Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 69.9% → 99.3%

Time: 22.0s

Alternatives: 21

Speedup: 7.8×

• 47.1% 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: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around inf

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.5

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

   8. Simplified 99.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 100.0

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

   5. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

         \[\leadsto 6 \cdot {x1}^{\color{blue}{\left(2 + 2\right)}} + x1 \]

      3. pow-prod-up N/A

         \[\leadsto 6 \cdot \color{blue}{\left({x1}^{2} \cdot {x1}^{2}\right)} + x1 \]

      4. pow2 N/A

         \[\leadsto 6 \cdot \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot {x1}^{2}\right) + x1 \]

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.6%

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

Alternative 2: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot \left(x2 \cdot \left(x2 \cdot 8\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{+255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{+251}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -2\right), x2 \cdot -6\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;x1 + t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* x1 (* x2 (* x2 8.0))))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_0 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -4e+255)
     t_1
     (if (<= t_4 1e+251)
       (+ x1 (fma x1 (fma x1 9.0 -2.0) (* x2 -6.0)))
       (if (<= t_4 INFINITY) (+ x1 t_1) (+ x1 (* x1 (fma x1 9.0 -2.0))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x1 * (x2 * (x2 * 8.0));
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_0 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -4e+255) {
		tmp = t_1;
	} else if (t_4 <= 1e+251) {
		tmp = x1 + fma(x1, fma(x1, 9.0, -2.0), (x2 * -6.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = x1 + t_1;
	} else {
		tmp = x1 + (x1 * fma(x1, 9.0, -2.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x1 * Float64(x2 * Float64(x2 * 8.0)))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_0 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -4e+255)
		tmp = t_1;
	elseif (t_4 <= 1e+251)
		tmp = Float64(x1 + fma(x1, fma(x1, 9.0, -2.0), Float64(x2 * -6.0)));
	elseif (t_4 <= Inf)
		tmp = Float64(x1 + t_1);
	else
		tmp = Float64(x1 + Float64(x1 * fma(x1, 9.0, -2.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x2 * N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e+255], t$95$1, If[LessEqual[t$95$4, 1e+251], N[(x1 + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(x1 + t$95$1), $MachinePrecision], N[(x1 + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -3.99999999999999995e255

   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 inf

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x1 around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 63.0

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

   11. Simplified 63.0%

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

   12. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 63.0

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

   14. Simplified 63.0%

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

   if -3.99999999999999995e255 < (+.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)))))) < 1e251

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 75.4%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 73.7

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

   8. Simplified 73.7%

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

   if 1e251 < (+.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 0

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

   5. Simplified 46.8%

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

   6. Taylor expanded in x2 around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 45.2

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

   8. Simplified 45.2%

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

   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

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

   5. Simplified 66.6%

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

   6. Taylor expanded in x2 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 91.0

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

   8. Simplified 91.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -4 \cdot 10^{+255}:\\ \;\;\;\;x1 \cdot \left(x2 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+251}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -2\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x1 + x1 \cdot \left(x2 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot \left(x2 \cdot \left(x2 \cdot 8\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{+255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{+251}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), -x1\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;x1 + t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* x1 (* x2 (* x2 8.0))))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_0 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -4e+255)
     t_1
     (if (<= t_4 1e+251)
       (fma x2 (fma x1 -12.0 -6.0) (- x1))
       (if (<= t_4 INFINITY) (+ x1 t_1) (+ x1 (* x1 (fma x1 9.0 -2.0))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x1 * (x2 * (x2 * 8.0));
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_0 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -4e+255) {
		tmp = t_1;
	} else if (t_4 <= 1e+251) {
		tmp = fma(x2, fma(x1, -12.0, -6.0), -x1);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = x1 + t_1;
	} else {
		tmp = x1 + (x1 * fma(x1, 9.0, -2.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x1 * Float64(x2 * Float64(x2 * 8.0)))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_0 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -4e+255)
		tmp = t_1;
	elseif (t_4 <= 1e+251)
		tmp = fma(x2, fma(x1, -12.0, -6.0), Float64(-x1));
	elseif (t_4 <= Inf)
		tmp = Float64(x1 + t_1);
	else
		tmp = Float64(x1 + Float64(x1 * fma(x1, 9.0, -2.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x2 * N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e+255], t$95$1, If[LessEqual[t$95$4, 1e+251], N[(x2 * N[(x1 * -12.0 + -6.0), $MachinePrecision] + (-x1)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(x1 + t$95$1), $MachinePrecision], N[(x1 + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -3.99999999999999995e255

   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 inf

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x1 around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 63.0

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

   11. Simplified 63.0%

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

   12. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 63.0

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

   14. Simplified 63.0%

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

   if -3.99999999999999995e255 < (+.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)))))) < 1e251

   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 inf

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.3

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x1 around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 73.9

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

   11. Simplified 73.9%

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

   12. Taylor expanded in x2 around 0

       \[\leadsto \color{blue}{-3 \cdot x1 + \left(2 \cdot x1 + x2 \cdot \left(-12 \cdot x1 - 6\right)\right)} \]
   13. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(-3 \cdot x1 + 2 \cdot x1\right) + x2 \cdot \left(-12 \cdot x1 - 6\right)} \]

       2. distribute-rgt-out N/A

          \[\leadsto \color{blue}{x1 \cdot \left(-3 + 2\right)} + x2 \cdot \left(-12 \cdot x1 - 6\right) \]

       3. metadata-eval N/A

          \[\leadsto x1 \cdot \color{blue}{-1} + x2 \cdot \left(-12 \cdot x1 - 6\right) \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot x1} + x2 \cdot \left(-12 \cdot x1 - 6\right) \]

       5. +-commutative N/A

          \[\leadsto \color{blue}{x2 \cdot \left(-12 \cdot x1 - 6\right) + -1 \cdot x1} \]

       6. accelerator-lowering-fma.f64 N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. neg-lowering-neg.f64 72.5

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

   14. Simplified 72.5%

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

   if 1e251 < (+.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 0

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

   5. Simplified 46.8%

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

   6. Taylor expanded in x2 around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 45.2

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

   8. Simplified 45.2%

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

   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

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

   5. Simplified 66.6%

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

   6. Taylor expanded in x2 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 91.0

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

   8. Simplified 91.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -4 \cdot 10^{+255}:\\ \;\;\;\;x1 \cdot \left(x2 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+251}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), -x1\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x1 + x1 \cdot \left(x2 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot \left(x2 \cdot \left(x2 \cdot 8\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{+255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{+251}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), -x1\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* x1 (* x2 (* x2 8.0))))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_0 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -4e+255)
     t_1
     (if (<= t_4 1e+251)
       (fma x2 (fma x1 -12.0 -6.0) (- x1))
       (if (<= t_4 INFINITY) t_1 (+ x1 (* x1 (fma x1 9.0 -2.0))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x1 * (x2 * (x2 * 8.0));
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_0 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -4e+255) {
		tmp = t_1;
	} else if (t_4 <= 1e+251) {
		tmp = fma(x2, fma(x1, -12.0, -6.0), -x1);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 + (x1 * fma(x1, 9.0, -2.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x1 * Float64(x2 * Float64(x2 * 8.0)))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_0 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -4e+255)
		tmp = t_1;
	elseif (t_4 <= 1e+251)
		tmp = fma(x2, fma(x1, -12.0, -6.0), Float64(-x1));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 + Float64(x1 * fma(x1, 9.0, -2.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x2 * N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e+255], t$95$1, If[LessEqual[t$95$4, 1e+251], N[(x2 * N[(x1 * -12.0 + -6.0), $MachinePrecision] + (-x1)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 + N[(x1 * N[(x1 * 9.0 + -2.0), $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)))))) < -3.99999999999999995e255 or 1e251 < (+.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 inf

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x1 around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 50.2

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

   11. Simplified 50.2%

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

   12. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 50.1

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

   14. Simplified 50.1%

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

   if -3.99999999999999995e255 < (+.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)))))) < 1e251

   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 inf

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.3

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x1 around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 73.9

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

   11. Simplified 73.9%

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

   12. Taylor expanded in x2 around 0

       \[\leadsto \color{blue}{-3 \cdot x1 + \left(2 \cdot x1 + x2 \cdot \left(-12 \cdot x1 - 6\right)\right)} \]
   13. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(-3 \cdot x1 + 2 \cdot x1\right) + x2 \cdot \left(-12 \cdot x1 - 6\right)} \]

       2. distribute-rgt-out N/A

          \[\leadsto \color{blue}{x1 \cdot \left(-3 + 2\right)} + x2 \cdot \left(-12 \cdot x1 - 6\right) \]

       3. metadata-eval N/A

          \[\leadsto x1 \cdot \color{blue}{-1} + x2 \cdot \left(-12 \cdot x1 - 6\right) \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot x1} + x2 \cdot \left(-12 \cdot x1 - 6\right) \]

       5. +-commutative N/A

          \[\leadsto \color{blue}{x2 \cdot \left(-12 \cdot x1 - 6\right) + -1 \cdot x1} \]

       6. accelerator-lowering-fma.f64 N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. neg-lowering-neg.f64 72.5

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

   14. Simplified 72.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 66.6%

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

   6. Taylor expanded in x2 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 91.0

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

   8. Simplified 91.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -4 \cdot 10^{+255}:\\ \;\;\;\;x1 \cdot \left(x2 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+251}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), -x1\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x1 \cdot \left(x2 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), -x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (if (<=
        (+
         x1
         (+
          (+
           x1
           (+
            (+
             (*
              t_1
              (+
               (* (* (* x1 2.0) t_2) (- t_2 3.0))
               (* (* x1 x1) (- (* t_2 4.0) 6.0))))
             (* t_0 t_2))
            (* x1 (* x1 x1))))
          (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))
        INFINITY)
     (fma x2 (fma x1 -12.0 -6.0) (- x1))
     (+ x1 (* x1 (fma x1 9.0 -2.0))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= ((double) INFINITY)) {
		tmp = fma(x2, fma(x1, -12.0, -6.0), -x1);
	} else {
		tmp = x1 + (x1 * fma(x1, 9.0, -2.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1)))) <= Inf)
		tmp = fma(x2, fma(x1, -12.0, -6.0), Float64(-x1));
	else
		tmp = Float64(x1 + Float64(x1 * fma(x1, 9.0, -2.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * 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]}, If[LessEqual[N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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], Infinity], N[(x2 * N[(x1 * -12.0 + -6.0), $MachinePrecision] + (-x1)), $MachinePrecision], N[(x1 + N[(x1 * N[(x1 * 9.0 + -2.0), $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 99.5%

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

   3. Taylor expanded in x1 around inf

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.5

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

   8. Simplified 99.5%

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

   9. Taylor expanded in x1 around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 66.3

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

   11. Simplified 66.3%

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

   12. Taylor expanded in x2 around 0

       \[\leadsto \color{blue}{-3 \cdot x1 + \left(2 \cdot x1 + x2 \cdot \left(-12 \cdot x1 - 6\right)\right)} \]
   13. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(-3 \cdot x1 + 2 \cdot x1\right) + x2 \cdot \left(-12 \cdot x1 - 6\right)} \]

       2. distribute-rgt-out N/A

          \[\leadsto \color{blue}{x1 \cdot \left(-3 + 2\right)} + x2 \cdot \left(-12 \cdot x1 - 6\right) \]

       3. metadata-eval N/A

          \[\leadsto x1 \cdot \color{blue}{-1} + x2 \cdot \left(-12 \cdot x1 - 6\right) \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot x1} + x2 \cdot \left(-12 \cdot x1 - 6\right) \]

       5. +-commutative N/A

          \[\leadsto \color{blue}{x2 \cdot \left(-12 \cdot x1 - 6\right) + -1 \cdot x1} \]

       6. accelerator-lowering-fma.f64 N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. neg-lowering-neg.f64 52.2

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

   14. Simplified 52.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 66.6%

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

   6. Taylor expanded in x2 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 91.0

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

   8. Simplified 91.0%

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

4. Final simplification 61.6%

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

Alternative 6: 96.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\\ \mathbf{if}\;x1 \leq -5800:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{t\_0 - \frac{\mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 2100:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{t\_0}{x1} - 3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma 4.0 (fma x2 2.0 -3.0) 9.0)))
   (if (<= x1 -5800.0)
     (+
      x1
      (*
       (pow x1 4.0)
       (+
        6.0
        (/ (- (/ (- t_0 (/ (* (fma x2 2.0 -3.0) -6.0) x1)) x1) 3.0) x1))))
     (if (<= x1 2100.0)
       (+
        x1
        (fma
         x2
         (fma x1 (* x2 8.0) (fma x1 (fma x1 12.0 -12.0) -6.0))
         (* x1 (fma x1 9.0 -2.0))))
       (+ x1 (* (pow x1 4.0) (+ 6.0 (/ (- (/ t_0 x1) 3.0) x1))))))))

C

double code(double x1, double x2) {
	double t_0 = fma(4.0, fma(x2, 2.0, -3.0), 9.0);
	double tmp;
	if (x1 <= -5800.0) {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + ((((t_0 - ((fma(x2, 2.0, -3.0) * -6.0) / x1)) / x1) - 3.0) / x1)));
	} else if (x1 <= 2100.0) {
		tmp = x1 + fma(x2, fma(x1, (x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), (x1 * fma(x1, 9.0, -2.0)));
	} else {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + (((t_0 / x1) - 3.0) / x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(4.0, fma(x2, 2.0, -3.0), 9.0)
	tmp = 0.0
	if (x1 <= -5800.0)
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(Float64(t_0 - Float64(Float64(fma(x2, 2.0, -3.0) * -6.0) / x1)) / x1) - 3.0) / x1))));
	elseif (x1 <= 2100.0)
		tmp = Float64(x1 + fma(x2, fma(x1, Float64(x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), Float64(x1 * fma(x1, 9.0, -2.0))));
	else
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(t_0 / x1) - 3.0) / x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]}, If[LessEqual[x1, -5800.0], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(t$95$0 - N[(N[(N[(x2 * 2.0 + -3.0), $MachinePrecision] * -6.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2100.0], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0), $MachinePrecision] + N[(x1 * N[(x1 * 12.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(t$95$0 / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -5800

   1. Initial program 36.6%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   5. Simplified 99.9%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   if -5800 < x1 < 2100

   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

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

   5. Simplified 86.2%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. metadata-eval N/A

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

      21. accelerator-lowering-fma.f64 98.5

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

   8. Simplified 98.5%

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

   if 2100 < x1

   1. Initial program 56.4%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-lowering-*.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. pow-lowering-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. --lowering--.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. /-lowering-/.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. Simplified 96.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -5800:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{\mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 2100:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{if}\;x1 \leq -8500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2100000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          x1
          (*
           (pow x1 4.0)
           (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))))
   (if (<= x1 -8500.0)
     t_0
     (if (<= x1 2100000.0)
       (+
        x1
        (fma
         x2
         (fma x1 (* x2 8.0) (fma x1 (fma x1 12.0 -12.0) -6.0))
         (* x1 (fma x1 9.0 -2.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1)));
	double tmp;
	if (x1 <= -8500.0) {
		tmp = t_0;
	} else if (x1 <= 2100000.0) {
		tmp = x1 + fma(x2, fma(x1, (x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), (x1 * fma(x1, 9.0, -2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = 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))))
	tmp = 0.0
	if (x1 <= -8500.0)
		tmp = t_0;
	elseif (x1 <= 2100000.0)
		tmp = Float64(x1 + fma(x2, fma(x1, Float64(x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), Float64(x1 * fma(x1, 9.0, -2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = 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]}, If[LessEqual[x1, -8500.0], t$95$0, If[LessEqual[x1, 2100000.0], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0), $MachinePrecision] + N[(x1 * N[(x1 * 12.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -8500 or 2.1e6 < x1

   1. Initial program 46.8%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-lowering-*.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. pow-lowering-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. --lowering--.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. /-lowering-/.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. Simplified 97.8%

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

   if -8500 < x1 < 2.1e6

   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

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

   5. Simplified 86.2%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. metadata-eval N/A

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

      21. accelerator-lowering-fma.f64 98.5

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

   8. Simplified 98.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -8500:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 2100000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -5600:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{-3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 30000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(3 \cdot \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1} - 4}{x1}\right)\right)\right)\right)\right) + 3 \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -5600.0)
   (+ x1 (* (pow x1 4.0) (+ 6.0 (/ -3.0 x1))))
   (if (<= x1 30000.0)
     (+
      x1
      (fma
       x2
       (fma x1 (* x2 8.0) (fma x1 (fma x1 12.0 -12.0) -6.0))
       (* x1 (fma x1 9.0 -2.0))))
     (+
      x1
      (+
       (+
        x1
        (+
         (* x1 (* x1 x1))
         (+
          (* 3.0 (* x1 (* x1 3.0)))
          (*
           (+ (* x1 x1) 1.0)
           (*
            (* x1 x1)
            (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) -6.0) x1) 4.0) x1)))))))
       (* 3.0 3.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -5600.0) {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + (-3.0 / x1)));
	} else if (x1 <= 30000.0) {
		tmp = x1 + fma(x2, fma(x1, (x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), (x1 * fma(x1, 9.0, -2.0)));
	} else {
		tmp = x1 + ((x1 + ((x1 * (x1 * x1)) + ((3.0 * (x1 * (x1 * 3.0))) + (((x1 * x1) + 1.0) * ((x1 * x1) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1) - 4.0) / x1))))))) + (3.0 * 3.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -5600.0)
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(-3.0 / x1))));
	elseif (x1 <= 30000.0)
		tmp = Float64(x1 + fma(x2, fma(x1, Float64(x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), Float64(x1 * fma(x1, 9.0, -2.0))));
	else
		tmp = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(Float64(3.0 * Float64(x1 * Float64(x1 * 3.0))) + Float64(Float64(Float64(x1 * x1) + 1.0) * Float64(Float64(x1 * x1) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1) - 4.0) / x1))))))) + Float64(3.0 * 3.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -5600.0], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(-3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 30000.0], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0), $MachinePrecision] + N[(x1 * N[(x1 * 12.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 * N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + -6.0), $MachinePrecision] / x1), $MachinePrecision] - 4.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -5600

   1. Initial program 36.6%

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

   3. Taylor expanded in x1 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. /-lowering-/.f64 N/A

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

      9. metadata-eval 93.1

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

   5. Simplified 93.1%

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

   if -5600 < x1 < 3e4

   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

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

   5. Simplified 86.2%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. metadata-eval N/A

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

      21. accelerator-lowering-fma.f64 98.5

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

   8. Simplified 98.5%

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

   if 3e4 < x1

   1. Initial program 56.4%

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

   3. Taylor expanded in x1 around inf

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

   5. Simplified 56.4%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

   8. Simplified 52.5%

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

   9. Taylor expanded in x1 around inf

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

   11. Simplified 95.8%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -5600:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{-3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 30000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(3 \cdot \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1} - 4}{x1}\right)\right)\right)\right)\right) + 3 \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1550:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 2300:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(3 \cdot \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1} - 4}{x1}\right)\right)\right)\right)\right) + 3 \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1550.0)
   (* 6.0 (pow x1 4.0))
   (if (<= x1 2300.0)
     (+
      x1
      (fma
       x2
       (fma x1 (* x2 8.0) (fma x1 (fma x1 12.0 -12.0) -6.0))
       (* x1 (fma x1 9.0 -2.0))))
     (+
      x1
      (+
       (+
        x1
        (+
         (* x1 (* x1 x1))
         (+
          (* 3.0 (* x1 (* x1 3.0)))
          (*
           (+ (* x1 x1) 1.0)
           (*
            (* x1 x1)
            (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) -6.0) x1) 4.0) x1)))))))
       (* 3.0 3.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1550.0) {
		tmp = 6.0 * pow(x1, 4.0);
	} else if (x1 <= 2300.0) {
		tmp = x1 + fma(x2, fma(x1, (x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), (x1 * fma(x1, 9.0, -2.0)));
	} else {
		tmp = x1 + ((x1 + ((x1 * (x1 * x1)) + ((3.0 * (x1 * (x1 * 3.0))) + (((x1 * x1) + 1.0) * ((x1 * x1) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1) - 4.0) / x1))))))) + (3.0 * 3.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1550.0)
		tmp = Float64(6.0 * (x1 ^ 4.0));
	elseif (x1 <= 2300.0)
		tmp = Float64(x1 + fma(x2, fma(x1, Float64(x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), Float64(x1 * fma(x1, 9.0, -2.0))));
	else
		tmp = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(Float64(3.0 * Float64(x1 * Float64(x1 * 3.0))) + Float64(Float64(Float64(x1 * x1) + 1.0) * Float64(Float64(x1 * x1) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1) - 4.0) / x1))))))) + Float64(3.0 * 3.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1550.0], N[(6.0 * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2300.0], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0), $MachinePrecision] + N[(x1 * N[(x1 * 12.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 * N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + -6.0), $MachinePrecision] / x1), $MachinePrecision] - 4.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1550

   1. Initial program 36.6%

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

   3. Taylor expanded in x1 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 92.0

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

   5. Simplified 92.0%

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

   6. Taylor expanded in x1 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 92.0

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

   8. Simplified 92.0%

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

   if -1550 < x1 < 2300

   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

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

   5. Simplified 86.2%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. metadata-eval N/A

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

      21. accelerator-lowering-fma.f64 98.5

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

   8. Simplified 98.5%

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

   if 2300 < x1

   1. Initial program 56.4%

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

   3. Taylor expanded in x1 around inf

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

   5. Simplified 56.4%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

   8. Simplified 52.5%

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

   9. Taylor expanded in x1 around inf

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

   11. Simplified 95.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1550:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 2300:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(3 \cdot \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1} - 4}{x1}\right)\right)\right)\right)\right) + 3 \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -54000:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 3200:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(3 \cdot \left(x1 \cdot \left(x1 \cdot 3\right)\right) - \left(x1 \cdot \left(x1 \cdot \left(6 + \frac{-4}{x1}\right)\right)\right) \cdot \left(-1 - x1 \cdot x1\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -54000.0)
   (* 6.0 (pow x1 4.0))
   (if (<= x1 3200.0)
     (+
      x1
      (fma
       x2
       (fma x1 (* x2 8.0) (fma x1 (fma x1 12.0 -12.0) -6.0))
       (* x1 (fma x1 9.0 -2.0))))
     (+
      x1
      (+
       (* 3.0 (- (* x2 -2.0) x1))
       (+
        x1
        (+
         (* x1 (* x1 x1))
         (-
          (* 3.0 (* x1 (* x1 3.0)))
          (* (* x1 (* x1 (+ 6.0 (/ -4.0 x1)))) (- -1.0 (* x1 x1)))))))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -54000.0) {
		tmp = 6.0 * pow(x1, 4.0);
	} else if (x1 <= 3200.0) {
		tmp = x1 + fma(x2, fma(x1, (x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), (x1 * fma(x1, 9.0, -2.0)));
	} else {
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + ((x1 * (x1 * x1)) + ((3.0 * (x1 * (x1 * 3.0))) - ((x1 * (x1 * (6.0 + (-4.0 / x1)))) * (-1.0 - (x1 * x1)))))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -54000.0)
		tmp = Float64(6.0 * (x1 ^ 4.0));
	elseif (x1 <= 3200.0)
		tmp = Float64(x1 + fma(x2, fma(x1, Float64(x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), Float64(x1 * fma(x1, 9.0, -2.0))));
	else
		tmp = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(x2 * -2.0) - x1)) + Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(Float64(3.0 * Float64(x1 * Float64(x1 * 3.0))) - Float64(Float64(x1 * Float64(x1 * Float64(6.0 + Float64(-4.0 / x1)))) * Float64(-1.0 - Float64(x1 * x1))))))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -54000.0], N[(6.0 * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 3200.0], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0), $MachinePrecision] + N[(x1 * N[(x1 * 12.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(3.0 * N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 * N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * N[(x1 * N[(6.0 + N[(-4.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -54000

   1. Initial program 36.6%

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

   3. Taylor expanded in x1 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 92.0

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

   5. Simplified 92.0%

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

   6. Taylor expanded in x1 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 92.0

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

   8. Simplified 92.0%

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

   if -54000 < x1 < 3200

   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

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

   5. Simplified 86.2%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. metadata-eval N/A

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

      21. accelerator-lowering-fma.f64 98.5

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

   8. Simplified 98.5%

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

   if 3200 < x1

   1. Initial program 56.4%

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

   3. Taylor expanded in x1 around inf

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

   5. Simplified 56.4%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 56.4

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

   8. Simplified 56.4%

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

   9. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. sub-neg N/A

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

       6. +-lowering-+.f64 N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-frac N/A

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

       10. metadata-eval N/A

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

       11. /-lowering-/.f64 90.1

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

   11. Simplified 90.1%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -54000:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 3200:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(3 \cdot \left(x1 \cdot \left(x1 \cdot 3\right)\right) - \left(x1 \cdot \left(x1 \cdot \left(6 + \frac{-4}{x1}\right)\right)\right) \cdot \left(-1 - x1 \cdot x1\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -72000:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x1 \cdot 6, x1\right)\\ \mathbf{elif}\;x1 \leq 25000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(t\_0 + \left(3 \cdot \left(x1 \cdot \left(x1 \cdot 3\right)\right) - \left(x1 \cdot \left(x1 \cdot \left(6 + \frac{-4}{x1}\right)\right)\right) \cdot \left(-1 - x1 \cdot x1\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 x1))))
   (if (<= x1 -72000.0)
     (fma t_0 (* x1 6.0) x1)
     (if (<= x1 25000.0)
       (+
        x1
        (fma
         x2
         (fma x1 (* x2 8.0) (fma x1 (fma x1 12.0 -12.0) -6.0))
         (* x1 (fma x1 9.0 -2.0))))
       (+
        x1
        (+
         (* 3.0 (- (* x2 -2.0) x1))
         (+
          x1
          (+
           t_0
           (-
            (* 3.0 (* x1 (* x1 3.0)))
            (* (* x1 (* x1 (+ 6.0 (/ -4.0 x1)))) (- -1.0 (* x1 x1))))))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * x1);
	double tmp;
	if (x1 <= -72000.0) {
		tmp = fma(t_0, (x1 * 6.0), x1);
	} else if (x1 <= 25000.0) {
		tmp = x1 + fma(x2, fma(x1, (x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), (x1 * fma(x1, 9.0, -2.0)));
	} else {
		tmp = x1 + ((3.0 * ((x2 * -2.0) - x1)) + (x1 + (t_0 + ((3.0 * (x1 * (x1 * 3.0))) - ((x1 * (x1 * (6.0 + (-4.0 / x1)))) * (-1.0 - (x1 * x1)))))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -72000.0)
		tmp = fma(t_0, Float64(x1 * 6.0), x1);
	elseif (x1 <= 25000.0)
		tmp = Float64(x1 + fma(x2, fma(x1, Float64(x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), Float64(x1 * fma(x1, 9.0, -2.0))));
	else
		tmp = Float64(x1 + Float64(Float64(3.0 * Float64(Float64(x2 * -2.0) - x1)) + Float64(x1 + Float64(t_0 + Float64(Float64(3.0 * Float64(x1 * Float64(x1 * 3.0))) - Float64(Float64(x1 * Float64(x1 * Float64(6.0 + Float64(-4.0 / x1)))) * Float64(-1.0 - Float64(x1 * x1))))))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -72000.0], N[(t$95$0 * N[(x1 * 6.0), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 25000.0], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0), $MachinePrecision] + N[(x1 * N[(x1 * 12.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(3.0 * N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] + N[(x1 + N[(t$95$0 + N[(N[(3.0 * N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * N[(x1 * N[(6.0 + N[(-4.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -72000

   1. Initial program 36.6%

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

   3. Taylor expanded in x1 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 92.0

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

   5. Simplified 92.0%

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

      1. metadata-eval N/A

         \[\leadsto x1 + 6 \cdot {x1}^{\color{blue}{\left(2 + 2\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x1 + 6 \cdot \color{blue}{\left({x1}^{2} \cdot {x1}^{2}\right)} \]

      3. pow2 N/A

         \[\leadsto x1 + 6 \cdot \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot {x1}^{2}\right) \]

      4. pow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 91.9

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

   7. Applied egg-rr 91.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 91.9

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

   9. Applied egg-rr 91.9%

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

   if -72000 < x1 < 25000

   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

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

   5. Simplified 86.2%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. metadata-eval N/A

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

      21. accelerator-lowering-fma.f64 98.5

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

   8. Simplified 98.5%

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

   if 25000 < x1

   1. Initial program 56.4%

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

   3. Taylor expanded in x1 around inf

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

   5. Simplified 56.4%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 56.4

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

   8. Simplified 56.4%

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

   9. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. sub-neg N/A

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

       6. +-lowering-+.f64 N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-frac N/A

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

       10. metadata-eval N/A

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

       11. /-lowering-/.f64 90.1

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

   11. Simplified 90.1%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -72000:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), x1 \cdot 6, x1\right)\\ \mathbf{elif}\;x1 \leq 25000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(3 \cdot \left(x2 \cdot -2 - x1\right) + \left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(3 \cdot \left(x1 \cdot \left(x1 \cdot 3\right)\right) - \left(x1 \cdot \left(x1 \cdot \left(6 + \frac{-4}{x1}\right)\right)\right) \cdot \left(-1 - x1 \cdot x1\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), x1 \cdot 6, x1\right)\\ \mathbf{if}\;x1 \leq -70000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 340000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, x2 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 12, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* x1 (* x1 x1)) (* x1 6.0) x1)))
   (if (<= x1 -70000.0)
     t_0
     (if (<= x1 340000.0)
       (+
        x1
        (fma
         x2
         (fma x1 (* x2 8.0) (fma x1 (fma x1 12.0 -12.0) -6.0))
         (* x1 (fma x1 9.0 -2.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * (x1 * x1)), (x1 * 6.0), x1);
	double tmp;
	if (x1 <= -70000.0) {
		tmp = t_0;
	} else if (x1 <= 340000.0) {
		tmp = x1 + fma(x2, fma(x1, (x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), (x1 * fma(x1, 9.0, -2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * Float64(x1 * x1)), Float64(x1 * 6.0), x1)
	tmp = 0.0
	if (x1 <= -70000.0)
		tmp = t_0;
	elseif (x1 <= 340000.0)
		tmp = Float64(x1 + fma(x2, fma(x1, Float64(x2 * 8.0), fma(x1, fma(x1, 12.0, -12.0), -6.0)), Float64(x1 * fma(x1, 9.0, -2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * 6.0), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -70000.0], t$95$0, If[LessEqual[x1, 340000.0], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0), $MachinePrecision] + N[(x1 * N[(x1 * 12.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -7e4 or 3.4e5 < x1

   1. Initial program 46.8%

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

   3. Taylor expanded in x1 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 90.7

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

   5. Simplified 90.7%

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

      1. metadata-eval N/A

         \[\leadsto x1 + 6 \cdot {x1}^{\color{blue}{\left(2 + 2\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x1 + 6 \cdot \color{blue}{\left({x1}^{2} \cdot {x1}^{2}\right)} \]

      3. pow2 N/A

         \[\leadsto x1 + 6 \cdot \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot {x1}^{2}\right) \]

      4. pow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 90.6

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

   7. Applied egg-rr 90.6%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 90.7

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

   9. Applied egg-rr 90.7%

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

   if -7e4 < x1 < 3.4e5

   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

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

   5. Simplified 86.2%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. metadata-eval N/A

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

      21. accelerator-lowering-fma.f64 98.5

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

   8. Simplified 98.5%

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

Alternative 13: 81.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), x1 \cdot 6, x1\right)\\ \mathbf{if}\;x1 \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 7.5 \cdot 10^{-77}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -2\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 \leq 1750000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, x2 \cdot \left(x2 \cdot 8\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* x1 (* x1 x1)) (* x1 6.0) x1)))
   (if (<= x1 -5.5)
     t_0
     (if (<= x1 7.5e-77)
       (+ x1 (fma x1 (fma x1 9.0 -2.0) (* x2 -6.0)))
       (if (<= x1 1750000000.0)
         (+ x1 (fma x1 (* x2 (* x2 8.0)) (* x2 -6.0)))
         t_0)))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * (x1 * x1)), (x1 * 6.0), x1);
	double tmp;
	if (x1 <= -5.5) {
		tmp = t_0;
	} else if (x1 <= 7.5e-77) {
		tmp = x1 + fma(x1, fma(x1, 9.0, -2.0), (x2 * -6.0));
	} else if (x1 <= 1750000000.0) {
		tmp = x1 + fma(x1, (x2 * (x2 * 8.0)), (x2 * -6.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * Float64(x1 * x1)), Float64(x1 * 6.0), x1)
	tmp = 0.0
	if (x1 <= -5.5)
		tmp = t_0;
	elseif (x1 <= 7.5e-77)
		tmp = Float64(x1 + fma(x1, fma(x1, 9.0, -2.0), Float64(x2 * -6.0)));
	elseif (x1 <= 1750000000.0)
		tmp = Float64(x1 + fma(x1, Float64(x2 * Float64(x2 * 8.0)), Float64(x2 * -6.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * 6.0), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -5.5], t$95$0, If[LessEqual[x1, 7.5e-77], N[(x1 + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1750000000.0], N[(x1 + N[(x1 * N[(x2 * N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -5.5 or 1.75e9 < x1

   1. Initial program 47.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 90.0

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

   5. Simplified 90.0%

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

      1. metadata-eval N/A

         \[\leadsto x1 + 6 \cdot {x1}^{\color{blue}{\left(2 + 2\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x1 + 6 \cdot \color{blue}{\left({x1}^{2} \cdot {x1}^{2}\right)} \]

      3. pow2 N/A

         \[\leadsto x1 + 6 \cdot \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot {x1}^{2}\right) \]

      4. pow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 89.9

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

   7. Applied egg-rr 89.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 89.9

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

   9. Applied egg-rr 89.9%

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

   if -5.5 < x1 < 7.5000000000000006e-77

   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

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

   5. Simplified 85.6%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 77.7

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

   8. Simplified 77.7%

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

   if 7.5000000000000006e-77 < x1 < 1.75e9

   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

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

   5. Simplified 89.6%

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

   6. Taylor expanded in x2 around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 73.2

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

   8. Simplified 73.2%

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

Alternative 14: 80.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), x1 \cdot 6, x1\right)\\ \mathbf{if}\;x1 \leq -6.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.42 \cdot 10^{-76}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -2\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 \leq 13500000:\\ \;\;\;\;x1 \cdot \left(x2 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* x1 (* x1 x1)) (* x1 6.0) x1)))
   (if (<= x1 -6.2)
     t_0
     (if (<= x1 2.42e-76)
       (+ x1 (fma x1 (fma x1 9.0 -2.0) (* x2 -6.0)))
       (if (<= x1 13500000.0) (* x1 (* x2 (* x2 8.0))) t_0)))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * (x1 * x1)), (x1 * 6.0), x1);
	double tmp;
	if (x1 <= -6.2) {
		tmp = t_0;
	} else if (x1 <= 2.42e-76) {
		tmp = x1 + fma(x1, fma(x1, 9.0, -2.0), (x2 * -6.0));
	} else if (x1 <= 13500000.0) {
		tmp = x1 * (x2 * (x2 * 8.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * Float64(x1 * x1)), Float64(x1 * 6.0), x1)
	tmp = 0.0
	if (x1 <= -6.2)
		tmp = t_0;
	elseif (x1 <= 2.42e-76)
		tmp = Float64(x1 + fma(x1, fma(x1, 9.0, -2.0), Float64(x2 * -6.0)));
	elseif (x1 <= 13500000.0)
		tmp = Float64(x1 * Float64(x2 * Float64(x2 * 8.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * 6.0), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -6.2], t$95$0, If[LessEqual[x1, 2.42e-76], N[(x1 + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 13500000.0], N[(x1 * N[(x2 * N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -6.20000000000000018 or 1.35e7 < x1

   1. Initial program 47.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 90.0

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

   5. Simplified 90.0%

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

      1. metadata-eval N/A

         \[\leadsto x1 + 6 \cdot {x1}^{\color{blue}{\left(2 + 2\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x1 + 6 \cdot \color{blue}{\left({x1}^{2} \cdot {x1}^{2}\right)} \]

      3. pow2 N/A

         \[\leadsto x1 + 6 \cdot \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot {x1}^{2}\right) \]

      4. pow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 89.9

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

   7. Applied egg-rr 89.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 89.9

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

   9. Applied egg-rr 89.9%

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

   if -6.20000000000000018 < x1 < 2.42e-76

   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

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

   5. Simplified 85.6%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 77.7

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

   8. Simplified 77.7%

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

   if 2.42e-76 < x1 < 1.35e7

   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 inf

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.6

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x1 around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 89.4

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

   11. Simplified 89.4%

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

   12. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 64.9

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

   14. Simplified 64.9%

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

Alternative 15: 80.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot x1, \left(x1 \cdot x1\right) \cdot 6, x1\right)\\ \mathbf{if}\;x1 \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.42 \cdot 10^{-76}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -2\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 \leq 200000:\\ \;\;\;\;x1 \cdot \left(x2 \cdot \left(x2 \cdot 8\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* x1 x1) (* (* x1 x1) 6.0) x1)))
   (if (<= x1 -5.5)
     t_0
     (if (<= x1 2.42e-76)
       (+ x1 (fma x1 (fma x1 9.0 -2.0) (* x2 -6.0)))
       (if (<= x1 200000.0) (* x1 (* x2 (* x2 8.0))) t_0)))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * x1), ((x1 * x1) * 6.0), x1);
	double tmp;
	if (x1 <= -5.5) {
		tmp = t_0;
	} else if (x1 <= 2.42e-76) {
		tmp = x1 + fma(x1, fma(x1, 9.0, -2.0), (x2 * -6.0));
	} else if (x1 <= 200000.0) {
		tmp = x1 * (x2 * (x2 * 8.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * x1), Float64(Float64(x1 * x1) * 6.0), x1)
	tmp = 0.0
	if (x1 <= -5.5)
		tmp = t_0;
	elseif (x1 <= 2.42e-76)
		tmp = Float64(x1 + fma(x1, fma(x1, 9.0, -2.0), Float64(x2 * -6.0)));
	elseif (x1 <= 200000.0)
		tmp = Float64(x1 * Float64(x2 * Float64(x2 * 8.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -5.5], t$95$0, If[LessEqual[x1, 2.42e-76], N[(x1 + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 200000.0], N[(x1 * N[(x2 * N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -5.5 or 2e5 < x1

   1. Initial program 47.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 90.0

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

   5. Simplified 90.0%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

         \[\leadsto 6 \cdot {x1}^{\color{blue}{\left(2 + 2\right)}} + x1 \]

      3. pow-prod-up N/A

         \[\leadsto 6 \cdot \color{blue}{\left({x1}^{2} \cdot {x1}^{2}\right)} + x1 \]

      4. pow2 N/A

         \[\leadsto 6 \cdot \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot {x1}^{2}\right) + x1 \]

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 89.9

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

   7. Applied egg-rr 89.9%

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

   if -5.5 < x1 < 2.42e-76

   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

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

   5. Simplified 85.6%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 77.7

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

   8. Simplified 77.7%

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

   if 2.42e-76 < x1 < 2e5

   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 inf

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.6

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x1 around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 89.4

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

   11. Simplified 89.4%

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

   12. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 64.9

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

   14. Simplified 64.9%

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

Alternative 16: 93.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), x1 \cdot 6, x1\right)\\ \mathbf{if}\;x1 \leq -72000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 450000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(x1, x2 \cdot 8, -6\right)\right), -x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* x1 (* x1 x1)) (* x1 6.0) x1)))
   (if (<= x1 -72000.0)
     t_0
     (if (<= x1 450000.0)
       (fma x2 (fma x1 -12.0 (fma x1 (* x2 8.0) -6.0)) (- x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * (x1 * x1)), (x1 * 6.0), x1);
	double tmp;
	if (x1 <= -72000.0) {
		tmp = t_0;
	} else if (x1 <= 450000.0) {
		tmp = fma(x2, fma(x1, -12.0, fma(x1, (x2 * 8.0), -6.0)), -x1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * Float64(x1 * x1)), Float64(x1 * 6.0), x1)
	tmp = 0.0
	if (x1 <= -72000.0)
		tmp = t_0;
	elseif (x1 <= 450000.0)
		tmp = fma(x2, fma(x1, -12.0, fma(x1, Float64(x2 * 8.0), -6.0)), Float64(-x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * 6.0), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -72000.0], t$95$0, If[LessEqual[x1, 450000.0], N[(x2 * N[(x1 * -12.0 + N[(x1 * N[(x2 * 8.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + (-x1)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -72000 or 4.5e5 < x1

   1. Initial program 46.8%

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

   3. Taylor expanded in x1 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 90.7

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

   5. Simplified 90.7%

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

      1. metadata-eval N/A

         \[\leadsto x1 + 6 \cdot {x1}^{\color{blue}{\left(2 + 2\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x1 + 6 \cdot \color{blue}{\left({x1}^{2} \cdot {x1}^{2}\right)} \]

      3. pow2 N/A

         \[\leadsto x1 + 6 \cdot \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot {x1}^{2}\right) \]

      4. pow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 90.6

         \[\leadsto x1 + 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \color{blue}{\left(x1 \cdot x1\right)}\right) \]

   7. Applied egg-rr 90.6%

      \[\leadsto x1 + 6 \cdot \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) + x1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot 6} + x1 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right)} \cdot 6 + x1 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(x1 \cdot 6\right)} + x1 \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x1 \cdot x1\right) \cdot x1, x1 \cdot 6, x1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot \left(x1 \cdot x1\right)}, x1 \cdot 6, x1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot \left(x1 \cdot x1\right)}, x1 \cdot 6, x1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(x1 \cdot x1\right)}, x1 \cdot 6, x1\right) \]

      9. *-lowering-*.f64 90.7

         \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), \color{blue}{x1 \cdot 6}, x1\right) \]

   9. Applied egg-rr 90.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), x1 \cdot 6, x1\right)} \]

   if -72000 < x1 < 4.5e5

   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 inf

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{3}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   4. Step-by-step derivation

   5. Simplified 98.2%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{3}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   6. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(-2 \cdot x2 + -1 \cdot x1\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \left(-2 \cdot x2 + \color{blue}{\left(\mathsf{neg}\left(x1\right)\right)}\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(-2 \cdot x2 - x1\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(-2 \cdot x2 - x1\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \left(\color{blue}{x2 \cdot -2} - x1\right)\right) \]

      5. *-lowering-*.f64 99.4

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \left(\color{blue}{x2 \cdot -2} - x1\right)\right) \]

   8. Simplified 99.4%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(x2 \cdot -2 - x1\right)}\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\color{blue}{4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x1 + \left(\left(\color{blue}{\left(4 \cdot x1\right) \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       2. *-commutative N/A

          \[\leadsto x1 + \left(\left(\color{blue}{\left(x1 \cdot 4\right)} \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto x1 + \left(\left(\color{blue}{x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x1 + \left(\left(\color{blue}{x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \color{blue}{\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)}\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       7. sub-neg N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       8. *-commutative N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(\color{blue}{x2 \cdot 2} + \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(x2 \cdot 2 + \color{blue}{-3}\right)\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       10. accelerator-lowering-fma.f64 85.2

           \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \color{blue}{\mathsf{fma}\left(x2, 2, -3\right)}\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

   11. Simplified 85.2%

       \[\leadsto x1 + \left(\left(\color{blue}{x1 \cdot \left(4 \cdot \left(x2 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

   12. Taylor expanded in x2 around 0

       \[\leadsto \color{blue}{-3 \cdot x1 + \left(2 \cdot x1 + x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)} \]
   13. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(-3 \cdot x1 + 2 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)} \]

       2. distribute-rgt-out N/A

          \[\leadsto \color{blue}{x1 \cdot \left(-3 + 2\right)} + x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) \]

       3. metadata-eval N/A

          \[\leadsto x1 \cdot \color{blue}{-1} + x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot x1} + x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) \]

       5. +-commutative N/A

          \[\leadsto \color{blue}{x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + -1 \cdot x1} \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6, -1 \cdot x1\right)} \]

       7. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(x2, \color{blue}{-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) - 6\right)}, -1 \cdot x1\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x2, \color{blue}{x1 \cdot -12} + \left(8 \cdot \left(x1 \cdot x2\right) - 6\right), -1 \cdot x1\right) \]

       9. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, -12, 8 \cdot \left(x1 \cdot x2\right) - 6\right)}, -1 \cdot x1\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \color{blue}{8 \cdot \left(x1 \cdot x2\right) + \left(\mathsf{neg}\left(6\right)\right)}\right), -1 \cdot x1\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \color{blue}{\left(x1 \cdot x2\right) \cdot 8} + \left(\mathsf{neg}\left(6\right)\right)\right), -1 \cdot x1\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \color{blue}{x1 \cdot \left(x2 \cdot 8\right)} + \left(\mathsf{neg}\left(6\right)\right)\right), -1 \cdot x1\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, x1 \cdot \color{blue}{\left(8 \cdot x2\right)} + \left(\mathsf{neg}\left(6\right)\right)\right), -1 \cdot x1\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, x1 \cdot \left(8 \cdot x2\right) + \color{blue}{-6}\right), -1 \cdot x1\right) \]

       15. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \color{blue}{\mathsf{fma}\left(x1, 8 \cdot x2, -6\right)}\right), -1 \cdot x1\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(x1, \color{blue}{x2 \cdot 8}, -6\right)\right), -1 \cdot x1\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(x1, \color{blue}{x2 \cdot 8}, -6\right)\right), -1 \cdot x1\right) \]

       18. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(x1, x2 \cdot 8, -6\right)\right), \color{blue}{\mathsf{neg}\left(x1\right)}\right) \]

       19. neg-lowering-neg.f64 97.9

           \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(x1, x2 \cdot 8, -6\right)\right), \color{blue}{-x1}\right) \]

   14. Simplified 97.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(x1, x2 \cdot 8, -6\right)\right), -x1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 31.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -2 \cdot 10^{-240}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;2 \cdot x2 \leq 10^{-165}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* 2.0 x2) -2e-240)
   (* x2 -6.0)
   (if (<= (* 2.0 x2) 1e-165) (- x1) (fma x2 -6.0 x1))))

C

double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -2e-240) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 1e-165) {
		tmp = -x1;
	} else {
		tmp = fma(x2, -6.0, x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(2.0 * x2) <= -2e-240)
		tmp = Float64(x2 * -6.0);
	elseif (Float64(2.0 * x2) <= 1e-165)
		tmp = Float64(-x1);
	else
		tmp = fma(x2, -6.0, x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(2.0 * x2), $MachinePrecision], -2e-240], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[N[(2.0 * x2), $MachinePrecision], 1e-165], (-x1), N[(x2 * -6.0 + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -1.9999999999999999e-240

   1. Initial program 77.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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. *-lowering-*.f64 35.1

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Simplified 35.1%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   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. *-lowering-*.f64 35.3

         \[\leadsto \color{blue}{x2 \cdot -6} \]

   8. Simplified 35.3%

      \[\leadsto \color{blue}{x2 \cdot -6} \]

   if -1.9999999999999999e-240 < (*.f64 #s(literal 2 binary64) x2) < 1e-165

   1. Initial program 82.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{3}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   4. Step-by-step derivation

   5. Simplified 79.7%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{3}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   6. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(-2 \cdot x2 + -1 \cdot x1\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \left(-2 \cdot x2 + \color{blue}{\left(\mathsf{neg}\left(x1\right)\right)}\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(-2 \cdot x2 - x1\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(-2 \cdot x2 - x1\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \left(\color{blue}{x2 \cdot -2} - x1\right)\right) \]

      5. *-lowering-*.f64 82.1

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \left(\color{blue}{x2 \cdot -2} - x1\right)\right) \]

   8. Simplified 82.1%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(x2 \cdot -2 - x1\right)}\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\color{blue}{4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x1 + \left(\left(\color{blue}{\left(4 \cdot x1\right) \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       2. *-commutative N/A

          \[\leadsto x1 + \left(\left(\color{blue}{\left(x1 \cdot 4\right)} \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto x1 + \left(\left(\color{blue}{x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x1 + \left(\left(\color{blue}{x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \color{blue}{\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)}\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       7. sub-neg N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       8. *-commutative N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(\color{blue}{x2 \cdot 2} + \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(x2 \cdot 2 + \color{blue}{-3}\right)\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       10. accelerator-lowering-fma.f64 47.7

           \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \color{blue}{\mathsf{fma}\left(x2, 2, -3\right)}\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

   11. Simplified 47.7%

       \[\leadsto x1 + \left(\left(\color{blue}{x1 \cdot \left(4 \cdot \left(x2 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

   12. Taylor expanded in x2 around 0

       \[\leadsto \color{blue}{-3 \cdot x1 + 2 \cdot x1} \]
   13. Step-by-step derivation

       1. distribute-rgt-out N/A

          \[\leadsto \color{blue}{x1 \cdot \left(-3 + 2\right)} \]

       2. metadata-eval N/A

          \[\leadsto x1 \cdot \color{blue}{-1} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot x1} \]

       4. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

       5. neg-lowering-neg.f64 46.2

          \[\leadsto \color{blue}{-x1} \]

   14. Simplified 46.2%

       \[\leadsto \color{blue}{-x1} \]

   if 1e-165 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 69.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

      2. *-lowering-*.f64 31.8

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Simplified 31.8%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x2 \cdot -6 + x1} \]

      2. accelerator-lowering-fma.f64 31.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1\right)} \]

   7. Applied egg-rr 31.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 30.8% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -2 \cdot 10^{-240}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;2 \cdot x2 \leq 10^{-156}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;x2 \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* 2.0 x2) -2e-240)
   (* x2 -6.0)
   (if (<= (* 2.0 x2) 1e-156) (- x1) (* x2 -6.0))))

C

double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -2e-240) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 1e-156) {
		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) <= (-2d-240)) then
        tmp = x2 * (-6.0d0)
    else if ((2.0d0 * x2) <= 1d-156) 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) <= -2e-240) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 1e-156) {
		tmp = -x1;
	} else {
		tmp = x2 * -6.0;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (2.0 * x2) <= -2e-240:
		tmp = x2 * -6.0
	elif (2.0 * x2) <= 1e-156:
		tmp = -x1
	else:
		tmp = x2 * -6.0
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(2.0 * x2) <= -2e-240)
		tmp = Float64(x2 * -6.0);
	elseif (Float64(2.0 * x2) <= 1e-156)
		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) <= -2e-240)
		tmp = x2 * -6.0;
	elseif ((2.0 * x2) <= 1e-156)
		tmp = -x1;
	else
		tmp = x2 * -6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(2.0 * x2), $MachinePrecision], -2e-240], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[N[(2.0 * x2), $MachinePrecision], 1e-156], (-x1), N[(x2 * -6.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -1.9999999999999999e-240 or 1.00000000000000004e-156 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 74.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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. *-lowering-*.f64 33.9

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Simplified 33.9%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   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. *-lowering-*.f64 33.6

         \[\leadsto \color{blue}{x2 \cdot -6} \]

   8. Simplified 33.6%

      \[\leadsto \color{blue}{x2 \cdot -6} \]

   if -1.9999999999999999e-240 < (*.f64 #s(literal 2 binary64) x2) < 1.00000000000000004e-156

   1. Initial program 79.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{3}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   4. Step-by-step derivation

   5. Simplified 77.2%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{3}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   6. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(-2 \cdot x2 + -1 \cdot x1\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \left(-2 \cdot x2 + \color{blue}{\left(\mathsf{neg}\left(x1\right)\right)}\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(-2 \cdot x2 - x1\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(-2 \cdot x2 - x1\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \left(\color{blue}{x2 \cdot -2} - x1\right)\right) \]

      5. *-lowering-*.f64 79.5

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \left(\color{blue}{x2 \cdot -2} - x1\right)\right) \]

   8. Simplified 79.5%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(x2 \cdot -2 - x1\right)}\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\color{blue}{4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x1 + \left(\left(\color{blue}{\left(4 \cdot x1\right) \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       2. *-commutative N/A

          \[\leadsto x1 + \left(\left(\color{blue}{\left(x1 \cdot 4\right)} \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto x1 + \left(\left(\color{blue}{x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x1 + \left(\left(\color{blue}{x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \color{blue}{\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)}\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       7. sub-neg N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       8. *-commutative N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(\color{blue}{x2 \cdot 2} + \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(x2 \cdot 2 + \color{blue}{-3}\right)\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

       10. accelerator-lowering-fma.f64 45.2

           \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \color{blue}{\mathsf{fma}\left(x2, 2, -3\right)}\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

   11. Simplified 45.2%

       \[\leadsto x1 + \left(\left(\color{blue}{x1 \cdot \left(4 \cdot \left(x2 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

   12. Taylor expanded in x2 around 0

       \[\leadsto \color{blue}{-3 \cdot x1 + 2 \cdot x1} \]
   13. Step-by-step derivation

       1. distribute-rgt-out N/A

          \[\leadsto \color{blue}{x1 \cdot \left(-3 + 2\right)} \]

       2. metadata-eval N/A

          \[\leadsto x1 \cdot \color{blue}{-1} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot x1} \]

       4. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

       5. neg-lowering-neg.f64 43.8

          \[\leadsto \color{blue}{-x1} \]

   14. Simplified 43.8%

       \[\leadsto \color{blue}{-x1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 55.5% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)\\ \mathbf{if}\;x1 \leq -7.4 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 3.7 \cdot 10^{-92}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (* x1 (fma x1 9.0 -2.0)))))
   (if (<= x1 -7.4e-90) t_0 (if (<= x1 3.7e-92) (* x2 -6.0) t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (x1 * fma(x1, 9.0, -2.0));
	double tmp;
	if (x1 <= -7.4e-90) {
		tmp = t_0;
	} else if (x1 <= 3.7e-92) {
		tmp = x2 * -6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64(x1 * fma(x1, 9.0, -2.0)))
	tmp = 0.0
	if (x1 <= -7.4e-90)
		tmp = t_0;
	elseif (x1 <= 3.7e-92)
		tmp = Float64(x2 * -6.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(x1 * N[(x1 * 9.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -7.4e-90], t$95$0, If[LessEqual[x1, 3.7e-92], N[(x2 * -6.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -7.40000000000000035e-90 or 3.69999999999999977e-92 < x1

   1. Initial program 60.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Simplified 57.2%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \color{blue}{x1 \cdot \left(9 \cdot x1 - 2\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x1 + \color{blue}{x1 \cdot \left(9 \cdot x1 - 2\right)} \]

      2. sub-neg N/A

         \[\leadsto x1 + x1 \cdot \color{blue}{\left(9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto x1 + x1 \cdot \left(\color{blue}{x1 \cdot 9} + \left(\mathsf{neg}\left(2\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto x1 + x1 \cdot \left(x1 \cdot 9 + \color{blue}{-2}\right) \]

      5. accelerator-lowering-fma.f64 48.1

         \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -2\right)} \]

   8. Simplified 48.1%

      \[\leadsto x1 + \color{blue}{x1 \cdot \mathsf{fma}\left(x1, 9, -2\right)} \]

   if -7.40000000000000035e-90 < x1 < 3.69999999999999977e-92

   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. *-lowering-*.f64 64.8

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Simplified 64.8%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   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. *-lowering-*.f64 65.2

         \[\leadsto \color{blue}{x2 \cdot -6} \]

   8. Simplified 65.2%

      \[\leadsto \color{blue}{x2 \cdot -6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 13.8% 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 75.4%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x1 around inf

   \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{3}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
4. Step-by-step derivation

5. Simplified 74.7%

   \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{3}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

6. Taylor expanded in x1 around 0

   \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(-2 \cdot x2 + -1 \cdot x1\right)}\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \left(-2 \cdot x2 + \color{blue}{\left(\mathsf{neg}\left(x1\right)\right)}\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(-2 \cdot x2 - x1\right)}\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(-2 \cdot x2 - x1\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \left(\color{blue}{x2 \cdot -2} - x1\right)\right) \]

   5. *-lowering-*.f64 75.4

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \left(\color{blue}{x2 \cdot -2} - x1\right)\right) \]

8. Simplified 75.4%

   \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \color{blue}{\left(x2 \cdot -2 - x1\right)}\right) \]

9. Taylor expanded in x1 around 0

   \[\leadsto x1 + \left(\left(\color{blue}{4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto x1 + \left(\left(\color{blue}{\left(4 \cdot x1\right) \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

    2. *-commutative N/A

       \[\leadsto x1 + \left(\left(\color{blue}{\left(x1 \cdot 4\right)} \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto x1 + \left(\left(\color{blue}{x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto x1 + \left(\left(\color{blue}{x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto x1 + \left(\left(x1 \cdot \color{blue}{\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)}\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

    7. sub-neg N/A

       \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

    8. *-commutative N/A

       \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(\color{blue}{x2 \cdot 2} + \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

    9. metadata-eval N/A

       \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(x2 \cdot 2 + \color{blue}{-3}\right)\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

    10. accelerator-lowering-fma.f64 55.7

        \[\leadsto x1 + \left(\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \color{blue}{\mathsf{fma}\left(x2, 2, -3\right)}\right)\right) + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

11. Simplified 55.7%

    \[\leadsto x1 + \left(\left(\color{blue}{x1 \cdot \left(4 \cdot \left(x2 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right)\right)} + x1\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right) \]

12. Taylor expanded in x2 around 0

    \[\leadsto \color{blue}{-3 \cdot x1 + 2 \cdot x1} \]
13. Step-by-step derivation

    1. distribute-rgt-out N/A

       \[\leadsto \color{blue}{x1 \cdot \left(-3 + 2\right)} \]

    2. metadata-eval N/A

       \[\leadsto x1 \cdot \color{blue}{-1} \]

    3. *-commutative N/A

       \[\leadsto \color{blue}{-1 \cdot x1} \]

    4. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

    5. neg-lowering-neg.f64 13.9

       \[\leadsto \color{blue}{-x1} \]

14. Simplified 13.9%

    \[\leadsto \color{blue}{-x1} \]
15. Add Preprocessing

Alternative 21: 3.2% accurate, 298.0× speedup

Math

\[\begin{array}{l} \\ x1 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 x1)

C

double code(double x1, double x2) {
	return x1;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = x1
end function

Java

public static double code(double x1, double x2) {
	return x1;
}

Python

def code(x1, x2):
	return x1

Julia

function code(x1, x2)
	return x1
end

MATLAB

function tmp = code(x1, x2)
	tmp = x1;
end

Wolfram

code[x1_, x2_] := x1

Derivation

1. Initial program 75.4%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
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. *-lowering-*.f64 27.1

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

5. Simplified 27.1%

   \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

6. Taylor expanded in x1 around inf

   \[\leadsto \color{blue}{x1} \]
7. Step-by-step derivation

8. Simplified 3.1%

   \[\leadsto \color{blue}{x1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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))))))