdot 10^{-224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq 5.8 \cdot 10^{-273}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(9, x1, -2\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 750000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1))
        (t_1 (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))))
   (if (<= x1 -33000.0)
     t_0
     (if (<= x1 -6.2e-224)
       t_1
       (if (<= x1 5.8e-273)
         (+ (fma (fma 9.0 x1 -2.0) x1 (* -6.0 x2)) x1)
         (if (<= x1 750000000.0) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	double t_1 = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, (-6.0 * x2));
	double tmp;
	if (x1 <= -33000.0) {
		tmp = t_0;
	} else if (x1 <= -6.2e-224) {
		tmp = t_1;
	} else if (x1 <= 5.8e-273) {
		tmp = fma(fma(9.0, x1, -2.0), x1, (-6.0 * x2)) + x1;
	} else if (x1 <= 750000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	t_1 = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2))
	tmp = 0.0
	if (x1 <= -33000.0)
		tmp = t_0;
	elseif (x1 <= -6.2e-224)
		tmp = t_1;
	elseif (x1 <= 5.8e-273)
		tmp = Float64(fma(fma(9.0, x1, -2.0), x1, Float64(-6.0 * x2)) + x1);
	elseif (x1 <= 750000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -33000.0], t$95$0, If[LessEqual[x1, -6.2e-224], t$95$1, If[LessEqual[x1, 5.8e-273], N[(N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 750000000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -33000 or 7.5e8 < x1

   1. Initial program 38.4%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 92.6

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

   5. Applied rewrites 92.6%

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

   7. Applied rewrites 92.6%

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

   if -33000 < x1 < -6.20000000000000017e-224 or 5.79999999999999973e-273 < x1 < 7.5e8

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 34.8

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

   5. Applied rewrites 34.8%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.7

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

   8. Applied rewrites 89.7%

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

   if -6.20000000000000017e-224 < x1 < 5.79999999999999973e-273

   1. Initial program 96.6%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 88.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -33000:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq -6.2 \cdot 10^{-224}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 5.8 \cdot 10^{-273}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(9, x1, -2\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 750000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.5% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 9 \cdot \left(x1 \cdot x1\right) + x1\\ t_1 := -2 \cdot x1 + x1\\ \mathbf{if}\;x1 \leq -1.9 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -1.62 \cdot 10^{-152}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq 2.4 \cdot 10^{-63}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 1.4:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* 9.0 (* x1 x1)) x1)) (t_1 (+ (* -2.0 x1) x1)))
   (if (<= x1 -1.9e-5)
     t_0
     (if (<= x1 -1.62e-152)
       t_1
       (if (<= x1 2.4e-63) (* -6.0 x2) (if (<= x1 1.4) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = (9.0 * (x1 * x1)) + x1;
	double t_1 = (-2.0 * x1) + x1;
	double tmp;
	if (x1 <= -1.9e-5) {
		tmp = t_0;
	} else if (x1 <= -1.62e-152) {
		tmp = t_1;
	} else if (x1 <= 2.4e-63) {
		tmp = -6.0 * x2;
	} else if (x1 <= 1.4) {
		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 = (9.0d0 * (x1 * x1)) + x1
    t_1 = ((-2.0d0) * x1) + x1
    if (x1 <= (-1.9d-5)) then
        tmp = t_0
    else if (x1 <= (-1.62d-152)) then
        tmp = t_1
    else if (x1 <= 2.4d-63) then
        tmp = (-6.0d0) * x2
    else if (x1 <= 1.4d0) 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 = (9.0 * (x1 * x1)) + x1;
	double t_1 = (-2.0 * x1) + x1;
	double tmp;
	if (x1 <= -1.9e-5) {
		tmp = t_0;
	} else if (x1 <= -1.62e-152) {
		tmp = t_1;
	} else if (x1 <= 2.4e-63) {
		tmp = -6.0 * x2;
	} else if (x1 <= 1.4) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (9.0 * (x1 * x1)) + x1
	t_1 = (-2.0 * x1) + x1
	tmp = 0
	if x1 <= -1.9e-5:
		tmp = t_0
	elif x1 <= -1.62e-152:
		tmp = t_1
	elif x1 <= 2.4e-63:
		tmp = -6.0 * x2
	elif x1 <= 1.4:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(9.0 * Float64(x1 * x1)) + x1)
	t_1 = Float64(Float64(-2.0 * x1) + x1)
	tmp = 0.0
	if (x1 <= -1.9e-5)
		tmp = t_0;
	elseif (x1 <= -1.62e-152)
		tmp = t_1;
	elseif (x1 <= 2.4e-63)
		tmp = Float64(-6.0 * x2);
	elseif (x1 <= 1.4)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (9.0 * (x1 * x1)) + x1;
	t_1 = (-2.0 * x1) + x1;
	tmp = 0.0;
	if (x1 <= -1.9e-5)
		tmp = t_0;
	elseif (x1 <= -1.62e-152)
		tmp = t_1;
	elseif (x1 <= 2.4e-63)
		tmp = -6.0 * x2;
	elseif (x1 <= 1.4)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * x1), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -1.9e-5], t$95$0, If[LessEqual[x1, -1.62e-152], t$95$1, If[LessEqual[x1, 2.4e-63], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x1, 1.4], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.9000000000000001e-5 or 1.3999999999999999 < x1

   1. Initial program 39.3%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 55.9%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 55.9%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]

   if -1.9000000000000001e-5 < x1 < -1.61999999999999995e-152 or 2.4000000000000001e-63 < x1 < 1.3999999999999999

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 53.5%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + -2 \cdot x1 \]
   10. Step-by-step derivation

   11. Applied rewrites 50.8%

       \[\leadsto x1 + -2 \cdot x1 \]

   if -1.61999999999999995e-152 < x1 < 2.4000000000000001e-63

   1. Initial program 98.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.3

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

   8. Applied rewrites 68.3%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.9 \cdot 10^{-5}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq -1.62 \cdot 10^{-152}:\\ \;\;\;\;-2 \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 2.4 \cdot 10^{-63}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 1.4:\\ \;\;\;\;-2 \cdot x1 + x1\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.3% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.62 \cdot 10^{-152}:\\ \;\;\;\;\mathsf{fma}\left(9, x1, -2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 1.45 \cdot 10^{-72}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.62e-152)
   (+ (* (fma 9.0 x1 -2.0) x1) x1)
   (if (<= x1 1.45e-72)
     (* -6.0 x2)
     (if (<= x1 1.05e+101)
       (+ (* (* (* x2 x2) x1) 8.0) x1)
       (+ (fma (* x1 x1) x1 (* -6.0 x2)) x1)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.62e-152) {
		tmp = (fma(9.0, x1, -2.0) * x1) + x1;
	} else if (x1 <= 1.45e-72) {
		tmp = -6.0 * x2;
	} else if (x1 <= 1.05e+101) {
		tmp = (((x2 * x2) * x1) * 8.0) + x1;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.62e-152)
		tmp = Float64(Float64(fma(9.0, x1, -2.0) * x1) + x1);
	elseif (x1 <= 1.45e-72)
		tmp = Float64(-6.0 * x2);
	elseif (x1 <= 1.05e+101)
		tmp = Float64(Float64(Float64(Float64(x2 * x2) * x1) * 8.0) + x1);
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(-6.0 * x2)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.62e-152], N[(N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1.45e-72], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x1, 1.05e+101], N[(N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.61999999999999995e-152

   1. Initial program 53.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 56.1%

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

   if -1.61999999999999995e-152 < x1 < 1.44999999999999999e-72

   1. Initial program 98.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

   if 1.44999999999999999e-72 < x1 < 1.05e101

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 37.8%

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

   if 1.05e101 < x1

   1. Initial program 30.0%

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

   4. Applied rewrites 36.0%

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

   5. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 94.0

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

   7. Applied rewrites 94.0%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.62 \cdot 10^{-152}:\\ \;\;\;\;\mathsf{fma}\left(9, x1, -2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 1.45 \cdot 10^{-72}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.8% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.62 \cdot 10^{-152}:\\ \;\;\;\;\mathsf{fma}\left(9, x1, -2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 1.45 \cdot 10^{-72}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 4.3 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.62e-152)
   (+ (* (fma 9.0 x1 -2.0) x1) x1)
   (if (<= x1 1.45e-72)
     (* -6.0 x2)
     (if (<= x1 4.3e+153) (* (* (* x2 x2) x1) 8.0) (+ (* 9.0 (* x1 x1)) x1)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.62e-152) {
		tmp = (fma(9.0, x1, -2.0) * x1) + x1;
	} else if (x1 <= 1.45e-72) {
		tmp = -6.0 * x2;
	} else if (x1 <= 4.3e+153) {
		tmp = ((x2 * x2) * x1) * 8.0;
	} else {
		tmp = (9.0 * (x1 * x1)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.62e-152)
		tmp = Float64(Float64(fma(9.0, x1, -2.0) * x1) + x1);
	elseif (x1 <= 1.45e-72)
		tmp = Float64(-6.0 * x2);
	elseif (x1 <= 4.3e+153)
		tmp = Float64(Float64(Float64(x2 * x2) * x1) * 8.0);
	else
		tmp = Float64(Float64(9.0 * Float64(x1 * x1)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.62e-152], N[(N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1.45e-72], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x1, 4.3e+153], N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision], N[(N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.61999999999999995e-152

   1. Initial program 53.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 56.1%

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

   if -1.61999999999999995e-152 < x1 < 1.44999999999999999e-72

   1. Initial program 98.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

   if 1.44999999999999999e-72 < x1 < 4.2999999999999998e153

   1. Initial program 92.5%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.8

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

   5. Applied rewrites 3.8%

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

   6. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 29.5

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

   8. Applied rewrites 29.5%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 37.1%

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

   if 4.2999999999999998e153 < x1

   1. Initial program 3.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 100.0%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.62 \cdot 10^{-152}:\\ \;\;\;\;\mathsf{fma}\left(9, x1, -2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 1.45 \cdot 10^{-72}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 4.3 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 31.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -5 \cdot 10^{-193}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \cdot 2 \leq 2 \cdot 10^{-182}:\\ \;\;\;\;-2 \cdot x1 + x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2 + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* x2 2.0) -5e-193)
   (* -6.0 x2)
   (if (<= (* x2 2.0) 2e-182) (+ (* -2.0 x1) x1) (+ (* -6.0 x2) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x2 * 2.0) <= -5e-193) {
		tmp = -6.0 * x2;
	} else if ((x2 * 2.0) <= 2e-182) {
		tmp = (-2.0 * x1) + x1;
	} else {
		tmp = (-6.0 * x2) + x1;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((x2 * 2.0d0) <= (-5d-193)) then
        tmp = (-6.0d0) * x2
    else if ((x2 * 2.0d0) <= 2d-182) then
        tmp = ((-2.0d0) * x1) + x1
    else
        tmp = ((-6.0d0) * x2) + x1
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((x2 * 2.0) <= -5e-193) {
		tmp = -6.0 * x2;
	} else if ((x2 * 2.0) <= 2e-182) {
		tmp = (-2.0 * x1) + x1;
	} else {
		tmp = (-6.0 * x2) + x1;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x2 * 2.0) <= -5e-193:
		tmp = -6.0 * x2
	elif (x2 * 2.0) <= 2e-182:
		tmp = (-2.0 * x1) + x1
	else:
		tmp = (-6.0 * x2) + x1
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(x2 * 2.0) <= -5e-193)
		tmp = Float64(-6.0 * x2);
	elseif (Float64(x2 * 2.0) <= 2e-182)
		tmp = Float64(Float64(-2.0 * x1) + x1);
	else
		tmp = Float64(Float64(-6.0 * x2) + x1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x2 * 2.0) <= -5e-193)
		tmp = -6.0 * x2;
	elseif ((x2 * 2.0) <= 2e-182)
		tmp = (-2.0 * x1) + x1;
	else
		tmp = (-6.0 * x2) + x1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(x2 * 2.0), $MachinePrecision], -5e-193], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[N[(x2 * 2.0), $MachinePrecision], 2e-182], N[(N[(-2.0 * x1), $MachinePrecision] + x1), $MachinePrecision], N[(N[(-6.0 * x2), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -5.0000000000000005e-193

   1. Initial program 65.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 29.9

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.4

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

   8. Applied rewrites 30.4%

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

   if -5.0000000000000005e-193 < (*.f64 #s(literal 2 binary64) x2) < 2.0000000000000001e-182

   1. Initial program 69.8%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + -2 \cdot x1 \]
   10. Step-by-step derivation

   11. Applied rewrites 42.0%

       \[\leadsto x1 + -2 \cdot x1 \]

   if 2.0000000000000001e-182 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 68.9%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 30.0

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

   5. Applied rewrites 30.0%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -5 \cdot 10^{-193}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \cdot 2 \leq 2 \cdot 10^{-182}:\\ \;\;\;\;-2 \cdot x1 + x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2 + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 31.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -5 \cdot 10^{-193}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \cdot 2 \leq 2 \cdot 10^{-182}:\\ \;\;\;\;-2 \cdot x1 + x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* x2 2.0) -5e-193)
   (* -6.0 x2)
   (if (<= (* x2 2.0) 2e-182) (+ (* -2.0 x1) x1) (* -6.0 x2))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x2 * 2.0) <= -5e-193) {
		tmp = -6.0 * x2;
	} else if ((x2 * 2.0) <= 2e-182) {
		tmp = (-2.0 * x1) + x1;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((x2 * 2.0d0) <= (-5d-193)) then
        tmp = (-6.0d0) * x2
    else if ((x2 * 2.0d0) <= 2d-182) then
        tmp = ((-2.0d0) * x1) + x1
    else
        tmp = (-6.0d0) * x2
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((x2 * 2.0) <= -5e-193) {
		tmp = -6.0 * x2;
	} else if ((x2 * 2.0) <= 2e-182) {
		tmp = (-2.0 * x1) + x1;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x2 * 2.0) <= -5e-193:
		tmp = -6.0 * x2
	elif (x2 * 2.0) <= 2e-182:
		tmp = (-2.0 * x1) + x1
	else:
		tmp = -6.0 * x2
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(x2 * 2.0) <= -5e-193)
		tmp = Float64(-6.0 * x2);
	elseif (Float64(x2 * 2.0) <= 2e-182)
		tmp = Float64(Float64(-2.0 * x1) + x1);
	else
		tmp = Float64(-6.0 * x2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x2 * 2.0) <= -5e-193)
		tmp = -6.0 * x2;
	elseif ((x2 * 2.0) <= 2e-182)
		tmp = (-2.0 * x1) + x1;
	else
		tmp = -6.0 * x2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(x2 * 2.0), $MachinePrecision], -5e-193], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[N[(x2 * 2.0), $MachinePrecision], 2e-182], N[(N[(-2.0 * x1), $MachinePrecision] + x1), $MachinePrecision], N[(-6.0 * x2), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -5.0000000000000005e-193 or 2.0000000000000001e-182 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 66.8%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 29.9

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 29.9

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

   8. Applied rewrites 29.9%

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

   if -5.0000000000000005e-193 < (*.f64 #s(literal 2 binary64) x2) < 2.0000000000000001e-182

   1. Initial program 69.8%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + -2 \cdot x1 \]
   10. Step-by-step derivation

   11. Applied rewrites 42.0%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -5 \cdot 10^{-193}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \cdot 2 \leq 2 \cdot 10^{-182}:\\ \;\;\;\;-2 \cdot x1 + x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 53.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(9, x1, -2\right) \cdot x1 + x1\\ \mathbf{if}\;x1 \leq -1.62 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.4 \cdot 10^{-63}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (fma 9.0 x1 -2.0) x1) x1)))
   (if (<= x1 -1.62e-152) t_0 (if (<= x1 2.4e-63) (* -6.0 x2) t_0))))

C

double code(double x1, double x2) {
	double t_0 = (fma(9.0, x1, -2.0) * x1) + x1;
	double tmp;
	if (x1 <= -1.62e-152) {
		tmp = t_0;
	} else if (x1 <= 2.4e-63) {
		tmp = -6.0 * x2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(fma(9.0, x1, -2.0) * x1) + x1)
	tmp = 0.0
	if (x1 <= -1.62e-152)
		tmp = t_0;
	elseif (x1 <= 2.4e-63)
		tmp = Float64(-6.0 * x2);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -1.62e-152], t$95$0, If[LessEqual[x1, 2.4e-63], N[(-6.0 * x2), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.61999999999999995e-152 or 2.4000000000000001e-63 < x1

   1. Initial program 53.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 55.3%

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

   if -1.61999999999999995e-152 < x1 < 2.4000000000000001e-63

   1. Initial program 98.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.3

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

   8. Applied rewrites 68.3%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.62 \cdot 10^{-152}:\\ \;\;\;\;\mathsf{fma}\left(9, x1, -2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 2.4 \cdot 10^{-63}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, x1, -2\right) \cdot x1 + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 25.6% accurate, 49.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

3. Taylor expanded in x1 around 0

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

   1. lower-*.f64 25.1

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

5. Applied rewrites 25.1%

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

6. Taylor expanded in x1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.0

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

8. Applied rewrites 25.0%

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

9. Final simplification 25.0%

   \[\leadsto -6 \cdot x2 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(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))))))