Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 69.9% → 99.0%

Time: 21.3s

Alternatives: 20

Speedup: 7.6×

• 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: 69.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\\ t_2 := -1 - x1 \cdot x1\\ t_3 := x1 \cdot \left(x1 \cdot 3\right)\\ t_4 := \left(t\_3 + 2 \cdot x2\right) - x1\\ t_5 := \frac{t\_4}{t\_0}\\ t_6 := \frac{t\_4}{t\_2}\\ t_7 := x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_6\right) + \left(\left(x1 \cdot 2\right) \cdot t\_5\right) \cdot \left(3 + t\_6\right)\right) \cdot t\_2 + t\_3 \cdot t\_5\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\\ t_8 := x1 + \left(t\_7 + 3 \cdot \frac{\left(t\_3 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ t_9 := \frac{t\_1}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;t\_8 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{t\_3 - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(t\_9, 4, -6\right), \frac{\left(-3 + t\_9\right) \cdot \left(\left(x1 \cdot 2\right) \cdot t\_1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot t\_9, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;t\_8 \leq \infty:\\ \;\;\;\;x1 + \left(t\_7 + 3 \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (* x1 (* x1 3.0)))
        (t_4 (- (+ t_3 (* 2.0 x2)) x1))
        (t_5 (/ t_4 t_0))
        (t_6 (/ t_4 t_2))
        (t_7
         (+
          x1
          (+
           (+
            (*
             (+
              (* (* x1 x1) (+ 6.0 (* 4.0 t_6)))
              (* (* (* x1 2.0) t_5) (+ 3.0 t_6)))
             t_2)
            (* t_3 t_5))
           (* x1 (* x1 x1)))))
        (t_8 (+ x1 (+ t_7 (* 3.0 (/ (- (- t_3 (* 2.0 x2)) x1) t_0)))))
        (t_9 (/ t_1 (fma x1 x1 1.0))))
   (if (<= t_8 5e+262)
     (+
      x1
      (fma
       (/ (- t_3 (fma 2.0 x2 x1)) (fma x1 x1 1.0))
       3.0
       (fma
        (fma x1 x1 1.0)
        (fma
         x1
         (* x1 (fma t_9 4.0 -6.0))
         (/ (* (+ -3.0 t_9) (* (* x1 2.0) t_1)) (fma x1 x1 1.0)))
        (fma x1 (* (* x1 3.0) t_9) (fma x1 (* x1 x1) x1)))))
     (if (<= t_8 INFINITY)
       (+ x1 (+ t_7 (* 3.0 3.0)))
       (+ x1 (* (* x1 x1) (* x1 (* x1 6.0))))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = fma(2.0, x2, fma(x1, (x1 * 3.0), -x1));
	double t_2 = -1.0 - (x1 * x1);
	double t_3 = x1 * (x1 * 3.0);
	double t_4 = (t_3 + (2.0 * x2)) - x1;
	double t_5 = t_4 / t_0;
	double t_6 = t_4 / t_2;
	double t_7 = x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_6))) + (((x1 * 2.0) * t_5) * (3.0 + t_6))) * t_2) + (t_3 * t_5)) + (x1 * (x1 * x1)));
	double t_8 = x1 + (t_7 + (3.0 * (((t_3 - (2.0 * x2)) - x1) / t_0)));
	double t_9 = t_1 / fma(x1, x1, 1.0);
	double tmp;
	if (t_8 <= 5e+262) {
		tmp = x1 + fma(((t_3 - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), fma(x1, (x1 * fma(t_9, 4.0, -6.0)), (((-3.0 + t_9) * ((x1 * 2.0) * t_1)) / fma(x1, x1, 1.0))), fma(x1, ((x1 * 3.0) * t_9), fma(x1, (x1 * x1), x1))));
	} else if (t_8 <= ((double) INFINITY)) {
		tmp = x1 + (t_7 + (3.0 * 3.0));
	} else {
		tmp = x1 + ((x1 * x1) * (x1 * (x1 * 6.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1)))
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(x1 * Float64(x1 * 3.0))
	t_4 = Float64(Float64(t_3 + Float64(2.0 * x2)) - x1)
	t_5 = Float64(t_4 / t_0)
	t_6 = Float64(t_4 / t_2)
	t_7 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_6))) + Float64(Float64(Float64(x1 * 2.0) * t_5) * Float64(3.0 + t_6))) * t_2) + Float64(t_3 * t_5)) + Float64(x1 * Float64(x1 * x1))))
	t_8 = Float64(x1 + Float64(t_7 + Float64(3.0 * Float64(Float64(Float64(t_3 - Float64(2.0 * x2)) - x1) / t_0))))
	t_9 = Float64(t_1 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (t_8 <= 5e+262)
		tmp = Float64(x1 + fma(Float64(Float64(t_3 - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(t_9, 4.0, -6.0)), Float64(Float64(Float64(-3.0 + t_9) * Float64(Float64(x1 * 2.0) * t_1)) / fma(x1, x1, 1.0))), fma(x1, Float64(Float64(x1 * 3.0) * t_9), fma(x1, Float64(x1 * x1), x1)))));
	elseif (t_8 <= Inf)
		tmp = Float64(x1 + Float64(t_7 + Float64(3.0 * 3.0)));
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * Float64(x1 * Float64(x1 * 6.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / t$95$0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 / t$95$2), $MachinePrecision]}, Block[{t$95$7 = N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(3.0 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$3 * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(x1 + N[(t$95$7 + N[(3.0 * N[(N[(N[(t$95$3 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$8, 5e+262], N[(x1 + N[(N[(N[(t$95$3 - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(t$95$9 * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-3.0 + t$95$9), $MachinePrecision] * N[(N[(x1 * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * t$95$9), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$8, Infinity], N[(x1 + N[(t$95$7 + N[(3.0 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 5.00000000000000008e262

   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%

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

   if 5.00000000000000008e262 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.9%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), 8 \cdot x2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+262}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(-3 + \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ t_5 := -1 - x1 \cdot x1\\ t_6 := \frac{t\_2}{t\_5}\\ t_7 := x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_6\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_6\right)\right) \cdot t\_5 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ t_8 := x2 \cdot \left(x2 \cdot \left(x1 \cdot 8\right)\right)\\ \mathbf{if}\;t\_7 \leq -5 \cdot 10^{+263}:\\ \;\;\;\;t\_8\\ \mathbf{elif}\;t\_7 \leq -1 \cdot 10^{-58}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;t\_7 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_7 \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;t\_8\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (* x1 (fma 9.0 x1 -1.0)))
        (t_5 (- -1.0 (* x1 x1)))
        (t_6 (/ t_2 t_5))
        (t_7
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_6)))
                (* (* (* x1 2.0) t_3) (+ 3.0 t_6)))
               t_5)
              (* t_1 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0)))))
        (t_8 (* x2 (* x2 (* x1 8.0)))))
   (if (<= t_7 -5e+263)
     t_8
     (if (<= t_7 -1e-58)
       (* x2 -6.0)
       (if (<= t_7 2e-6)
         t_4
         (if (<= t_7 5e+186) (* x2 -6.0) (if (<= t_7 INFINITY) t_8 t_4)))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = x1 * fma(9.0, x1, -1.0);
	double t_5 = -1.0 - (x1 * x1);
	double t_6 = t_2 / t_5;
	double t_7 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_6))) + (((x1 * 2.0) * t_3) * (3.0 + t_6))) * t_5) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double t_8 = x2 * (x2 * (x1 * 8.0));
	double tmp;
	if (t_7 <= -5e+263) {
		tmp = t_8;
	} else if (t_7 <= -1e-58) {
		tmp = x2 * -6.0;
	} else if (t_7 <= 2e-6) {
		tmp = t_4;
	} else if (t_7 <= 5e+186) {
		tmp = x2 * -6.0;
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = t_8;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(x1 * fma(9.0, x1, -1.0))
	t_5 = Float64(-1.0 - Float64(x1 * x1))
	t_6 = Float64(t_2 / t_5)
	t_7 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_6))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_6))) * t_5) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	t_8 = Float64(x2 * Float64(x2 * Float64(x1 * 8.0)))
	tmp = 0.0
	if (t_7 <= -5e+263)
		tmp = t_8;
	elseif (t_7 <= -1e-58)
		tmp = Float64(x2 * -6.0);
	elseif (t_7 <= 2e-6)
		tmp = t_4;
	elseif (t_7 <= 5e+186)
		tmp = Float64(x2 * -6.0);
	elseif (t_7 <= Inf)
		tmp = t_8;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 / t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(x1 + N[(N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, -5e+263], t$95$8, If[LessEqual[t$95$7, -1e-58], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[t$95$7, 2e-6], t$95$4, If[LessEqual[t$95$7, 5e+186], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[t$95$7, Infinity], t$95$8, t$95$4]]]]]]]]]]]]]]

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)))))) < -5.00000000000000022e263 or 4.99999999999999954e186 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 61.2%

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

   12. Taylor expanded in x2 around inf

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

   14. Applied rewrites 60.0%

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

   if -5.00000000000000022e263 < (+.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)))))) < -1e-58 or 1.99999999999999991e-6 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.99999999999999954e186

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 66.7

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

   8. Applied rewrites 66.7%

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

   if -1e-58 < (+.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)))))) < 1.99999999999999991e-6 or +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 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. *-commutative N/A

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

      2. lower-*.f64 16.3

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

   5. Applied rewrites 16.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 75.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{+263}:\\ \;\;\;\;x2 \cdot \left(x2 \cdot \left(x1 \cdot 8\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -1 \cdot 10^{-58}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x2 \cdot \left(x2 \cdot \left(x1 \cdot 8\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ t_5 := -1 - x1 \cdot x1\\ t_6 := \frac{t\_2}{t\_5}\\ t_7 := x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_6\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_6\right)\right) \cdot t\_5 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ t_8 := 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{if}\;t\_7 \leq -5 \cdot 10^{+263}:\\ \;\;\;\;t\_8\\ \mathbf{elif}\;t\_7 \leq -1 \cdot 10^{-58}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;t\_7 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_7 \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;t\_8\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (* x1 (fma 9.0 x1 -1.0)))
        (t_5 (- -1.0 (* x1 x1)))
        (t_6 (/ t_2 t_5))
        (t_7
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_6)))
                (* (* (* x1 2.0) t_3) (+ 3.0 t_6)))
               t_5)
              (* t_1 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0)))))
        (t_8 (* 8.0 (* x1 (* x2 x2)))))
   (if (<= t_7 -5e+263)
     t_8
     (if (<= t_7 -1e-58)
       (* x2 -6.0)
       (if (<= t_7 2e-6)
         t_4
         (if (<= t_7 5e+186) (* x2 -6.0) (if (<= t_7 INFINITY) t_8 t_4)))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = x1 * fma(9.0, x1, -1.0);
	double t_5 = -1.0 - (x1 * x1);
	double t_6 = t_2 / t_5;
	double t_7 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_6))) + (((x1 * 2.0) * t_3) * (3.0 + t_6))) * t_5) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double t_8 = 8.0 * (x1 * (x2 * x2));
	double tmp;
	if (t_7 <= -5e+263) {
		tmp = t_8;
	} else if (t_7 <= -1e-58) {
		tmp = x2 * -6.0;
	} else if (t_7 <= 2e-6) {
		tmp = t_4;
	} else if (t_7 <= 5e+186) {
		tmp = x2 * -6.0;
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = t_8;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(x1 * fma(9.0, x1, -1.0))
	t_5 = Float64(-1.0 - Float64(x1 * x1))
	t_6 = Float64(t_2 / t_5)
	t_7 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_6))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_6))) * t_5) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	t_8 = Float64(8.0 * Float64(x1 * Float64(x2 * x2)))
	tmp = 0.0
	if (t_7 <= -5e+263)
		tmp = t_8;
	elseif (t_7 <= -1e-58)
		tmp = Float64(x2 * -6.0);
	elseif (t_7 <= 2e-6)
		tmp = t_4;
	elseif (t_7 <= 5e+186)
		tmp = Float64(x2 * -6.0);
	elseif (t_7 <= Inf)
		tmp = t_8;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 / t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(x1 + N[(N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, -5e+263], t$95$8, If[LessEqual[t$95$7, -1e-58], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[t$95$7, 2e-6], t$95$4, If[LessEqual[t$95$7, 5e+186], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[t$95$7, Infinity], t$95$8, t$95$4]]]]]]]]]]]]]]

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)))))) < -5.00000000000000022e263 or 4.99999999999999954e186 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 46.5%

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

   if -5.00000000000000022e263 < (+.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)))))) < -1e-58 or 1.99999999999999991e-6 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.99999999999999954e186

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 66.7

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

   8. Applied rewrites 66.7%

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

   if -1e-58 < (+.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)))))) < 1.99999999999999991e-6 or +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 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. *-commutative N/A

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

      2. lower-*.f64 16.3

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

   5. Applied rewrites 16.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 75.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{+263}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -1 \cdot 10^{-58}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ t_5 := -1 - x1 \cdot x1\\ t_6 := \frac{t\_2}{t\_5}\\ t_7 := x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_6\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_6\right)\right) \cdot t\_5 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ t_8 := x2 \cdot \left(x2 \cdot \left(x1 \cdot 8\right)\right)\\ \mathbf{if}\;t\_7 \leq -5 \cdot 10^{+263}:\\ \;\;\;\;t\_8\\ \mathbf{elif}\;t\_7 \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, t\_4\right)\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;t\_8\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (* x1 (fma 9.0 x1 -1.0)))
        (t_5 (- -1.0 (* x1 x1)))
        (t_6 (/ t_2 t_5))
        (t_7
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_6)))
                (* (* (* x1 2.0) t_3) (+ 3.0 t_6)))
               t_5)
              (* t_1 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0)))))
        (t_8 (* x2 (* x2 (* x1 8.0)))))
   (if (<= t_7 -5e+263)
     t_8
     (if (<= t_7 5e+186) (fma x2 -6.0 t_4) (if (<= t_7 INFINITY) t_8 t_4)))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = x1 * fma(9.0, x1, -1.0);
	double t_5 = -1.0 - (x1 * x1);
	double t_6 = t_2 / t_5;
	double t_7 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_6))) + (((x1 * 2.0) * t_3) * (3.0 + t_6))) * t_5) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double t_8 = x2 * (x2 * (x1 * 8.0));
	double tmp;
	if (t_7 <= -5e+263) {
		tmp = t_8;
	} else if (t_7 <= 5e+186) {
		tmp = fma(x2, -6.0, t_4);
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = t_8;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(x1 * fma(9.0, x1, -1.0))
	t_5 = Float64(-1.0 - Float64(x1 * x1))
	t_6 = Float64(t_2 / t_5)
	t_7 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_6))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_6))) * t_5) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	t_8 = Float64(x2 * Float64(x2 * Float64(x1 * 8.0)))
	tmp = 0.0
	if (t_7 <= -5e+263)
		tmp = t_8;
	elseif (t_7 <= 5e+186)
		tmp = fma(x2, -6.0, t_4);
	elseif (t_7 <= Inf)
		tmp = t_8;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 / t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(x1 + N[(N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, -5e+263], t$95$8, If[LessEqual[t$95$7, 5e+186], N[(x2 * -6.0 + t$95$4), $MachinePrecision], If[LessEqual[t$95$7, Infinity], t$95$8, t$95$4]]]]]]]]]]]]

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)))))) < -5.00000000000000022e263 or 4.99999999999999954e186 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 61.2%

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

   12. Taylor expanded in x2 around inf

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

   14. Applied rewrites 60.0%

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

   if -5.00000000000000022e263 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.99999999999999954e186

   1. Initial program 99.2%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 53.7

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 80.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 89.5%

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

   12. Taylor expanded in x1 around 0

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

   14. Applied rewrites 84.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.9

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

   5. Applied rewrites 3.9%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 72.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 85.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{+263}:\\ \;\;\;\;x2 \cdot \left(x2 \cdot \left(x1 \cdot 8\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x2 \cdot \left(x2 \cdot \left(x1 \cdot 8\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := -1 - x1 \cdot x1\\ t_5 := \frac{t\_2}{t\_4}\\ t_6 := x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_5\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_5\right)\right) \cdot t\_4 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_6 \leq -1 \cdot 10^{-58}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;t\_6 \leq 10^{-25}:\\ \;\;\;\;-x1\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (- -1.0 (* x1 x1)))
        (t_5 (/ t_2 t_4))
        (t_6
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_5)))
                (* (* (* x1 2.0) t_3) (+ 3.0 t_5)))
               t_4)
              (* t_1 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_6 -1e-58)
     (* x2 -6.0)
     (if (<= t_6 1e-25)
       (- x1)
       (if (<= t_6 5e+186) (* x2 -6.0) (* x1 (* x1 9.0)))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = -1.0 - (x1 * x1);
	double t_5 = t_2 / t_4;
	double t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_6 <= -1e-58) {
		tmp = x2 * -6.0;
	} else if (t_6 <= 1e-25) {
		tmp = -x1;
	} else if (t_6 <= 5e+186) {
		tmp = x2 * -6.0;
	} else {
		tmp = x1 * (x1 * 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_0 = (x1 * x1) + 1.0d0
    t_1 = x1 * (x1 * 3.0d0)
    t_2 = (t_1 + (2.0d0 * x2)) - x1
    t_3 = t_2 / t_0
    t_4 = (-1.0d0) - (x1 * x1)
    t_5 = t_2 / t_4
    t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0d0 + (4.0d0 * t_5))) + (((x1 * 2.0d0) * t_3) * (3.0d0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0d0 * (((t_1 - (2.0d0 * x2)) - x1) / t_0)))
    if (t_6 <= (-1d-58)) then
        tmp = x2 * (-6.0d0)
    else if (t_6 <= 1d-25) then
        tmp = -x1
    else if (t_6 <= 5d+186) then
        tmp = x2 * (-6.0d0)
    else
        tmp = x1 * (x1 * 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = -1.0 - (x1 * x1);
	double t_5 = t_2 / t_4;
	double t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_6 <= -1e-58) {
		tmp = x2 * -6.0;
	} else if (t_6 <= 1e-25) {
		tmp = -x1;
	} else if (t_6 <= 5e+186) {
		tmp = x2 * -6.0;
	} else {
		tmp = x1 * (x1 * 9.0);
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * x1) + 1.0
	t_1 = x1 * (x1 * 3.0)
	t_2 = (t_1 + (2.0 * x2)) - x1
	t_3 = t_2 / t_0
	t_4 = -1.0 - (x1 * x1)
	t_5 = t_2 / t_4
	t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)))
	tmp = 0
	if t_6 <= -1e-58:
		tmp = x2 * -6.0
	elif t_6 <= 1e-25:
		tmp = -x1
	elif t_6 <= 5e+186:
		tmp = x2 * -6.0
	else:
		tmp = x1 * (x1 * 9.0)
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(-1.0 - Float64(x1 * x1))
	t_5 = Float64(t_2 / t_4)
	t_6 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_5))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_5))) * t_4) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_6 <= -1e-58)
		tmp = Float64(x2 * -6.0);
	elseif (t_6 <= 1e-25)
		tmp = Float64(-x1);
	elseif (t_6 <= 5e+186)
		tmp = Float64(x2 * -6.0);
	else
		tmp = Float64(x1 * Float64(x1 * 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * x1) + 1.0;
	t_1 = x1 * (x1 * 3.0);
	t_2 = (t_1 + (2.0 * x2)) - x1;
	t_3 = t_2 / t_0;
	t_4 = -1.0 - (x1 * x1);
	t_5 = t_2 / t_4;
	t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	tmp = 0.0;
	if (t_6 <= -1e-58)
		tmp = x2 * -6.0;
	elseif (t_6 <= 1e-25)
		tmp = -x1;
	elseif (t_6 <= 5e+186)
		tmp = x2 * -6.0;
	else
		tmp = x1 * (x1 * 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 + N[(N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -1e-58], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[t$95$6, 1e-25], (-x1), If[LessEqual[t$95$6, 5e+186], N[(x2 * -6.0), $MachinePrecision], N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1e-58 or 1.00000000000000004e-25 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.99999999999999954e186

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 54.7

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

   8. Applied rewrites 54.7%

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

   if -1e-58 < (+.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)))))) < 1.00000000000000004e-25

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 35.5

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 61.3%

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

   12. Taylor expanded in x1 around 0

       \[\leadsto -1 \cdot x1 \]
   13. Step-by-step derivation

   14. Applied rewrites 61.3%

       \[\leadsto -x1 \]

   if 4.99999999999999954e186 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 37.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 55.6%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 55.7%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -1 \cdot 10^{-58}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{-25}:\\ \;\;\;\;-x1\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := -1 - x1 \cdot x1\\ t_5 := \frac{t\_2}{t\_4}\\ t_6 := x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_5\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_5\right)\right) \cdot t\_4 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\\ t_7 := x1 + \left(t\_6 + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_7 \leq 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;x1 + \left(t\_6 + 3 \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (- -1.0 (* x1 x1)))
        (t_5 (/ t_2 t_4))
        (t_6
         (+
          x1
          (+
           (+
            (*
             (+
              (* (* x1 x1) (+ 6.0 (* 4.0 t_5)))
              (* (* (* x1 2.0) t_3) (+ 3.0 t_5)))
             t_4)
            (* t_1 t_3))
           (* x1 (* x1 x1)))))
        (t_7 (+ x1 (+ t_6 (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_7 1e+22)
     (fma
      x2
      (fma x1 (fma 12.0 x1 (fma x2 8.0 -12.0)) -6.0)
      (* x1 (fma 9.0 x1 -1.0)))
     (if (<= t_7 INFINITY)
       (+ x1 (+ t_6 (* 3.0 3.0)))
       (+ x1 (* (* x1 x1) (* x1 (* x1 6.0))))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = -1.0 - (x1 * x1);
	double t_5 = t_2 / t_4;
	double t_6 = x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)));
	double t_7 = x1 + (t_6 + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_7 <= 1e+22) {
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), (x1 * fma(9.0, x1, -1.0)));
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = x1 + (t_6 + (3.0 * 3.0));
	} else {
		tmp = x1 + ((x1 * x1) * (x1 * (x1 * 6.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(-1.0 - Float64(x1 * x1))
	t_5 = Float64(t_2 / t_4)
	t_6 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_5))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_5))) * t_4) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1))))
	t_7 = Float64(x1 + Float64(t_6 + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_7 <= 1e+22)
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), Float64(x1 * fma(9.0, x1, -1.0)));
	elseif (t_7 <= Inf)
		tmp = Float64(x1 + Float64(t_6 + Float64(3.0 * 3.0)));
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * Float64(x1 * Float64(x1 * 6.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(x1 + N[(t$95$6 + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 1e+22], N[(x2 * N[(x1 * N[(12.0 * x1 + N[(x2 * 8.0 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, Infinity], N[(x1 + N[(t$95$6 + N[(3.0 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1e22

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 99.6%

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

   if 1e22 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 88.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), 8 \cdot x2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 100.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{t\_0}\\ t_4 := -1 - x1 \cdot x1\\ t_5 := \frac{t\_2}{t\_4}\\ t_6 := x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_5\right) + \left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(3 + t\_5\right)\right) \cdot t\_4 + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_6 \leq -5 \cdot 10^{+263}:\\ \;\;\;\;x2 \cdot \left(x2 \cdot \left(x1 \cdot 8\right)\right)\\ \mathbf{elif}\;t\_6 \leq 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 t_0))
        (t_4 (- -1.0 (* x1 x1)))
        (t_5 (/ t_2 t_4))
        (t_6
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_5)))
                (* (* (* x1 2.0) t_3) (+ 3.0 t_5)))
               t_4)
              (* t_1 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_6 -5e+263)
     (* x2 (* x2 (* x1 8.0)))
     (if (<= t_6 1e+78)
       (fma x2 -6.0 (* x1 (fma 9.0 x1 -1.0)))
       (+ x1 (* (* x1 x1) (* x1 (* x1 6.0))))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / t_0;
	double t_4 = -1.0 - (x1 * x1);
	double t_5 = t_2 / t_4;
	double t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_6 <= -5e+263) {
		tmp = x2 * (x2 * (x1 * 8.0));
	} else if (t_6 <= 1e+78) {
		tmp = fma(x2, -6.0, (x1 * fma(9.0, x1, -1.0)));
	} else {
		tmp = x1 + ((x1 * x1) * (x1 * (x1 * 6.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / t_0)
	t_4 = Float64(-1.0 - Float64(x1 * x1))
	t_5 = Float64(t_2 / t_4)
	t_6 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_5))) + Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(3.0 + t_5))) * t_4) + Float64(t_1 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_6 <= -5e+263)
		tmp = Float64(x2 * Float64(x2 * Float64(x1 * 8.0)));
	elseif (t_6 <= 1e+78)
		tmp = fma(x2, -6.0, Float64(x1 * fma(9.0, x1, -1.0)));
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * Float64(x1 * Float64(x1 * 6.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 + N[(N[(x1 + N[(N[(N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(3.0 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -5e+263], N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1e+78], N[(x2 * -6.0 + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.00000000000000022e263

   1. Initial program 100.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x2 around inf

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

   14. Applied rewrites 100.0%

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

   if -5.00000000000000022e263 < (+.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)))))) < 1.00000000000000001e78

   1. Initial program 99.2%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 96.8%

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

   12. Taylor expanded in x1 around 0

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

   14. Applied rewrites 94.9%

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

   if 1.00000000000000001e78 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 46.0%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 81.5%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), 8 \cdot x2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 81.6%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 80.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{+263}:\\ \;\;\;\;x2 \cdot \left(x2 \cdot \left(x1 \cdot 8\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x1, x2):
	t_0 = (x1 * x1) + 1.0
	t_1 = x1 * (x1 * 3.0)
	t_2 = (t_1 + (2.0 * x2)) - x1
	t_3 = t_2 / t_0
	t_4 = -1.0 - (x1 * x1)
	t_5 = t_2 / t_4
	t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)))
	tmp = 0
	if t_6 <= math.inf:
		tmp = t_6
	else:
		tmp = x1 + ((x1 * x1) * (x1 * (x1 * 6.0)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * x1) + 1.0;
	t_1 = x1 * (x1 * 3.0);
	t_2 = (t_1 + (2.0 * x2)) - x1;
	t_3 = t_2 / t_0;
	t_4 = -1.0 - (x1 * x1);
	t_5 = t_2 / t_4;
	t_6 = x1 + ((x1 + ((((((x1 * x1) * (6.0 + (4.0 * t_5))) + (((x1 * 2.0) * t_3) * (3.0 + t_5))) * t_4) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	tmp = 0.0;
	if (t_6 <= Inf)
		tmp = t_6;
	else
		tmp = x1 + ((x1 * x1) * (x1 * (x1 * 6.0)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), 8 \cdot x2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 100.0%

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

4. Final simplification 99.6%

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

Alternative 9: 95.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x2, 2, -3\right), \frac{4}{x1 \cdot x1}, \frac{9}{x1 \cdot x1}\right) + \left(6 + \frac{-3}{x1}\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -31000.0)
   (+
    x1
    (*
     (pow x1 4.0)
     (+
      (fma (fma x2 2.0 -3.0) (/ 4.0 (* x1 x1)) (/ 9.0 (* x1 x1)))
      (+ 6.0 (/ -3.0 x1)))))
   (if (<= x1 4.8e+21)
     (fma
      x2
      (fma x1 (fma 12.0 x1 (fma x2 8.0 -12.0)) -6.0)
      (* x1 (fma 9.0 x1 -1.0)))
     (+
      x1
      (*
       (pow x1 4.0)
       (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -31000.0) {
		tmp = x1 + (pow(x1, 4.0) * (fma(fma(x2, 2.0, -3.0), (4.0 / (x1 * x1)), (9.0 / (x1 * x1))) + (6.0 + (-3.0 / x1))));
	} else if (x1 <= 4.8e+21) {
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), (x1 * fma(9.0, x1, -1.0)));
	} else {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -31000.0)
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(fma(fma(x2, 2.0, -3.0), Float64(4.0 / Float64(x1 * x1)), Float64(9.0 / Float64(x1 * x1))) + Float64(6.0 + Float64(-3.0 / x1)))));
	elseif (x1 <= 4.8e+21)
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), Float64(x1 * fma(9.0, x1, -1.0)));
	else
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -31000.0], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(N[(N[(x2 * 2.0 + -3.0), $MachinePrecision] * N[(4.0 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(9.0 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 + N[(-3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.8e+21], N[(x2 * N[(x1 * N[(12.0 * x1 + N[(x2 * 8.0 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -31000

   1. Initial program 37.3%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 93.6%

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

   if -31000 < x1 < 4.8e21

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.5%

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

   if 4.8e21 < x1

   1. Initial program 41.1%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 95.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x2, 2, -3\right), \frac{4}{x1 \cdot x1}, \frac{9}{x1 \cdot x1}\right) + \left(6 + \frac{-3}{x1}\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(\frac{1}{x1 \cdot x1}, \mathsf{fma}\left(x2, 8, -3\right), 6\right)\right), x1 \cdot x1, -3 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -31000.0)
   (+
    x1
    (fma
     (* x1 (* x1 (fma (/ 1.0 (* x1 x1)) (fma x2 8.0 -3.0) 6.0)))
     (* x1 x1)
     (* -3.0 (* x1 (* x1 x1)))))
   (if (<= x1 4.8e+21)
     (fma
      x2
      (fma x1 (fma 12.0 x1 (fma x2 8.0 -12.0)) -6.0)
      (* x1 (fma 9.0 x1 -1.0)))
     (+
      x1
      (*
       (pow x1 4.0)
       (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -31000.0) {
		tmp = x1 + fma((x1 * (x1 * fma((1.0 / (x1 * x1)), fma(x2, 8.0, -3.0), 6.0))), (x1 * x1), (-3.0 * (x1 * (x1 * x1))));
	} else if (x1 <= 4.8e+21) {
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), (x1 * fma(9.0, x1, -1.0)));
	} else {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -31000.0)
		tmp = Float64(x1 + fma(Float64(x1 * Float64(x1 * fma(Float64(1.0 / Float64(x1 * x1)), fma(x2, 8.0, -3.0), 6.0))), Float64(x1 * x1), Float64(-3.0 * Float64(x1 * Float64(x1 * x1)))));
	elseif (x1 <= 4.8e+21)
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), Float64(x1 * fma(9.0, x1, -1.0)));
	else
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -31000.0], N[(x1 + N[(N[(x1 * N[(x1 * N[(N[(1.0 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x2 * 8.0 + -3.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(-3.0 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.8e+21], N[(x2 * N[(x1 * N[(12.0 * x1 + N[(x2 * 8.0 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -31000

   1. Initial program 37.3%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 93.6%

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

   7. Applied rewrites 92.0%

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

   9. Applied rewrites 92.0%

      \[\leadsto x1 + \mathsf{fma}\left(\left(x1 \cdot \left(6 + \left(\frac{\mathsf{fma}\left(x2, 8, -12\right)}{x1 \cdot x1} + \frac{9}{x1 \cdot x1}\right)\right)\right) \cdot \left(x1 \cdot x1\right), \color{blue}{x1}, \frac{x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)}{x1 \cdot -0.3333333333333333}\right) \]

   11. Applied rewrites 93.6%

       \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(\frac{1}{x1 \cdot x1}, \mathsf{fma}\left(x2, 8, -3\right), 6\right)\right), \color{blue}{x1 \cdot x1}, 1 \cdot \left(\left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot -3\right)\right) \]

   if -31000 < x1 < 4.8e21

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.5%

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

   if 4.8e21 < x1

   1. Initial program 41.1%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 95.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(\frac{1}{x1 \cdot x1}, \mathsf{fma}\left(x2, 8, -3\right), 6\right)\right), x1 \cdot x1, -3 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(\frac{1}{x1 \cdot x1}, \mathsf{fma}\left(x2, 8, -3\right), 6\right)\right), x1 \cdot x1, -3 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), x2 \cdot 8\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -31000.0)
   (+
    x1
    (fma
     (* x1 (* x1 (fma (/ 1.0 (* x1 x1)) (fma x2 8.0 -3.0) 6.0)))
     (* x1 x1)
     (* -3.0 (* x1 (* x1 x1)))))
   (if (<= x1 4.8e+21)
     (fma
      x2
      (fma x1 (fma 12.0 x1 (fma x2 8.0 -12.0)) -6.0)
      (* x1 (fma 9.0 x1 -1.0)))
     (+ x1 (* (* x1 x1) (fma x1 (fma x1 6.0 -3.0) (* x2 8.0)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -31000.0) {
		tmp = x1 + fma((x1 * (x1 * fma((1.0 / (x1 * x1)), fma(x2, 8.0, -3.0), 6.0))), (x1 * x1), (-3.0 * (x1 * (x1 * x1))));
	} else if (x1 <= 4.8e+21) {
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), (x1 * fma(9.0, x1, -1.0)));
	} else {
		tmp = x1 + ((x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), (x2 * 8.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -31000.0)
		tmp = Float64(x1 + fma(Float64(x1 * Float64(x1 * fma(Float64(1.0 / Float64(x1 * x1)), fma(x2, 8.0, -3.0), 6.0))), Float64(x1 * x1), Float64(-3.0 * Float64(x1 * Float64(x1 * x1)))));
	elseif (x1 <= 4.8e+21)
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), Float64(x1 * fma(9.0, x1, -1.0)));
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), Float64(x2 * 8.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -31000.0], N[(x1 + N[(N[(x1 * N[(x1 * N[(N[(1.0 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x2 * 8.0 + -3.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(-3.0 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.8e+21], N[(x2 * N[(x1 * N[(12.0 * x1 + N[(x2 * 8.0 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -31000

   1. Initial program 37.3%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 93.6%

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

   7. Applied rewrites 92.0%

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

   9. Applied rewrites 92.0%

      \[\leadsto x1 + \mathsf{fma}\left(\left(x1 \cdot \left(6 + \left(\frac{\mathsf{fma}\left(x2, 8, -12\right)}{x1 \cdot x1} + \frac{9}{x1 \cdot x1}\right)\right)\right) \cdot \left(x1 \cdot x1\right), \color{blue}{x1}, \frac{x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)}{x1 \cdot -0.3333333333333333}\right) \]

   11. Applied rewrites 93.6%

       \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(\frac{1}{x1 \cdot x1}, \mathsf{fma}\left(x2, 8, -3\right), 6\right)\right), \color{blue}{x1 \cdot x1}, 1 \cdot \left(\left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot -3\right)\right) \]

   if -31000 < x1 < 4.8e21

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.5%

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

   if 4.8e21 < x1

   1. Initial program 41.1%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 95.6%

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

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), 8 \cdot x2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 95.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(\frac{1}{x1 \cdot x1}, \mathsf{fma}\left(x2, 8, -3\right), 6\right)\right), x1 \cdot x1, -3 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), x2 \cdot 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -1\right)\right)\\ \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(6, x1, -3\right), -3\right)\right)\\ \mathbf{elif}\;x1 \leq -1.62 \cdot 10^{-174}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.3 \cdot 10^{-209}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x1 + 6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma x2 -6.0 (* x1 (fma x2 (fma x2 8.0 -12.0) -1.0)))))
   (if (<= x1 -31000.0)
     (+ x1 (* x1 (* x1 (fma x1 (fma 6.0 x1 -3.0) -3.0))))
     (if (<= x1 -1.62e-174)
       t_0
       (if (<= x1 1.3e-209)
         (fma x2 -6.0 (* x1 (fma 9.0 x1 -1.0)))
         (if (<= x1 4.8e+21) t_0 (+ x1 (* 6.0 (* x1 (* x1 (* x1 x1)))))))))))

C

double code(double x1, double x2) {
	double t_0 = fma(x2, -6.0, (x1 * fma(x2, fma(x2, 8.0, -12.0), -1.0)));
	double tmp;
	if (x1 <= -31000.0) {
		tmp = x1 + (x1 * (x1 * fma(x1, fma(6.0, x1, -3.0), -3.0)));
	} else if (x1 <= -1.62e-174) {
		tmp = t_0;
	} else if (x1 <= 1.3e-209) {
		tmp = fma(x2, -6.0, (x1 * fma(9.0, x1, -1.0)));
	} else if (x1 <= 4.8e+21) {
		tmp = t_0;
	} else {
		tmp = x1 + (6.0 * (x1 * (x1 * (x1 * x1))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(x2, -6.0, Float64(x1 * fma(x2, fma(x2, 8.0, -12.0), -1.0)))
	tmp = 0.0
	if (x1 <= -31000.0)
		tmp = Float64(x1 + Float64(x1 * Float64(x1 * fma(x1, fma(6.0, x1, -3.0), -3.0))));
	elseif (x1 <= -1.62e-174)
		tmp = t_0;
	elseif (x1 <= 1.3e-209)
		tmp = fma(x2, -6.0, Float64(x1 * fma(9.0, x1, -1.0)));
	elseif (x1 <= 4.8e+21)
		tmp = t_0;
	else
		tmp = Float64(x1 + Float64(6.0 * Float64(x1 * Float64(x1 * Float64(x1 * x1)))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x2 * -6.0 + N[(x1 * N[(x2 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -31000.0], N[(x1 + N[(x1 * N[(x1 * N[(x1 * N[(6.0 * x1 + -3.0), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.62e-174], t$95$0, If[LessEqual[x1, 1.3e-209], N[(x2 * -6.0 + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.8e+21], t$95$0, N[(x1 + N[(6.0 * N[(x1 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -31000

   1. Initial program 37.3%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 93.6%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 90.7%

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

   if -31000 < x1 < -1.6200000000000001e-174 or 1.29999999999999992e-209 < x1 < 4.8e21

   1. Initial program 99.2%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 34.6

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

   5. Applied rewrites 34.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 87.8%

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

   if -1.6200000000000001e-174 < x1 < 1.29999999999999992e-209

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 99.9%

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

   12. Taylor expanded in x1 around 0

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

   14. Applied rewrites 90.7%

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

   if 4.8e21 < x1

   1. Initial program 41.1%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 94.8

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 94.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(6, x1, -3\right), -3\right)\right)\\ \mathbf{elif}\;x1 \leq -1.62 \cdot 10^{-174}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -1\right)\right)\\ \mathbf{elif}\;x1 \leq 1.3 \cdot 10^{-209}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + 6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), x2 \cdot 8\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -31000.0)
   (+
    x1
    (* (* x1 x1) (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))))
   (if (<= x1 4.8e+21)
     (fma
      x2
      (fma x1 (fma 12.0 x1 (fma x2 8.0 -12.0)) -6.0)
      (* x1 (fma 9.0 x1 -1.0)))
     (+ x1 (* (* x1 x1) (fma x1 (fma x1 6.0 -3.0) (* x2 8.0)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -31000.0) {
		tmp = x1 + ((x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)));
	} else if (x1 <= 4.8e+21) {
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), (x1 * fma(9.0, x1, -1.0)));
	} else {
		tmp = x1 + ((x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), (x2 * 8.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -31000.0)
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0))));
	elseif (x1 <= 4.8e+21)
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), Float64(x1 * fma(9.0, x1, -1.0)));
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), Float64(x2 * 8.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -31000.0], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.8e+21], N[(x2 * N[(x1 * N[(12.0 * x1 + N[(x2 * 8.0 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -31000

   1. Initial program 37.3%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 93.6%

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

   if -31000 < x1 < 4.8e21

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.5%

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

   if 4.8e21 < x1

   1. Initial program 41.1%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 95.6%

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

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), 8 \cdot x2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 95.6%

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

Alternative 14: 95.5% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), x2 \cdot 8\right)\\ \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (* (* x1 x1) (fma x1 (fma x1 6.0 -3.0) (* x2 8.0))))))
   (if (<= x1 -31000.0)
     t_0
     (if (<= x1 4.8e+21)
       (fma
        x2
        (fma x1 (fma 12.0 x1 (fma x2 8.0 -12.0)) -6.0)
        (* x1 (fma 9.0 x1 -1.0)))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + ((x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), (x2 * 8.0)));
	double tmp;
	if (x1 <= -31000.0) {
		tmp = t_0;
	} else if (x1 <= 4.8e+21) {
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), (x1 * fma(9.0, x1, -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64(Float64(x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), Float64(x2 * 8.0))))
	tmp = 0.0
	if (x1 <= -31000.0)
		tmp = t_0;
	elseif (x1 <= 4.8e+21)
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), Float64(x1 * fma(9.0, x1, -1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -31000.0], t$95$0, If[LessEqual[x1, 4.8e+21], N[(x2 * N[(x1 * N[(12.0 * x1 + N[(x2 * 8.0 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -31000 or 4.8e21 < x1

   1. Initial program 39.2%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 94.6%

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

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), 8 \cdot x2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.5%

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

   if -31000 < x1 < 4.8e21

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.5%

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

Alternative 15: 95.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), x2 \cdot 8\right)\\ \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), -x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (* (* x1 x1) (fma x1 (fma x1 6.0 -3.0) (* x2 8.0))))))
   (if (<= x1 -31000.0)
     t_0
     (if (<= x1 4.8e+21)
       (fma x2 (fma x1 (fma 12.0 x1 (fma x2 8.0 -12.0)) -6.0) (- x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + ((x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), (x2 * 8.0)));
	double tmp;
	if (x1 <= -31000.0) {
		tmp = t_0;
	} else if (x1 <= 4.8e+21) {
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), -x1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64(Float64(x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), Float64(x2 * 8.0))))
	tmp = 0.0
	if (x1 <= -31000.0)
		tmp = t_0;
	elseif (x1 <= 4.8e+21)
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), Float64(-x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -31000.0], t$95$0, If[LessEqual[x1, 4.8e+21], N[(x2 * N[(x1 * N[(12.0 * x1 + N[(x2 * 8.0 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] + (-x1)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -31000 or 4.8e21 < x1

   1. Initial program 39.2%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 94.6%

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

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), 8 \cdot x2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.5%

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

   if -31000 < x1 < 4.8e21

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.5%

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

   12. Taylor expanded in x1 around 0

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

   14. Applied rewrites 96.7%

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

Alternative 16: 92.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(6, x1, -3\right), -3\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), -x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + 6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -31000.0)
   (+ x1 (* x1 (* x1 (fma x1 (fma 6.0 x1 -3.0) -3.0))))
   (if (<= x1 4.8e+21)
     (fma x2 (fma x1 (fma 12.0 x1 (fma x2 8.0 -12.0)) -6.0) (- x1))
     (+ x1 (* 6.0 (* x1 (* x1 (* x1 x1))))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -31000.0) {
		tmp = x1 + (x1 * (x1 * fma(x1, fma(6.0, x1, -3.0), -3.0)));
	} else if (x1 <= 4.8e+21) {
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), -x1);
	} else {
		tmp = x1 + (6.0 * (x1 * (x1 * (x1 * x1))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -31000.0)
		tmp = Float64(x1 + Float64(x1 * Float64(x1 * fma(x1, fma(6.0, x1, -3.0), -3.0))));
	elseif (x1 <= 4.8e+21)
		tmp = fma(x2, fma(x1, fma(12.0, x1, fma(x2, 8.0, -12.0)), -6.0), Float64(-x1));
	else
		tmp = Float64(x1 + Float64(6.0 * Float64(x1 * Float64(x1 * Float64(x1 * x1)))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -31000.0], N[(x1 + N[(x1 * N[(x1 * N[(x1 * N[(6.0 * x1 + -3.0), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.8e+21], N[(x2 * N[(x1 * N[(12.0 * x1 + N[(x2 * 8.0 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] + (-x1)), $MachinePrecision], N[(x1 + N[(6.0 * N[(x1 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -31000

   1. Initial program 37.3%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 93.6%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 90.7%

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

   if -31000 < x1 < 4.8e21

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.5%

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

   12. Taylor expanded in x1 around 0

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

   14. Applied rewrites 96.7%

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

   if 4.8e21 < x1

   1. Initial program 41.1%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 94.8

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 94.8%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(6, x1, -3\right), -3\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, \mathsf{fma}\left(x2, 8, -12\right)\right), -6\right), -x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + 6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -5 \cdot 10^{-103}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;2 \cdot x2 \leq 2 \cdot 10^{-62}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;x1 + x2 \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* 2.0 x2) -5e-103)
   (* x2 -6.0)
   (if (<= (* 2.0 x2) 2e-62) (- x1) (+ x1 (* x2 -6.0)))))

C

double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -5e-103) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 2e-62) {
		tmp = -x1;
	} else {
		tmp = x1 + (x2 * -6.0);
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((2.0d0 * x2) <= (-5d-103)) then
        tmp = x2 * (-6.0d0)
    else if ((2.0d0 * x2) <= 2d-62) then
        tmp = -x1
    else
        tmp = x1 + (x2 * (-6.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -5e-103) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 2e-62) {
		tmp = -x1;
	} else {
		tmp = x1 + (x2 * -6.0);
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (2.0 * x2) <= -5e-103:
		tmp = x2 * -6.0
	elif (2.0 * x2) <= 2e-62:
		tmp = -x1
	else:
		tmp = x1 + (x2 * -6.0)
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(2.0 * x2) <= -5e-103)
		tmp = Float64(x2 * -6.0);
	elseif (Float64(2.0 * x2) <= 2e-62)
		tmp = Float64(-x1);
	else
		tmp = Float64(x1 + Float64(x2 * -6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((2.0 * x2) <= -5e-103)
		tmp = x2 * -6.0;
	elseif ((2.0 * x2) <= 2e-62)
		tmp = -x1;
	else
		tmp = x1 + (x2 * -6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(2.0 * x2), $MachinePrecision], -5e-103], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[N[(2.0 * x2), $MachinePrecision], 2e-62], (-x1), N[(x1 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -4.99999999999999966e-103

   1. Initial program 67.8%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 29.5

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 29.9

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

   8. Applied rewrites 29.9%

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

   if -4.99999999999999966e-103 < (*.f64 #s(literal 2 binary64) x2) < 2.0000000000000001e-62

   1. Initial program 61.2%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 12.1

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

   5. Applied rewrites 12.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 66.9%

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

   12. Taylor expanded in x1 around 0

       \[\leadsto -1 \cdot x1 \]
   13. Step-by-step derivation

   14. Applied rewrites 33.0%

       \[\leadsto -x1 \]

   if 2.0000000000000001e-62 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 81.5%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 43.0

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

   5. Applied rewrites 43.0%

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

Alternative 18: 30.1% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -5 \cdot 10^{-103}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;2 \cdot x2 \leq 2 \cdot 10^{-62}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;x2 \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* 2.0 x2) -5e-103)
   (* x2 -6.0)
   (if (<= (* 2.0 x2) 2e-62) (- x1) (* x2 -6.0))))

C

double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -5e-103) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 2e-62) {
		tmp = -x1;
	} else {
		tmp = x2 * -6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((2.0d0 * x2) <= (-5d-103)) then
        tmp = x2 * (-6.0d0)
    else if ((2.0d0 * x2) <= 2d-62) then
        tmp = -x1
    else
        tmp = x2 * (-6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -5e-103) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 2e-62) {
		tmp = -x1;
	} else {
		tmp = x2 * -6.0;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (2.0 * x2) <= -5e-103:
		tmp = x2 * -6.0
	elif (2.0 * x2) <= 2e-62:
		tmp = -x1
	else:
		tmp = x2 * -6.0
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(2.0 * x2) <= -5e-103)
		tmp = Float64(x2 * -6.0);
	elseif (Float64(2.0 * x2) <= 2e-62)
		tmp = Float64(-x1);
	else
		tmp = Float64(x2 * -6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((2.0 * x2) <= -5e-103)
		tmp = x2 * -6.0;
	elseif ((2.0 * x2) <= 2e-62)
		tmp = -x1;
	else
		tmp = x2 * -6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(2.0 * x2), $MachinePrecision], -5e-103], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[N[(2.0 * x2), $MachinePrecision], 2e-62], (-x1), N[(x2 * -6.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -4.99999999999999966e-103 or 2.0000000000000001e-62 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 75.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 36.9

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

   5. Applied rewrites 36.9%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 36.8

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

   8. Applied rewrites 36.8%

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

   if -4.99999999999999966e-103 < (*.f64 #s(literal 2 binary64) x2) < 2.0000000000000001e-62

   1. Initial program 61.2%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 12.1

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

   5. Applied rewrites 12.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 66.9%

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

   12. Taylor expanded in x1 around 0

       \[\leadsto -1 \cdot x1 \]
   13. Step-by-step derivation

   14. Applied rewrites 33.0%

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

Alternative 19: 54.1% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ \mathbf{if}\;x1 \leq -1.75 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 5.6 \cdot 10^{-86}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (fma 9.0 x1 -1.0))))
   (if (<= x1 -1.75e-48) t_0 (if (<= x1 5.6e-86) (* x2 -6.0) t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma(9.0, x1, -1.0);
	double tmp;
	if (x1 <= -1.75e-48) {
		tmp = t_0;
	} else if (x1 <= 5.6e-86) {
		tmp = x2 * -6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(9.0, x1, -1.0))
	tmp = 0.0
	if (x1 <= -1.75e-48)
		tmp = t_0;
	elseif (x1 <= 5.6e-86)
		tmp = Float64(x2 * -6.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.75e-48], t$95$0, If[LessEqual[x1, 5.6e-86], N[(x2 * -6.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.74999999999999996e-48 or 5.60000000000000019e-86 < x1

   1. Initial program 50.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.8

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

   5. Applied rewrites 3.8%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 54.1%

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

   if -1.74999999999999996e-48 < x1 < 5.60000000000000019e-86

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.8

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

   8. Applied rewrites 62.8%

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

Alternative 20: 13.7% accurate, 99.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x1, x2):
	return -x1

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

3. Taylor expanded in x1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 26.7

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

5. Applied rewrites 26.7%

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

6. Taylor expanded in x1 around 0

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

8. Applied rewrites 70.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \left(\mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right) + -6\right), \mathsf{fma}\left(\mathsf{fma}\left(4, x2 \cdot 2, -12\right), x2, -1\right)\right), x2 \cdot -6\right)} \]

9. Taylor expanded in x2 around 0

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

11. Applied rewrites 41.1%

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

12. Taylor expanded in x1 around 0

    \[\leadsto -1 \cdot x1 \]
13. Step-by-step derivation

14. Applied rewrites 15.1%

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

Reproduce

herbie shell --seed 2024219 
(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))))))