Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.9% → 99.5%

Time: 23.3s

Alternatives: 24

Speedup: 7.6×

• 46.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.9% 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: 99.5% accurate, 0.5× speedup

Math

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

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))
        (t_3
         (+
          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))))))
   (if (<= t_3 INFINITY)
     t_3
     (*
      x1
      (fma
       x1
       (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))
       1.0)))))

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 t_3 = 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 tmp;
	if (t_3 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = x1 * fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), 1.0);
	}
	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)
	t_3 = 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))))
	tmp = 0.0
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(x1 * fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), 1.0));
	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]}, Block[{t$95$3 = 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]}, If[LessEqual[t$95$3, Infinity], t$95$3, N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   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 100.0%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

4. Final simplification 99.7%

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

Alternative 2: 63.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := \mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+279}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -12, -2\right), x2 \cdot -6\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x1 \cdot 8\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (fma (* x2 (* x1 8.0)) x2 x1))
        (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 -5e+271)
     t_1
     (if (<= t_4 4e+279)
       (+ x1 (fma x1 (fma x2 -12.0 -2.0) (* x2 -6.0)))
       (if (<= t_4 INFINITY) t_1 (+ x1 (* x2 (* x1 (* x1 8.0)))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = fma((x2 * (x1 * 8.0)), x2, x1);
	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 <= -5e+271) {
		tmp = t_1;
	} else if (t_4 <= 4e+279) {
		tmp = x1 + fma(x1, fma(x2, -12.0, -2.0), (x2 * -6.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 + (x2 * (x1 * (x1 * 8.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = fma(Float64(x2 * Float64(x1 * 8.0)), x2, x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(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 <= -5e+271)
		tmp = t_1;
	elseif (t_4 <= 4e+279)
		tmp = Float64(x1 + fma(x1, fma(x2, -12.0, -2.0), Float64(x2 * -6.0)));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 + Float64(x2 * Float64(x1 * Float64(x1 * 8.0))));
	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[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision] * x2 + x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(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, -5e+271], t$95$1, If[LessEqual[t$95$4, 4e+279], N[(x1 + N[(x1 * N[(x2 * -12.0 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 + N[(x2 * N[(x1 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.0000000000000003e271 or 4.00000000000000023e279 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   3. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 53.8

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

   5. Simplified 53.8%

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

   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 54.3

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

   8. Simplified 54.3%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 57.7

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

   10. Applied egg-rr 57.7%

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

   if -5.0000000000000003e271 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.00000000000000023e279

   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

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 73.1

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

   5. Simplified 73.1%

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

   6. Taylor expanded in x2 around 0

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

   8. Simplified 79.3%

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

   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 x2 around inf

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

   5. Simplified 6.8%

      \[\leadsto x1 + \color{blue}{x2 \cdot \left(x2 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(2, \frac{x1 \cdot \left(2 \cdot \left(\left(\frac{\mathsf{fma}\left(3, x1 \cdot x1, 0 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) + \frac{\mathsf{fma}\left(3, x1 \cdot x1, 0 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 8 \cdot \left(x1 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right)\right), \frac{\mathsf{fma}\left(x1, x1, 1\right)}{x2}, \frac{x1 \cdot 8}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(6, x1 \cdot \frac{x1}{x2 \cdot \mathsf{fma}\left(x1, x1, 1\right)}, \frac{-6}{x2 \cdot \mathsf{fma}\left(x1, x1, 1\right)}\right)\right)\right)} \]

   6. Taylor expanded in x1 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 35.8

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

   8. Simplified 35.8%

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

   9. Taylor expanded in x2 around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 33.0

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

   11. Simplified 33.0%

       \[\leadsto x1 + x2 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot 8\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.5%

   \[\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 -5 \cdot 10^{+271}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, 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 4 \cdot 10^{+279}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -12, -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:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x1 \cdot 8\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := \mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{+273}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), 0 - x1\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x1 \cdot 8\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (fma (* x2 (* x1 8.0)) x2 x1))
        (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 -5e+271)
     t_1
     (if (<= t_4 1e+273)
       (fma x2 (fma x1 -12.0 -6.0) (- 0.0 x1))
       (if (<= t_4 INFINITY) t_1 (+ x1 (* x2 (* x1 (* x1 8.0)))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = fma((x2 * (x1 * 8.0)), x2, x1);
	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 <= -5e+271) {
		tmp = t_1;
	} else if (t_4 <= 1e+273) {
		tmp = fma(x2, fma(x1, -12.0, -6.0), (0.0 - x1));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 + (x2 * (x1 * (x1 * 8.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = fma(Float64(x2 * Float64(x1 * 8.0)), x2, x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(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 <= -5e+271)
		tmp = t_1;
	elseif (t_4 <= 1e+273)
		tmp = fma(x2, fma(x1, -12.0, -6.0), Float64(0.0 - x1));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 + Float64(x2 * Float64(x1 * Float64(x1 * 8.0))));
	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[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision] * x2 + x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(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, -5e+271], t$95$1, If[LessEqual[t$95$4, 1e+273], N[(x2 * N[(x1 * -12.0 + -6.0), $MachinePrecision] + N[(0.0 - x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 + N[(x2 * N[(x1 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.0000000000000003e271 or 9.99999999999999945e272 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   3. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 52.0

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

   5. Simplified 52.0%

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

   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 52.5

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

   8. Simplified 52.5%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 57.4

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

   10. Applied egg-rr 57.4%

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

   if -5.0000000000000003e271 < (+.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)))))) < 9.99999999999999945e272

   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

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 74.2

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

   5. Simplified 74.2%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft1-in N/A

         \[\leadsto x2 \cdot \left(-12 \cdot x1 - 6\right) + \color{blue}{\left(-2 + 1\right) \cdot x1} \]

      5. metadata-eval N/A

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

      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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), \color{blue}{0 - x1}\right) \]

      13. --lowering--.f64 79.7

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), \color{blue}{0 - x1}\right) \]

   8. Simplified 79.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), 0 - x1\right)} \]

   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 x2 around inf

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

   5. Simplified 6.8%

      \[\leadsto x1 + \color{blue}{x2 \cdot \left(x2 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(2, \frac{x1 \cdot \left(2 \cdot \left(\left(\frac{\mathsf{fma}\left(3, x1 \cdot x1, 0 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) + \frac{\mathsf{fma}\left(3, x1 \cdot x1, 0 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 8 \cdot \left(x1 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right)\right), \frac{\mathsf{fma}\left(x1, x1, 1\right)}{x2}, \frac{x1 \cdot 8}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(6, x1 \cdot \frac{x1}{x2 \cdot \mathsf{fma}\left(x1, x1, 1\right)}, \frac{-6}{x2 \cdot \mathsf{fma}\left(x1, x1, 1\right)}\right)\right)\right)} \]

   6. Taylor expanded in x1 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 35.8

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

   8. Simplified 35.8%

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

   9. Taylor expanded in x2 around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 33.0

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

   11. Simplified 33.0%

       \[\leadsto x1 + x2 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot 8\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.5%

   \[\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 -5 \cdot 10^{+271}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, 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 10^{+273}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), 0 - 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:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x2 \cdot \left(x1 \cdot \left(x1 \cdot 8\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := \mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{+273}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), 0 - x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (fma (* x2 (* x1 8.0)) x2 x1))
        (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 -5e+271)
     t_1
     (if (<= t_4 1e+273) (fma x2 (fma x1 -12.0 -6.0) (- 0.0 x1)) t_1))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = fma((x2 * (x1 * 8.0)), x2, x1);
	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 <= -5e+271) {
		tmp = t_1;
	} else if (t_4 <= 1e+273) {
		tmp = fma(x2, fma(x1, -12.0, -6.0), (0.0 - x1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = fma(Float64(x2 * Float64(x1 * 8.0)), x2, x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(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 <= -5e+271)
		tmp = t_1;
	elseif (t_4 <= 1e+273)
		tmp = fma(x2, fma(x1, -12.0, -6.0), Float64(0.0 - x1));
	else
		tmp = t_1;
	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[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision] * x2 + x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(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, -5e+271], t$95$1, If[LessEqual[t$95$4, 1e+273], N[(x2 * N[(x1 * -12.0 + -6.0), $MachinePrecision] + N[(0.0 - x1), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

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)))))) < -5.0000000000000003e271 or 9.99999999999999945e272 < (+.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 43.3%

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

   3. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 35.2

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

   5. Simplified 35.2%

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

   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 36.4

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

   8. Simplified 36.4%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 38.5

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

   10. Applied egg-rr 38.5%

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

   if -5.0000000000000003e271 < (+.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)))))) < 9.99999999999999945e272

   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

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 74.2

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

   5. Simplified 74.2%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft1-in N/A

         \[\leadsto x2 \cdot \left(-12 \cdot x1 - 6\right) + \color{blue}{\left(-2 + 1\right) \cdot x1} \]

      5. metadata-eval N/A

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

      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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), \color{blue}{0 - x1}\right) \]

      13. --lowering--.f64 79.7

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), \color{blue}{0 - x1}\right) \]

   8. Simplified 79.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), 0 - x1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{+271}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+273}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, -6\right), 0 - x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+289}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+279}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, -2, x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* 8.0 (* x1 (* x2 x2))))
        (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 -2e+289)
     t_1
     (if (<= t_4 4e+279) (+ x1 (fma x1 -2.0 (* x2 -6.0))) t_1))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = 8.0 * (x1 * (x2 * x2));
	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 <= -2e+289) {
		tmp = t_1;
	} else if (t_4 <= 4e+279) {
		tmp = x1 + fma(x1, -2.0, (x2 * -6.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(8.0 * Float64(x1 * Float64(x2 * x2)))
	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 <= -2e+289)
		tmp = t_1;
	elseif (t_4 <= 4e+279)
		tmp = Float64(x1 + fma(x1, -2.0, Float64(x2 * -6.0)));
	else
		tmp = t_1;
	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[(8.0 * N[(x1 * N[(x2 * x2), $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, -2e+289], t$95$1, If[LessEqual[t$95$4, 4e+279], N[(x1 + N[(x1 * -2.0 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

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)))))) < -2.0000000000000001e289 or 4.00000000000000023e279 < (+.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 41.1%

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

   3. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 36.3

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

   5. Simplified 36.3%

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

   6. Taylor expanded in x2 around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 36.8

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

   8. Simplified 36.8%

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

   if -2.0000000000000001e289 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.00000000000000023e279

   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

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 71.6

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

   5. Simplified 71.6%

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

   6. Taylor expanded in x2 around 0

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

   8. Simplified 78.3%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2 \cdot 10^{+289}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 4 \cdot 10^{+279}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, -2, x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_2}\\ \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_1 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_2}\right) \leq \infty:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{t\_1 - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(x1, 3 \cdot \frac{x1 \cdot t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(2, \frac{x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{\mathsf{fma}\left(x1, x1 \cdot 3, \mathsf{fma}\left(2, x2, 0 - x1\right)\right)}} \cdot \left(-3 + \frac{t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}\right), x1 \cdot \left(x1 \cdot \mathsf{fma}\left(3, 4, -6\right)\right)\right), x1 \cdot \mathsf{fma}\left(x1, x1, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (fma 3.0 (* x1 x1) (* 2.0 x2)) x1))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_1 (* 2.0 x2)) x1) t_2)))
   (if (<=
        (+
         x1
         (+
          (+
           x1
           (+
            (+
             (*
              t_2
              (+
               (* (* (* x1 2.0) t_3) (- t_3 3.0))
               (* (* x1 x1) (- (* t_3 4.0) 6.0))))
             (* t_1 t_3))
            (* x1 (* x1 x1))))
          (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_2))))
        INFINITY)
     (+
      x1
      (fma
       (/ (- t_1 (fma 2.0 x2 x1)) (fma x1 x1 1.0))
       3.0
       (fma
        x1
        (* 3.0 (/ (* x1 t_0) (fma x1 x1 1.0)))
        (fma
         (fma x1 x1 1.0)
         (fma
          2.0
          (*
           (/
            x1
            (/ (fma x1 x1 1.0) (fma x1 (* x1 3.0) (fma 2.0 x2 (- 0.0 x1)))))
           (+ -3.0 (/ t_0 (fma x1 x1 1.0))))
          (* x1 (* x1 (fma 3.0 4.0 -6.0))))
         (* x1 (fma x1 x1 1.0))))))
     (*
      x1
      (fma
       x1
       (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))
       1.0)))))

C

double code(double x1, double x2) {
	double t_0 = fma(3.0, (x1 * x1), (2.0 * x2)) - x1;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_1 + (2.0 * x2)) - x1) / t_2;
	double tmp;
	if ((x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_1 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)))) <= ((double) INFINITY)) {
		tmp = x1 + fma(((t_1 - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(x1, (3.0 * ((x1 * t_0) / fma(x1, x1, 1.0))), fma(fma(x1, x1, 1.0), fma(2.0, ((x1 / (fma(x1, x1, 1.0) / fma(x1, (x1 * 3.0), fma(2.0, x2, (0.0 - x1))))) * (-3.0 + (t_0 / fma(x1, x1, 1.0)))), (x1 * (x1 * fma(3.0, 4.0, -6.0)))), (x1 * fma(x1, x1, 1.0)))));
	} else {
		tmp = x1 * fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), 1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * N[(x1 * x1), $MachinePrecision] + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(x1 + N[(N[(N[(t$95$1 - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(x1 * N[(3.0 * N[(N[(x1 * t$95$0), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(2.0 * N[(N[(x1 / N[(N[(x1 * x1 + 1.0), $MachinePrecision] / N[(x1 * N[(x1 * 3.0), $MachinePrecision] + N[(2.0 * x2 + N[(0.0 - x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-3.0 + N[(t$95$0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * N[(3.0 * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision] + 1.0), $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

   4. Applied egg-rr 94.3%

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

   5. Taylor expanded in x1 around inf

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

   7. Simplified 93.1%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

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

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

      12. sub-neg N/A

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

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

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

      14. sub0-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(x1, 3 \cdot \frac{x1 \cdot \left(\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(2, \frac{x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{\mathsf{fma}\left(x1, x1 \cdot 3, \mathsf{fma}\left(2, x2, \color{blue}{0 - x1}\right)\right)}} \cdot \left(\frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} + -3\right), x1 \cdot \left(x1 \cdot \mathsf{fma}\left(3, 4, -6\right)\right)\right), \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right)\right) \]

      15. --lowering--.f64 98.5

          \[\leadsto x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(x1, 3 \cdot \frac{x1 \cdot \left(\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(2, \frac{x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{\mathsf{fma}\left(x1, x1 \cdot 3, \mathsf{fma}\left(2, x2, \color{blue}{0 - x1}\right)\right)}} \cdot \left(\frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} + -3\right), x1 \cdot \left(x1 \cdot \mathsf{fma}\left(3, 4, -6\right)\right)\right), \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right)\right) \]

   9. Applied egg-rr 98.5%

      \[\leadsto x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(x1, 3 \cdot \frac{x1 \cdot \left(\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(2, \color{blue}{\frac{x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{\mathsf{fma}\left(x1, x1 \cdot 3, \mathsf{fma}\left(2, x2, 0 - x1\right)\right)}}} \cdot \left(\frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} + -3\right), x1 \cdot \left(x1 \cdot \mathsf{fma}\left(3, 4, -6\right)\right)\right), \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right)\right) \]

   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

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   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 100.0%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(x1, 3 \cdot \frac{x1 \cdot \left(\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(2, \frac{x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{\mathsf{fma}\left(x1, x1 \cdot 3, \mathsf{fma}\left(2, x2, 0 - x1\right)\right)}} \cdot \left(-3 + \frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right), x1 \cdot \left(x1 \cdot \mathsf{fma}\left(3, 4, -6\right)\right)\right), x1 \cdot \mathsf{fma}\left(x1, x1, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1\\ t_1 := \frac{x1 \cdot t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_2 := x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 1\right)\\ \mathbf{if}\;x1 \leq -1.85 \cdot 10^{+68}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x1 \leq 1.7 \cdot 10^{+57}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(x1, 3 \cdot t\_1, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(2, t\_1 \cdot \left(-3 + \frac{t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}\right), x1 \cdot \left(x1 \cdot \mathsf{fma}\left(3, 4, -6\right)\right)\right), x1 \cdot \mathsf{fma}\left(x1, x1, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (fma 3.0 (* x1 x1) (* 2.0 x2)) x1))
        (t_1 (/ (* x1 t_0) (fma x1 x1 1.0)))
        (t_2
         (*
          x1
          (fma
           x1
           (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))
           1.0))))
   (if (<= x1 -1.85e+68)
     t_2
     (if (<= x1 1.7e+57)
       (+
        x1
        (fma
         (/ (- (* x1 (* x1 3.0)) (fma 2.0 x2 x1)) (fma x1 x1 1.0))
         3.0
         (fma
          x1
          (* 3.0 t_1)
          (fma
           (fma x1 x1 1.0)
           (fma
            2.0
            (* t_1 (+ -3.0 (/ t_0 (fma x1 x1 1.0))))
            (* x1 (* x1 (fma 3.0 4.0 -6.0))))
           (* x1 (fma x1 x1 1.0))))))
       t_2))))

C

double code(double x1, double x2) {
	double t_0 = fma(3.0, (x1 * x1), (2.0 * x2)) - x1;
	double t_1 = (x1 * t_0) / fma(x1, x1, 1.0);
	double t_2 = x1 * fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), 1.0);
	double tmp;
	if (x1 <= -1.85e+68) {
		tmp = t_2;
	} else if (x1 <= 1.7e+57) {
		tmp = x1 + fma((((x1 * (x1 * 3.0)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(x1, (3.0 * t_1), fma(fma(x1, x1, 1.0), fma(2.0, (t_1 * (-3.0 + (t_0 / fma(x1, x1, 1.0)))), (x1 * (x1 * fma(3.0, 4.0, -6.0)))), (x1 * fma(x1, x1, 1.0)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(3.0, Float64(x1 * x1), Float64(2.0 * x2)) - x1)
	t_1 = Float64(Float64(x1 * t_0) / fma(x1, x1, 1.0))
	t_2 = Float64(x1 * fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), 1.0))
	tmp = 0.0
	if (x1 <= -1.85e+68)
		tmp = t_2;
	elseif (x1 <= 1.7e+57)
		tmp = Float64(x1 + fma(Float64(Float64(Float64(x1 * Float64(x1 * 3.0)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(x1, Float64(3.0 * t_1), fma(fma(x1, x1, 1.0), fma(2.0, Float64(t_1 * Float64(-3.0 + Float64(t_0 / fma(x1, x1, 1.0)))), Float64(x1 * Float64(x1 * fma(3.0, 4.0, -6.0)))), Float64(x1 * fma(x1, x1, 1.0))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * N[(x1 * x1), $MachinePrecision] + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * t$95$0), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.85e+68], t$95$2, If[LessEqual[x1, 1.7e+57], N[(x1 + N[(N[(N[(N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(x1 * N[(3.0 * t$95$1), $MachinePrecision] + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(2.0 * N[(t$95$1 * N[(-3.0 + N[(t$95$0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * N[(3.0 * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.84999999999999999e68 or 1.69999999999999996e57 < x1

   1. Initial program 27.9%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   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 100.0%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   if -1.84999999999999999e68 < x1 < 1.69999999999999996e57

   1. Initial program 98.9%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x1 around inf

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

   7. Simplified 98.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.85 \cdot 10^{+68}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 1\right)\\ \mathbf{elif}\;x1 \leq 1.7 \cdot 10^{+57}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(x1, 3 \cdot \frac{x1 \cdot \left(\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(2, \frac{x1 \cdot \left(\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(-3 + \frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right), x1 \cdot \left(x1 \cdot \mathsf{fma}\left(3, 4, -6\right)\right)\right), x1 \cdot \mathsf{fma}\left(x1, x1, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 1\right)\\ t_1 := \mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1\\ \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 3.4 \cdot 10^{+57}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(x1, 3 \cdot \left(x1 \cdot 3\right), \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(2, \frac{x1 \cdot t\_1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(-3 + \frac{t\_1}{\mathsf{fma}\left(x1, x1, 1\right)}\right), x1 \cdot \left(x1 \cdot \mathsf{fma}\left(3, 4, -6\right)\right)\right), x1 \cdot \mathsf{fma}\left(x1, x1, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          x1
          (fma
           x1
           (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))
           1.0)))
        (t_1 (- (fma 3.0 (* x1 x1) (* 2.0 x2)) x1)))
   (if (<= x1 -3.6e+71)
     t_0
     (if (<= x1 3.4e+57)
       (+
        x1
        (fma
         (/ (- (* x1 (* x1 3.0)) (fma 2.0 x2 x1)) (fma x1 x1 1.0))
         3.0
         (fma
          x1
          (* 3.0 (* x1 3.0))
          (fma
           (fma x1 x1 1.0)
           (fma
            2.0
            (* (/ (* x1 t_1) (fma x1 x1 1.0)) (+ -3.0 (/ t_1 (fma x1 x1 1.0))))
            (* x1 (* x1 (fma 3.0 4.0 -6.0))))
           (* x1 (fma x1 x1 1.0))))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), 1.0);
	double t_1 = fma(3.0, (x1 * x1), (2.0 * x2)) - x1;
	double tmp;
	if (x1 <= -3.6e+71) {
		tmp = t_0;
	} else if (x1 <= 3.4e+57) {
		tmp = x1 + fma((((x1 * (x1 * 3.0)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(x1, (3.0 * (x1 * 3.0)), fma(fma(x1, x1, 1.0), fma(2.0, (((x1 * t_1) / fma(x1, x1, 1.0)) * (-3.0 + (t_1 / fma(x1, x1, 1.0)))), (x1 * (x1 * fma(3.0, 4.0, -6.0)))), (x1 * fma(x1, x1, 1.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), 1.0))
	t_1 = Float64(fma(3.0, Float64(x1 * x1), Float64(2.0 * x2)) - x1)
	tmp = 0.0
	if (x1 <= -3.6e+71)
		tmp = t_0;
	elseif (x1 <= 3.4e+57)
		tmp = Float64(x1 + fma(Float64(Float64(Float64(x1 * Float64(x1 * 3.0)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(x1, Float64(3.0 * Float64(x1 * 3.0)), fma(fma(x1, x1, 1.0), fma(2.0, Float64(Float64(Float64(x1 * t_1) / fma(x1, x1, 1.0)) * Float64(-3.0 + Float64(t_1 / fma(x1, x1, 1.0)))), Float64(x1 * Float64(x1 * fma(3.0, 4.0, -6.0)))), Float64(x1 * fma(x1, x1, 1.0))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * N[(x1 * x1), $MachinePrecision] + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, If[LessEqual[x1, -3.6e+71], t$95$0, If[LessEqual[x1, 3.4e+57], N[(x1 + N[(N[(N[(N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(x1 * N[(3.0 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(2.0 * N[(N[(N[(x1 * t$95$1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(-3.0 + N[(t$95$1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * N[(3.0 * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.6e71 or 3.39999999999999992e57 < x1

   1. Initial program 27.9%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   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 100.0%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   if -3.6e71 < x1 < 3.39999999999999992e57

   1. Initial program 98.9%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x1 around inf

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 98.3

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

   10. Simplified 98.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 1\right)\\ \mathbf{elif}\;x1 \leq 3.4 \cdot 10^{+57}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(x1, 3 \cdot \left(x1 \cdot 3\right), \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(2, \frac{x1 \cdot \left(\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(-3 + \frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right), x1 \cdot \left(x1 \cdot \mathsf{fma}\left(3, 4, -6\right)\right)\right), x1 \cdot \mathsf{fma}\left(x1, x1, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.9 \cdot 10^{+16}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right)\\ \mathbf{elif}\;x1 \leq 980:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), 0 - x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9 - \frac{\mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}\right)}{x1} - 3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.9e+16)
   (+
    x1
    (* (* x1 x1) (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))))
   (if (<= x1 980.0)
     (fma x2 (fma x1 -12.0 (fma 8.0 (* x1 x2) -6.0)) (- 0.0 x1))
     (+
      x1
      (*
       (pow x1 4.0)
       (+
        6.0
        (/
         (-
          (/
           (fma
            4.0
            (fma x2 2.0 -3.0)
            (- 9.0 (/ (* (fma x2 2.0 -3.0) -6.0) x1)))
           x1)
          3.0)
         x1)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.9e+16) {
		tmp = x1 + ((x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)));
	} else if (x1 <= 980.0) {
		tmp = fma(x2, fma(x1, -12.0, fma(8.0, (x1 * x2), -6.0)), (0.0 - x1));
	} else {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), (9.0 - ((fma(x2, 2.0, -3.0) * -6.0) / x1))) / x1) - 3.0) / x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.9e+16)
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0))));
	elseif (x1 <= 980.0)
		tmp = fma(x2, fma(x1, -12.0, fma(8.0, Float64(x1 * x2), -6.0)), Float64(0.0 - x1));
	else
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), Float64(9.0 - Float64(Float64(fma(x2, 2.0, -3.0) * -6.0) / x1))) / x1) - 3.0) / x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.9e+16], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 980.0], N[(x2 * N[(x1 * -12.0 + N[(8.0 * N[(x1 * x2), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(0.0 - x1), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + N[(9.0 - N[(N[(N[(x2 * 2.0 + -3.0), $MachinePrecision] * -6.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.9e16

   1. Initial program 27.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 97.9%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 97.9

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

   8. Simplified 97.9%

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

   if -1.9e16 < x1 < 980

   1. Initial program 98.8%

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

   3. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 83.5

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

   5. Simplified 83.5%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft1-in N/A

         \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{\left(-2 + 1\right) \cdot x1} \]

      5. metadata-eval N/A

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

      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. metadata-eval N/A

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

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

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

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

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

      14. mul-1-neg N/A

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

      15. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), \color{blue}{0 - x1}\right) \]

      16. --lowering--.f64 99.3

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), \color{blue}{0 - x1}\right) \]

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), 0 - x1\right)} \]

   if 980 < x1

   1. Initial program 49.1%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 94.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.9 \cdot 10^{+16}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right)\\ \mathbf{elif}\;x1 \leq 980:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), 0 - x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9 - \frac{\mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\\ \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+14}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), t\_0\right)\\ \mathbf{elif}\;x1 \leq 8500:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), 0 - x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{t\_0}{x1} - 3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma 4.0 (fma x2 2.0 -3.0) 9.0)))
   (if (<= x1 -1.4e+14)
     (+ x1 (* (* x1 x1) (fma x1 (fma x1 6.0 -3.0) t_0)))
     (if (<= x1 8500.0)
       (fma x2 (fma x1 -12.0 (fma 8.0 (* x1 x2) -6.0)) (- 0.0 x1))
       (+ 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 <= -1.4e+14) {
		tmp = x1 + ((x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), t_0));
	} else if (x1 <= 8500.0) {
		tmp = fma(x2, fma(x1, -12.0, fma(8.0, (x1 * x2), -6.0)), (0.0 - x1));
	} 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 <= -1.4e+14)
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * fma(x1, fma(x1, 6.0, -3.0), t_0)));
	elseif (x1 <= 8500.0)
		tmp = fma(x2, fma(x1, -12.0, fma(8.0, Float64(x1 * x2), -6.0)), Float64(0.0 - x1));
	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, -1.4e+14], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8500.0], N[(x2 * N[(x1 * -12.0 + N[(8.0 * N[(x1 * x2), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(0.0 - x1), $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 < -1.4e14

   1. Initial program 27.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 97.9%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 97.9

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

   8. Simplified 97.9%

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

   if -1.4e14 < x1 < 8500

   1. Initial program 98.8%

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

   3. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 83.5

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

   5. Simplified 83.5%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft1-in N/A

         \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{\left(-2 + 1\right) \cdot x1} \]

      5. metadata-eval N/A

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

      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. metadata-eval N/A

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

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

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

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

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

      14. mul-1-neg N/A

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

      15. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), \color{blue}{0 - x1}\right) \]

      16. --lowering--.f64 99.3

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), \color{blue}{0 - x1}\right) \]

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), 0 - x1\right)} \]

   if 8500 < x1

   1. Initial program 49.1%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-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 94.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+14}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right)\\ \mathbf{elif}\;x1 \leq 8500:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), 0 - x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, x1, 0\right)\right), 6, x1\right)\\ t_1 := \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -1\right), x2 \cdot -6\right)\\ \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -3.1 \cdot 10^{-226}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq 1.15 \cdot 10^{-237}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, -2, x2 \cdot -6\right)\\ \mathbf{elif}\;x1 \leq 6.5 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* x1 (* x1 (fma x1 x1 0.0))) 6.0 x1))
        (t_1 (fma x1 (fma x2 (fma x2 8.0 -12.0) -1.0) (* x2 -6.0))))
   (if (<= x1 -1.4e+14)
     t_0
     (if (<= x1 -3.1e-226)
       t_1
       (if (<= x1 1.15e-237)
         (+ x1 (fma x1 -2.0 (* x2 -6.0)))
         (if (<= x1 6.5e+24) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * (x1 * fma(x1, x1, 0.0))), 6.0, x1);
	double t_1 = fma(x1, fma(x2, fma(x2, 8.0, -12.0), -1.0), (x2 * -6.0));
	double tmp;
	if (x1 <= -1.4e+14) {
		tmp = t_0;
	} else if (x1 <= -3.1e-226) {
		tmp = t_1;
	} else if (x1 <= 1.15e-237) {
		tmp = x1 + fma(x1, -2.0, (x2 * -6.0));
	} else if (x1 <= 6.5e+24) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * Float64(x1 * fma(x1, x1, 0.0))), 6.0, x1)
	t_1 = fma(x1, fma(x2, fma(x2, 8.0, -12.0), -1.0), Float64(x2 * -6.0))
	tmp = 0.0
	if (x1 <= -1.4e+14)
		tmp = t_0;
	elseif (x1 <= -3.1e-226)
		tmp = t_1;
	elseif (x1 <= 1.15e-237)
		tmp = Float64(x1 + fma(x1, -2.0, Float64(x2 * -6.0)));
	elseif (x1 <= 6.5e+24)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * N[(x1 * N[(x1 * x1 + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0 + x1), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x2 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.4e+14], t$95$0, If[LessEqual[x1, -3.1e-226], t$95$1, If[LessEqual[x1, 1.15e-237], N[(x1 + N[(x1 * -2.0 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 6.5e+24], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.4e14 or 6.4999999999999996e24 < x1

   1. Initial program 37.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 94.1

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

   5. Simplified 94.1%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

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

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

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

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

      11. associate-*l* N/A

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

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

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

      13. +-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot \color{blue}{\left(0 + x1\right)}\right)\right), 6, x1\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 + 0\right)}\right)\right), 6, x1\right) \]

      15. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot x1 + 0 \cdot x1\right)}\right), 6, x1\right) \]

      16. mul0-lft N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1 + \color{blue}{0}\right)\right), 6, x1\right) \]

      17. accelerator-lowering-fma.f64 94.2

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \color{blue}{\mathsf{fma}\left(x1, x1, 0\right)}\right), 6, x1\right) \]

   7. Applied egg-rr 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, x1, 0\right)\right), 6, x1\right)} \]

   if -1.4e14 < x1 < -3.09999999999999989e-226 or 1.15000000000000006e-237 < x1 < 6.4999999999999996e24

   1. Initial program 98.5%

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

   3. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 87.4

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

   5. Simplified 87.4%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 87.4

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

   8. Simplified 87.4%

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

   if -3.09999999999999989e-226 < x1 < 1.15000000000000006e-237

   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

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 68.7

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

   5. Simplified 68.7%

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

   6. Taylor expanded in x2 around 0

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

   8. Simplified 99.3%

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

Alternative 12: 95.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right)\\ \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+14}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot t\_0\\ \mathbf{elif}\;x1 \leq 35:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), 0 - x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, t\_0, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))))
   (if (<= x1 -1.4e+14)
     (+ x1 (* (* x1 x1) t_0))
     (if (<= x1 35.0)
       (fma x2 (fma x1 -12.0 (fma 8.0 (* x1 x2) -6.0)) (- 0.0 x1))
       (* x1 (fma x1 t_0 1.0))))))

C

double code(double x1, double x2) {
	double t_0 = fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0));
	double tmp;
	if (x1 <= -1.4e+14) {
		tmp = x1 + ((x1 * x1) * t_0);
	} else if (x1 <= 35.0) {
		tmp = fma(x2, fma(x1, -12.0, fma(8.0, (x1 * x2), -6.0)), (0.0 - x1));
	} else {
		tmp = x1 * fma(x1, t_0, 1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0))
	tmp = 0.0
	if (x1 <= -1.4e+14)
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * t_0));
	elseif (x1 <= 35.0)
		tmp = fma(x2, fma(x1, -12.0, fma(8.0, Float64(x1 * x2), -6.0)), Float64(0.0 - x1));
	else
		tmp = Float64(x1 * fma(x1, t_0, 1.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.4e+14], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 35.0], N[(x2 * N[(x1 * -12.0 + N[(8.0 * N[(x1 * x2), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(0.0 - x1), $MachinePrecision]), $MachinePrecision], N[(x1 * N[(x1 * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.4e14

   1. Initial program 27.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 97.9%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 97.9

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

   8. Simplified 97.9%

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

   if -1.4e14 < x1 < 35

   1. Initial program 98.8%

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

   3. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 83.5

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

   5. Simplified 83.5%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft1-in N/A

         \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{\left(-2 + 1\right) \cdot x1} \]

      5. metadata-eval N/A

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

      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. metadata-eval N/A

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

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

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

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

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

      14. mul-1-neg N/A

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

      15. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), \color{blue}{0 - x1}\right) \]

      16. --lowering--.f64 99.3

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), \color{blue}{0 - x1}\right) \]

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), 0 - x1\right)} \]

   if 35 < x1

   1. Initial program 49.1%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-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 94.5%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 94.5

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

   8. Simplified 94.5%

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

Alternative 13: 95.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), 1\right)\\ \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2700:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), 0 - x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          x1
          (fma
           x1
           (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))
           1.0))))
   (if (<= x1 -1.4e+14)
     t_0
     (if (<= x1 2700.0)
       (fma x2 (fma x1 -12.0 (fma 8.0 (* x1 x2) -6.0)) (- 0.0 x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), 1.0);
	double tmp;
	if (x1 <= -1.4e+14) {
		tmp = t_0;
	} else if (x1 <= 2700.0) {
		tmp = fma(x2, fma(x1, -12.0, fma(8.0, (x1 * x2), -6.0)), (0.0 - x1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(x1, fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), 1.0))
	tmp = 0.0
	if (x1 <= -1.4e+14)
		tmp = t_0;
	elseif (x1 <= 2700.0)
		tmp = fma(x2, fma(x1, -12.0, fma(8.0, Float64(x1 * x2), -6.0)), Float64(0.0 - x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.4e+14], t$95$0, If[LessEqual[x1, 2700.0], N[(x2 * N[(x1 * -12.0 + N[(8.0 * N[(x1 * x2), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(0.0 - x1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.4e14 or 2700 < x1

   1. Initial program 40.3%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   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 95.9%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 95.9

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

   8. Simplified 95.9%

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

   if -1.4e14 < x1 < 2700

   1. Initial program 98.8%

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

   3. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 83.5

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

   5. Simplified 83.5%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft1-in N/A

         \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{\left(-2 + 1\right) \cdot x1} \]

      5. metadata-eval N/A

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

      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. metadata-eval N/A

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

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

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

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

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

      14. mul-1-neg N/A

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

      15. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), \color{blue}{0 - x1}\right) \]

      16. --lowering--.f64 99.3

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), \color{blue}{0 - x1}\right) \]

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), 0 - x1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 80.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, x1, 0\right)\right), 6, x1\right)\\ \mathbf{if}\;x1 \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -2.6 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ \mathbf{elif}\;x1 \leq 1.35:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -12, -2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* x1 (* x1 (fma x1 x1 0.0))) 6.0 x1)))
   (if (<= x1 -5.2e+14)
     t_0
     (if (<= x1 -2.6e-26)
       (fma (* x2 (* x1 8.0)) x2 x1)
       (if (<= x1 1.35)
         (+ x1 (fma x1 (fma x2 -12.0 -2.0) (* x2 -6.0)))
         t_0)))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * (x1 * fma(x1, x1, 0.0))), 6.0, x1);
	double tmp;
	if (x1 <= -5.2e+14) {
		tmp = t_0;
	} else if (x1 <= -2.6e-26) {
		tmp = fma((x2 * (x1 * 8.0)), x2, x1);
	} else if (x1 <= 1.35) {
		tmp = x1 + fma(x1, fma(x2, -12.0, -2.0), (x2 * -6.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * Float64(x1 * fma(x1, x1, 0.0))), 6.0, x1)
	tmp = 0.0
	if (x1 <= -5.2e+14)
		tmp = t_0;
	elseif (x1 <= -2.6e-26)
		tmp = fma(Float64(x2 * Float64(x1 * 8.0)), x2, x1);
	elseif (x1 <= 1.35)
		tmp = Float64(x1 + fma(x1, fma(x2, -12.0, -2.0), 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 * N[(x1 * x1 + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0 + x1), $MachinePrecision]}, If[LessEqual[x1, -5.2e+14], t$95$0, If[LessEqual[x1, -2.6e-26], N[(N[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision] * x2 + x1), $MachinePrecision], If[LessEqual[x1, 1.35], N[(x1 + N[(x1 * N[(x2 * -12.0 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -5.2e14 or 1.3500000000000001 < x1

   1. Initial program 40.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 90.1

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

   5. Simplified 90.1%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

         \[\leadsto 6 \cdot \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} + x1 \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot 6} + x1 \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right), 6, x1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)}, 6, x1\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(\left(x1 \cdot x1\right) \cdot x1\right)}, 6, x1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right)}, 6, x1\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot x1\right)\right)}, 6, x1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot x1\right)\right)}, 6, x1\right) \]

      13. +-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot \color{blue}{\left(0 + x1\right)}\right)\right), 6, x1\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 + 0\right)}\right)\right), 6, x1\right) \]

      15. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot x1 + 0 \cdot x1\right)}\right), 6, x1\right) \]

      16. mul0-lft N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1 + \color{blue}{0}\right)\right), 6, x1\right) \]

      17. accelerator-lowering-fma.f64 90.1

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \color{blue}{\mathsf{fma}\left(x1, x1, 0\right)}\right), 6, x1\right) \]

   7. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, x1, 0\right)\right), 6, x1\right)} \]

   if -5.2e14 < x1 < -2.6000000000000001e-26

   1. Initial program 100.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

      3. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      5. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      6. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

      9. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      10. distribute-lft-in N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      12. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

      15. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

      16. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

      17. *-lowering-*.f64 81.6

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   5. Simplified 81.6%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto x1 + \color{blue}{8 \cdot \left(x1 \cdot {x2}^{2}\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto x1 + 8 \cdot \color{blue}{\left(x1 \cdot {x2}^{2}\right)} \]

      3. unpow2 N/A

         \[\leadsto x1 + 8 \cdot \left(x1 \cdot \color{blue}{\left(x2 \cdot x2\right)}\right) \]

      4. *-lowering-*.f64 81.6

         \[\leadsto x1 + 8 \cdot \left(x1 \cdot \color{blue}{\left(x2 \cdot x2\right)}\right) \]

   8. Simplified 81.6%

      \[\leadsto x1 + \color{blue}{8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right) + x1} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(8 \cdot x1\right) \cdot \left(x2 \cdot x2\right)} + x1 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(8 \cdot x1\right) \cdot x2\right) \cdot x2} + x1 \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(8 \cdot x1\right) \cdot x2, x2, x1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(8 \cdot x1\right) \cdot x2}, x2, x1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x1 \cdot 8\right)} \cdot x2, x2, x1\right) \]

      7. *-lowering-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x1 \cdot 8\right)} \cdot x2, x2, x1\right) \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x1 \cdot 8\right) \cdot x2, x2, x1\right)} \]

   if -2.6000000000000001e-26 < x1 < 1.3500000000000001

   1. Initial program 98.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

      3. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      5. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      6. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

      9. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      10. distribute-lft-in N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      12. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

      15. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

      16. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

      17. *-lowering-*.f64 83.5

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   5. Simplified 83.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{-12}, -2\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Simplified 80.0%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{-12}, -2\right), x2 \cdot -6\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, x1, 0\right)\right), 6, x1\right)\\ \mathbf{elif}\;x1 \leq -2.6 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ \mathbf{elif}\;x1 \leq 1.35:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -12, -2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, x1, 0\right)\right), 6, x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.9% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(6, x1 \cdot \left(x1 \cdot x1\right), 1\right)\\ \mathbf{if}\;x1 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -1.7 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ \mathbf{elif}\;x1 \leq 1.35:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -12, -2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (fma 6.0 (* x1 (* x1 x1)) 1.0))))
   (if (<= x1 -2e+14)
     t_0
     (if (<= x1 -1.7e-25)
       (fma (* x2 (* x1 8.0)) x2 x1)
       (if (<= x1 1.35)
         (+ x1 (fma x1 (fma x2 -12.0 -2.0) (* x2 -6.0)))
         t_0)))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma(6.0, (x1 * (x1 * x1)), 1.0);
	double tmp;
	if (x1 <= -2e+14) {
		tmp = t_0;
	} else if (x1 <= -1.7e-25) {
		tmp = fma((x2 * (x1 * 8.0)), x2, x1);
	} else if (x1 <= 1.35) {
		tmp = x1 + fma(x1, fma(x2, -12.0, -2.0), (x2 * -6.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(6.0, Float64(x1 * Float64(x1 * x1)), 1.0))
	tmp = 0.0
	if (x1 <= -2e+14)
		tmp = t_0;
	elseif (x1 <= -1.7e-25)
		tmp = fma(Float64(x2 * Float64(x1 * 8.0)), x2, x1);
	elseif (x1 <= 1.35)
		tmp = Float64(x1 + fma(x1, fma(x2, -12.0, -2.0), Float64(x2 * -6.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(6.0 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -2e+14], t$95$0, If[LessEqual[x1, -1.7e-25], N[(N[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision] * x2 + x1), $MachinePrecision], If[LessEqual[x1, 1.35], N[(x1 + N[(x1 * N[(x2 * -12.0 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2e14 or 1.3500000000000001 < x1

   1. Initial program 40.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 90.1

         \[\leadsto x1 + 6 \cdot \color{blue}{{x1}^{4}} \]

   5. Simplified 90.1%

      \[\leadsto x1 + \color{blue}{6 \cdot {x1}^{4}} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{x1 \cdot \left(1 + 6 \cdot {x1}^{3}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x1 \cdot \left(1 + 6 \cdot {x1}^{3}\right)} \]

      2. +-commutative N/A

         \[\leadsto x1 \cdot \color{blue}{\left(6 \cdot {x1}^{3} + 1\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(6, {x1}^{3}, 1\right)} \]

      4. cube-mult N/A

         \[\leadsto x1 \cdot \mathsf{fma}\left(6, \color{blue}{x1 \cdot \left(x1 \cdot x1\right)}, 1\right) \]

      5. unpow2 N/A

         \[\leadsto x1 \cdot \mathsf{fma}\left(6, x1 \cdot \color{blue}{{x1}^{2}}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x1 \cdot \mathsf{fma}\left(6, \color{blue}{x1 \cdot {x1}^{2}}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto x1 \cdot \mathsf{fma}\left(6, x1 \cdot \color{blue}{\left(x1 \cdot x1\right)}, 1\right) \]

      8. *-lowering-*.f64 90.1

         \[\leadsto x1 \cdot \mathsf{fma}\left(6, x1 \cdot \color{blue}{\left(x1 \cdot x1\right)}, 1\right) \]

   8. Simplified 90.1%

      \[\leadsto \color{blue}{x1 \cdot \mathsf{fma}\left(6, x1 \cdot \left(x1 \cdot x1\right), 1\right)} \]

   if -2e14 < x1 < -1.70000000000000001e-25

   1. Initial program 100.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

      3. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      5. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      6. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

      9. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      10. distribute-lft-in N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      12. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

      15. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

      16. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

      17. *-lowering-*.f64 81.6

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   5. Simplified 81.6%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto x1 + \color{blue}{8 \cdot \left(x1 \cdot {x2}^{2}\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto x1 + 8 \cdot \color{blue}{\left(x1 \cdot {x2}^{2}\right)} \]

      3. unpow2 N/A

         \[\leadsto x1 + 8 \cdot \left(x1 \cdot \color{blue}{\left(x2 \cdot x2\right)}\right) \]

      4. *-lowering-*.f64 81.6

         \[\leadsto x1 + 8 \cdot \left(x1 \cdot \color{blue}{\left(x2 \cdot x2\right)}\right) \]

   8. Simplified 81.6%

      \[\leadsto x1 + \color{blue}{8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right) + x1} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(8 \cdot x1\right) \cdot \left(x2 \cdot x2\right)} + x1 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(8 \cdot x1\right) \cdot x2\right) \cdot x2} + x1 \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(8 \cdot x1\right) \cdot x2, x2, x1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(8 \cdot x1\right) \cdot x2}, x2, x1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x1 \cdot 8\right)} \cdot x2, x2, x1\right) \]

      7. *-lowering-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x1 \cdot 8\right)} \cdot x2, x2, x1\right) \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x1 \cdot 8\right) \cdot x2, x2, x1\right)} \]

   if -1.70000000000000001e-25 < x1 < 1.3500000000000001

   1. Initial program 98.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

      3. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      5. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      6. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

      9. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      10. distribute-lft-in N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      12. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

      15. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

      16. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

      17. *-lowering-*.f64 83.5

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   5. Simplified 83.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{-12}, -2\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Simplified 80.0%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{-12}, -2\right), x2 \cdot -6\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(6, x1 \cdot \left(x1 \cdot x1\right), 1\right)\\ \mathbf{elif}\;x1 \leq -1.7 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ \mathbf{elif}\;x1 \leq 1.35:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -12, -2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(6, x1 \cdot \left(x1 \cdot x1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 92.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, x1, 0\right)\right), 6, x1\right)\\ \mathbf{if}\;x1 \leq -1.9 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 6.5 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), 0 - x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* x1 (* x1 (fma x1 x1 0.0))) 6.0 x1)))
   (if (<= x1 -1.9e+16)
     t_0
     (if (<= x1 6.5e+24)
       (fma x2 (fma x1 -12.0 (fma 8.0 (* x1 x2) -6.0)) (- 0.0 x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * (x1 * fma(x1, x1, 0.0))), 6.0, x1);
	double tmp;
	if (x1 <= -1.9e+16) {
		tmp = t_0;
	} else if (x1 <= 6.5e+24) {
		tmp = fma(x2, fma(x1, -12.0, fma(8.0, (x1 * x2), -6.0)), (0.0 - x1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * Float64(x1 * fma(x1, x1, 0.0))), 6.0, x1)
	tmp = 0.0
	if (x1 <= -1.9e+16)
		tmp = t_0;
	elseif (x1 <= 6.5e+24)
		tmp = fma(x2, fma(x1, -12.0, fma(8.0, Float64(x1 * x2), -6.0)), Float64(0.0 - x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * N[(x1 * N[(x1 * x1 + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0 + x1), $MachinePrecision]}, If[LessEqual[x1, -1.9e+16], t$95$0, If[LessEqual[x1, 6.5e+24], N[(x2 * N[(x1 * -12.0 + N[(8.0 * N[(x1 * x2), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(0.0 - x1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.9e16 or 6.4999999999999996e24 < x1

   1. Initial program 37.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 94.1

         \[\leadsto x1 + 6 \cdot \color{blue}{{x1}^{4}} \]

   5. Simplified 94.1%

      \[\leadsto x1 + \color{blue}{6 \cdot {x1}^{4}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{6 \cdot {x1}^{4} + x1} \]

      2. metadata-eval N/A

         \[\leadsto 6 \cdot {x1}^{\color{blue}{\left(2 \cdot 2\right)}} + x1 \]

      3. pow-pow N/A

         \[\leadsto 6 \cdot \color{blue}{{\left({x1}^{2}\right)}^{2}} + x1 \]

      4. pow2 N/A

         \[\leadsto 6 \cdot {\color{blue}{\left(x1 \cdot x1\right)}}^{2} + x1 \]

      5. pow2 N/A

         \[\leadsto 6 \cdot \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} + x1 \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot 6} + x1 \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right), 6, x1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)}, 6, x1\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(\left(x1 \cdot x1\right) \cdot x1\right)}, 6, x1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right)}, 6, x1\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot x1\right)\right)}, 6, x1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot x1\right)\right)}, 6, x1\right) \]

      13. +-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot \color{blue}{\left(0 + x1\right)}\right)\right), 6, x1\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 + 0\right)}\right)\right), 6, x1\right) \]

      15. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot x1 + 0 \cdot x1\right)}\right), 6, x1\right) \]

      16. mul0-lft N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1 + \color{blue}{0}\right)\right), 6, x1\right) \]

      17. accelerator-lowering-fma.f64 94.2

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot \color{blue}{\mathsf{fma}\left(x1, x1, 0\right)}\right), 6, x1\right) \]

   7. Applied egg-rr 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, x1, 0\right)\right), 6, x1\right)} \]

   if -1.9e16 < x1 < 6.4999999999999996e24

   1. Initial program 98.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

      3. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      5. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      6. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

      9. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      10. distribute-lft-in N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      12. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

      15. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

      16. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

      17. *-lowering-*.f64 81.5

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   5. Simplified 81.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 + \left(-2 \cdot x1 + x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(-2 \cdot x1 + x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right) + x1} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + -2 \cdot x1\right)} + x1 \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \left(-2 \cdot x1 + x1\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{\left(-2 + 1\right) \cdot x1} \]

      5. metadata-eval N/A

         \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{-1} \cdot x1 \]

      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. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, 8 \cdot \left(x1 \cdot x2\right) + \color{blue}{-6}\right), -1 \cdot x1\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \color{blue}{\mathsf{fma}\left(8, x1 \cdot x2, -6\right)}\right), -1 \cdot x1\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, \color{blue}{x1 \cdot x2}, -6\right)\right), -1 \cdot x1\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), \color{blue}{\mathsf{neg}\left(x1\right)}\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), \color{blue}{0 - x1}\right) \]

      16. --lowering--.f64 96.6

          \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), \color{blue}{0 - x1}\right) \]

   8. Simplified 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, -12, \mathsf{fma}\left(8, x1 \cdot x2, -6\right)\right), 0 - x1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 60.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -3 \cdot 10^{+141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x1, x1, 0\right)}{0 - x1}\\ \mathbf{elif}\;x1 \leq 2.4 \cdot 10^{-12}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -12, -2\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ \mathbf{else}:\\ \;\;\;\;x2 \cdot \left(-6 + \frac{x1}{x2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -3e+141)
   (/ (fma x1 x1 0.0) (- 0.0 x1))
   (if (<= x1 2.4e-12)
     (+ x1 (fma x1 (fma x2 -12.0 -2.0) (* x2 -6.0)))
     (if (<= x1 2e+271)
       (fma (* x2 (* x1 8.0)) x2 x1)
       (* x2 (+ -6.0 (/ x1 x2)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -3e+141) {
		tmp = fma(x1, x1, 0.0) / (0.0 - x1);
	} else if (x1 <= 2.4e-12) {
		tmp = x1 + fma(x1, fma(x2, -12.0, -2.0), (x2 * -6.0));
	} else if (x1 <= 2e+271) {
		tmp = fma((x2 * (x1 * 8.0)), x2, x1);
	} else {
		tmp = x2 * (-6.0 + (x1 / x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -3e+141)
		tmp = Float64(fma(x1, x1, 0.0) / Float64(0.0 - x1));
	elseif (x1 <= 2.4e-12)
		tmp = Float64(x1 + fma(x1, fma(x2, -12.0, -2.0), Float64(x2 * -6.0)));
	elseif (x1 <= 2e+271)
		tmp = fma(Float64(x2 * Float64(x1 * 8.0)), x2, x1);
	else
		tmp = Float64(x2 * Float64(-6.0 + Float64(x1 / x2)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -3e+141], N[(N[(x1 * x1 + 0.0), $MachinePrecision] / N[(0.0 - x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.4e-12], N[(x1 + N[(x1 * N[(x2 * -12.0 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2e+271], N[(N[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision] * x2 + x1), $MachinePrecision], N[(x2 * N[(-6.0 + N[(x1 / x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -2.9999999999999999e141

   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

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

      3. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      5. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      6. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

      9. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      10. distribute-lft-in N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      12. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

      15. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

      16. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

      17. *-lowering-*.f64 3.2

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   5. Simplified 3.2%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 + -2 \cdot x1} \]
   7. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-2 + 1\right) \cdot x1} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{-1} \cdot x1 \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - x1} \]

      5. --lowering--.f64 6.9

         \[\leadsto \color{blue}{0 - x1} \]

   8. Simplified 6.9%

      \[\leadsto \color{blue}{0 - x1} \]
   9. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - x1 \cdot x1}{0 + x1}} \]

      2. +-lft-identity N/A

         \[\leadsto \frac{0 \cdot 0 - x1 \cdot x1}{\color{blue}{x1}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - x1 \cdot x1}{x1}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - x1 \cdot x1}{x1} \]

      5. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{0 - x1 \cdot x1}}{x1} \]

      6. +-lft-identity N/A

         \[\leadsto \frac{0 - x1 \cdot \color{blue}{\left(0 + x1\right)}}{x1} \]

      7. +-commutative N/A

         \[\leadsto \frac{0 - x1 \cdot \color{blue}{\left(x1 + 0\right)}}{x1} \]

      8. distribute-rgt-out N/A

         \[\leadsto \frac{0 - \color{blue}{\left(x1 \cdot x1 + 0 \cdot x1\right)}}{x1} \]

      9. mul0-lft N/A

         \[\leadsto \frac{0 - \left(x1 \cdot x1 + \color{blue}{0}\right)}{x1} \]

      10. accelerator-lowering-fma.f64 90.5

          \[\leadsto \frac{0 - \color{blue}{\mathsf{fma}\left(x1, x1, 0\right)}}{x1} \]

   10. Applied egg-rr 90.5%

       \[\leadsto \color{blue}{\frac{0 - \mathsf{fma}\left(x1, x1, 0\right)}{x1}} \]

   if -2.9999999999999999e141 < x1 < 2.39999999999999987e-12

   1. Initial program 95.6%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

      3. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      5. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      6. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

      9. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      10. distribute-lft-in N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      12. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

      15. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

      16. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

      17. *-lowering-*.f64 74.3

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   5. Simplified 74.3%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{-12}, -2\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Simplified 70.1%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{-12}, -2\right), x2 \cdot -6\right) \]

   if 2.39999999999999987e-12 < x1 < 1.99999999999999991e271

   1. Initial program 59.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

      3. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      5. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      6. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

      9. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      10. distribute-lft-in N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      12. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

      15. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

      16. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

      17. *-lowering-*.f64 36.5

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   5. Simplified 36.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto x1 + \color{blue}{8 \cdot \left(x1 \cdot {x2}^{2}\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto x1 + 8 \cdot \color{blue}{\left(x1 \cdot {x2}^{2}\right)} \]

      3. unpow2 N/A

         \[\leadsto x1 + 8 \cdot \left(x1 \cdot \color{blue}{\left(x2 \cdot x2\right)}\right) \]

      4. *-lowering-*.f64 39.7

         \[\leadsto x1 + 8 \cdot \left(x1 \cdot \color{blue}{\left(x2 \cdot x2\right)}\right) \]

   8. Simplified 39.7%

      \[\leadsto x1 + \color{blue}{8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right) + x1} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(8 \cdot x1\right) \cdot \left(x2 \cdot x2\right)} + x1 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(8 \cdot x1\right) \cdot x2\right) \cdot x2} + x1 \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(8 \cdot x1\right) \cdot x2, x2, x1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(8 \cdot x1\right) \cdot x2}, x2, x1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x1 \cdot 8\right)} \cdot x2, x2, x1\right) \]

      7. *-lowering-*.f64 39.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x1 \cdot 8\right)} \cdot x2, x2, x1\right) \]

   10. Applied egg-rr 39.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x1 \cdot 8\right) \cdot x2, x2, x1\right)} \]

   if 1.99999999999999991e271 < x1

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. 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 10.4

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Simplified 10.4%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x2 around inf

      \[\leadsto \color{blue}{x2 \cdot \left(\frac{x1}{x2} - 6\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x2 \cdot \left(\frac{x1}{x2} - 6\right)} \]

      2. sub-neg N/A

         \[\leadsto x2 \cdot \color{blue}{\left(\frac{x1}{x2} + \left(\mathsf{neg}\left(6\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto x2 \cdot \left(\frac{x1}{x2} + \color{blue}{-6}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto x2 \cdot \color{blue}{\left(\frac{x1}{x2} + -6\right)} \]

      5. /-lowering-/.f64 76.1

         \[\leadsto x2 \cdot \left(\color{blue}{\frac{x1}{x2}} + -6\right) \]

   8. Simplified 76.1%

      \[\leadsto \color{blue}{x2 \cdot \left(\frac{x1}{x2} + -6\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3 \cdot 10^{+141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x1, x1, 0\right)}{0 - x1}\\ \mathbf{elif}\;x1 \leq 2.4 \cdot 10^{-12}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -12, -2\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot \left(x1 \cdot 8\right), x2, x1\right)\\ \mathbf{else}:\\ \;\;\;\;x2 \cdot \left(-6 + \frac{x1}{x2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 32.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + -6 \cdot \left(x1 + x2\right)\\ \mathbf{if}\;2 \cdot x2 \leq -1 \cdot 10^{-193}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;2 \cdot x2 \leq 10^{-144}:\\ \;\;\;\;0 - x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (* -6.0 (+ x1 x2)))))
   (if (<= (* 2.0 x2) -1e-193)
     t_0
     (if (<= (* 2.0 x2) 1e-144) (- 0.0 x1) t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (-6.0 * (x1 + x2));
	double tmp;
	if ((2.0 * x2) <= -1e-193) {
		tmp = t_0;
	} else if ((2.0 * x2) <= 1e-144) {
		tmp = 0.0 - x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x1 + ((-6.0d0) * (x1 + x2))
    if ((2.0d0 * x2) <= (-1d-193)) then
        tmp = t_0
    else if ((2.0d0 * x2) <= 1d-144) then
        tmp = 0.0d0 - x1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = x1 + (-6.0 * (x1 + x2));
	double tmp;
	if ((2.0 * x2) <= -1e-193) {
		tmp = t_0;
	} else if ((2.0 * x2) <= 1e-144) {
		tmp = 0.0 - x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = x1 + (-6.0 * (x1 + x2))
	tmp = 0
	if (2.0 * x2) <= -1e-193:
		tmp = t_0
	elif (2.0 * x2) <= 1e-144:
		tmp = 0.0 - x1
	else:
		tmp = t_0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64(-6.0 * Float64(x1 + x2)))
	tmp = 0.0
	if (Float64(2.0 * x2) <= -1e-193)
		tmp = t_0;
	elseif (Float64(2.0 * x2) <= 1e-144)
		tmp = Float64(0.0 - x1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 + (-6.0 * (x1 + x2));
	tmp = 0.0;
	if ((2.0 * x2) <= -1e-193)
		tmp = t_0;
	elseif ((2.0 * x2) <= 1e-144)
		tmp = 0.0 - x1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(-6.0 * N[(x1 + x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * x2), $MachinePrecision], -1e-193], t$95$0, If[LessEqual[N[(2.0 * x2), $MachinePrecision], 1e-144], N[(0.0 - x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -1e-193 or 9.9999999999999995e-145 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 70.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \left(\left(\left(\left(\color{blue}{\left({x1}^{2} \cdot \left(6 - 4 \cdot \frac{1}{x1}\right)\right)} \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\color{blue}{\left(x1 \cdot x1\right)} \cdot \left(6 - 4 \cdot \frac{1}{x1}\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. associate-*l* N/A

         \[\leadsto x1 + \left(\left(\left(\left(\color{blue}{\left(x1 \cdot \left(x1 \cdot \left(6 - 4 \cdot \frac{1}{x1}\right)\right)\right)} \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\color{blue}{\left(x1 \cdot \left(x1 \cdot \left(6 - 4 \cdot \frac{1}{x1}\right)\right)\right)} \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(x1 \cdot \color{blue}{\left(x1 \cdot \left(6 - 4 \cdot \frac{1}{x1}\right)\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(x1 \cdot \left(x1 \cdot \color{blue}{\left(6 - 4 \cdot \frac{1}{x1}\right)}\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      6. associate-*r/ N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(x1 \cdot \left(x1 \cdot \left(6 - \color{blue}{\frac{4 \cdot 1}{x1}}\right)\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(x1 \cdot \left(x1 \cdot \left(6 - \frac{\color{blue}{4}}{x1}\right)\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      8. /-lowering-/.f64 49.1

         \[\leadsto x1 + \left(\left(\left(\left(\left(x1 \cdot \left(x1 \cdot \left(6 - \color{blue}{\frac{4}{x1}}\right)\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   5. Simplified 49.1%

      \[\leadsto x1 + \left(\left(\left(\left(\color{blue}{\left(x1 \cdot \left(x1 \cdot \left(6 - \frac{4}{x1}\right)\right)\right)} \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   6. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto x1 + \left(\left(\color{blue}{\left(\left(x1 \cdot \left(x1 \cdot \left(6 - \frac{4}{x1}\right)\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1\right) \cdot x1\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. *-commutative N/A

         \[\leadsto x1 + \left(\left(\left(\color{blue}{\left(x1 \cdot x1 + 1\right) \cdot \left(x1 \cdot \left(x1 \cdot \left(6 - \frac{4}{x1}\right)\right)\right)} + \left(\left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1\right) \cdot x1\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      3. associate-*r* N/A

         \[\leadsto x1 + \left(\left(\left(\color{blue}{\left(\left(x1 \cdot x1 + 1\right) \cdot x1\right) \cdot \left(x1 \cdot \left(6 - \frac{4}{x1}\right)\right)} + \left(\left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1\right) \cdot x1\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \left(\left(\color{blue}{\mathsf{fma}\left(\left(x1 \cdot x1 + 1\right) \cdot x1, x1 \cdot \left(6 - \frac{4}{x1}\right), \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} + \left(x1 \cdot x1\right) \cdot x1\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   7. Applied egg-rr 54.9%

      \[\leadsto x1 + \left(\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right) \cdot x1, x1 \cdot \left(6 + \frac{-4}{x1}\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, x1, 0\right), \frac{\mathsf{fma}\left(x1, x1 \cdot 3, 2 \cdot x2 - x1\right) \cdot \mathsf{fma}\left(x1, x1 \cdot 3, 0\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   8. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x1 + -6 \cdot x2\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto x1 + \color{blue}{-6 \cdot \left(x1 + x2\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto x1 + \color{blue}{-6 \cdot \left(x1 + x2\right)} \]

      3. +-lowering-+.f64 38.0

         \[\leadsto x1 + -6 \cdot \color{blue}{\left(x1 + x2\right)} \]

   10. Simplified 38.0%

       \[\leadsto x1 + \color{blue}{-6 \cdot \left(x1 + x2\right)} \]

   if -1e-193 < (*.f64 #s(literal 2 binary64) x2) < 9.9999999999999995e-145

   1. Initial program 73.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

      3. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      5. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      6. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

      9. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      10. distribute-lft-in N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      12. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

      15. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

      16. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

      17. *-lowering-*.f64 49.7

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   5. Simplified 49.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 + -2 \cdot x1} \]
   7. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-2 + 1\right) \cdot x1} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{-1} \cdot x1 \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - x1} \]

      5. --lowering--.f64 43.9

         \[\leadsto \color{blue}{0 - x1} \]

   8. Simplified 43.9%

      \[\leadsto \color{blue}{0 - x1} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

      2. neg-lowering-neg.f64 43.9

         \[\leadsto \color{blue}{-x1} \]

   10. Applied egg-rr 43.9%

       \[\leadsto \color{blue}{-x1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -1 \cdot 10^{-193}:\\ \;\;\;\;x1 + -6 \cdot \left(x1 + x2\right)\\ \mathbf{elif}\;2 \cdot x2 \leq 10^{-144}:\\ \;\;\;\;0 - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + -6 \cdot \left(x1 + x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 32.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + x2 \cdot -6\\ \mathbf{if}\;2 \cdot x2 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;2 \cdot x2 \leq 5 \cdot 10^{-150}:\\ \;\;\;\;0 - x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (* x2 -6.0))))
   (if (<= (* 2.0 x2) -5e-191)
     t_0
     (if (<= (* 2.0 x2) 5e-150) (- 0.0 x1) t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (x2 * -6.0);
	double tmp;
	if ((2.0 * x2) <= -5e-191) {
		tmp = t_0;
	} else if ((2.0 * x2) <= 5e-150) {
		tmp = 0.0 - x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x1 + (x2 * (-6.0d0))
    if ((2.0d0 * x2) <= (-5d-191)) then
        tmp = t_0
    else if ((2.0d0 * x2) <= 5d-150) then
        tmp = 0.0d0 - x1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = x1 + (x2 * -6.0);
	double tmp;
	if ((2.0 * x2) <= -5e-191) {
		tmp = t_0;
	} else if ((2.0 * x2) <= 5e-150) {
		tmp = 0.0 - x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = x1 + (x2 * -6.0)
	tmp = 0
	if (2.0 * x2) <= -5e-191:
		tmp = t_0
	elif (2.0 * x2) <= 5e-150:
		tmp = 0.0 - x1
	else:
		tmp = t_0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64(x2 * -6.0))
	tmp = 0.0
	if (Float64(2.0 * x2) <= -5e-191)
		tmp = t_0;
	elseif (Float64(2.0 * x2) <= 5e-150)
		tmp = Float64(0.0 - x1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 + (x2 * -6.0);
	tmp = 0.0;
	if ((2.0 * x2) <= -5e-191)
		tmp = t_0;
	elseif ((2.0 * x2) <= 5e-150)
		tmp = 0.0 - x1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * x2), $MachinePrecision], -5e-191], t$95$0, If[LessEqual[N[(2.0 * x2), $MachinePrecision], 5e-150], N[(0.0 - x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -5.0000000000000001e-191 or 4.9999999999999999e-150 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 70.9%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

      2. *-lowering-*.f64 37.5

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Simplified 37.5%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   if -5.0000000000000001e-191 < (*.f64 #s(literal 2 binary64) x2) < 4.9999999999999999e-150

   1. Initial program 72.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

      3. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      5. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      6. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

      9. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      10. distribute-lft-in N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      12. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

      15. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

      16. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

      17. *-lowering-*.f64 49.8

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   5. Simplified 49.8%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 + -2 \cdot x1} \]
   7. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-2 + 1\right) \cdot x1} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{-1} \cdot x1 \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - x1} \]

      5. --lowering--.f64 44.0

         \[\leadsto \color{blue}{0 - x1} \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{0 - x1} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

      2. neg-lowering-neg.f64 44.0

         \[\leadsto \color{blue}{-x1} \]

   10. Applied egg-rr 44.0%

       \[\leadsto \color{blue}{-x1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;x1 + x2 \cdot -6\\ \mathbf{elif}\;2 \cdot x2 \leq 5 \cdot 10^{-150}:\\ \;\;\;\;0 - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + x2 \cdot -6\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 32.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1\right)\\ \mathbf{elif}\;2 \cdot x2 \leq 2 \cdot 10^{-140}:\\ \;\;\;\;0 - x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* 2.0 x2) -5e-191)
   (fma x2 -6.0 x1)
   (if (<= (* 2.0 x2) 2e-140) (- 0.0 x1) (fma x2 -6.0 x1))))

C

double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -5e-191) {
		tmp = fma(x2, -6.0, x1);
	} else if ((2.0 * x2) <= 2e-140) {
		tmp = 0.0 - x1;
	} else {
		tmp = fma(x2, -6.0, x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(2.0 * x2) <= -5e-191)
		tmp = fma(x2, -6.0, x1);
	elseif (Float64(2.0 * x2) <= 2e-140)
		tmp = Float64(0.0 - x1);
	else
		tmp = fma(x2, -6.0, x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(2.0 * x2), $MachinePrecision], -5e-191], N[(x2 * -6.0 + x1), $MachinePrecision], If[LessEqual[N[(2.0 * x2), $MachinePrecision], 2e-140], N[(0.0 - x1), $MachinePrecision], N[(x2 * -6.0 + x1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -5.0000000000000001e-191 or 2e-140 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 70.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 37.6

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Simplified 37.6%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x2 \cdot -6 + x1} \]

      2. accelerator-lowering-fma.f64 37.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1\right)} \]

   7. Applied egg-rr 37.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1\right)} \]

   if -5.0000000000000001e-191 < (*.f64 #s(literal 2 binary64) x2) < 2e-140

   1. Initial program 73.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

      3. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      5. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      6. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

      9. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      10. distribute-lft-in N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      12. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

      15. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

      16. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

      17. *-lowering-*.f64 50.7

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   5. Simplified 50.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 + -2 \cdot x1} \]
   7. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-2 + 1\right) \cdot x1} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{-1} \cdot x1 \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - x1} \]

      5. --lowering--.f64 43.5

         \[\leadsto \color{blue}{0 - x1} \]

   8. Simplified 43.5%

      \[\leadsto \color{blue}{0 - x1} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

      2. neg-lowering-neg.f64 43.5

         \[\leadsto \color{blue}{-x1} \]

   10. Applied egg-rr 43.5%

       \[\leadsto \color{blue}{-x1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1\right)\\ \mathbf{elif}\;2 \cdot x2 \leq 2 \cdot 10^{-140}:\\ \;\;\;\;0 - x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 31.9% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;2 \cdot x2 \leq 5 \cdot 10^{-150}:\\ \;\;\;\;0 - x1\\ \mathbf{else}:\\ \;\;\;\;x2 \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* 2.0 x2) -5e-191)
   (* x2 -6.0)
   (if (<= (* 2.0 x2) 5e-150) (- 0.0 x1) (* x2 -6.0))))

C

double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -5e-191) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 5e-150) {
		tmp = 0.0 - x1;
	} else {
		tmp = x2 * -6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((2.0d0 * x2) <= (-5d-191)) then
        tmp = x2 * (-6.0d0)
    else if ((2.0d0 * x2) <= 5d-150) then
        tmp = 0.0d0 - x1
    else
        tmp = x2 * (-6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -5e-191) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 5e-150) {
		tmp = 0.0 - x1;
	} else {
		tmp = x2 * -6.0;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (2.0 * x2) <= -5e-191:
		tmp = x2 * -6.0
	elif (2.0 * x2) <= 5e-150:
		tmp = 0.0 - x1
	else:
		tmp = x2 * -6.0
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(2.0 * x2) <= -5e-191)
		tmp = Float64(x2 * -6.0);
	elseif (Float64(2.0 * x2) <= 5e-150)
		tmp = Float64(0.0 - x1);
	else
		tmp = Float64(x2 * -6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((2.0 * x2) <= -5e-191)
		tmp = x2 * -6.0;
	elseif ((2.0 * x2) <= 5e-150)
		tmp = 0.0 - x1;
	else
		tmp = x2 * -6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(2.0 * x2), $MachinePrecision], -5e-191], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[N[(2.0 * x2), $MachinePrecision], 5e-150], N[(0.0 - x1), $MachinePrecision], N[(x2 * -6.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -5.0000000000000001e-191 or 4.9999999999999999e-150 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 70.9%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

      2. *-lowering-*.f64 37.5

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Simplified 37.5%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x2 \cdot -6} \]

      2. *-lowering-*.f64 37.3

         \[\leadsto \color{blue}{x2 \cdot -6} \]

   8. Simplified 37.3%

      \[\leadsto \color{blue}{x2 \cdot -6} \]

   if -5.0000000000000001e-191 < (*.f64 #s(literal 2 binary64) x2) < 4.9999999999999999e-150

   1. Initial program 72.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

      3. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      5. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      6. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

      7. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

      9. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      10. distribute-lft-in N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

      12. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

      15. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

      16. *-commutative N/A

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

      17. *-lowering-*.f64 49.8

          \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   5. Simplified 49.8%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 + -2 \cdot x1} \]
   7. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-2 + 1\right) \cdot x1} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{-1} \cdot x1 \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - x1} \]

      5. --lowering--.f64 44.0

         \[\leadsto \color{blue}{0 - x1} \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{0 - x1} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

      2. neg-lowering-neg.f64 44.0

         \[\leadsto \color{blue}{-x1} \]

   10. Applied egg-rr 44.0%

       \[\leadsto \color{blue}{-x1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;2 \cdot x2 \leq 5 \cdot 10^{-150}:\\ \;\;\;\;0 - x1\\ \mathbf{else}:\\ \;\;\;\;x2 \cdot -6\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 38.8% accurate, 19.9× speedup

Math

\[\begin{array}{l} \\ x1 + \mathsf{fma}\left(x1, -2, x2 \cdot -6\right) \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (+ x1 (fma x1 -2.0 (* x2 -6.0))))

C

double code(double x1, double x2) {
	return x1 + fma(x1, -2.0, (x2 * -6.0));
}

Julia

function code(x1, x2)
	return Float64(x1 + fma(x1, -2.0, Float64(x2 * -6.0)))
end

Wolfram

code[x1_, x2_] := N[(x1 + N[(x1 * -2.0 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.2%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x1 around 0

   \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

   3. sub-neg N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

   4. *-commutative N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

   5. associate-*l* N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

   6. *-commutative N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

   7. metadata-eval N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

   9. sub-neg N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

   10. distribute-lft-in N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

   12. *-commutative N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

   14. metadata-eval N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

   15. metadata-eval N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

   16. *-commutative N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   17. *-lowering-*.f64 54.5

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

5. Simplified 54.5%

   \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

6. Taylor expanded in x2 around 0

   \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{-2}, x2 \cdot -6\right) \]
7. Step-by-step derivation

8. Simplified 41.8%

   \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{-2}, x2 \cdot -6\right) \]
9. Add Preprocessing

Alternative 23: 14.1% accurate, 74.5× speedup

Math

\[\begin{array}{l} \\ 0 - x1 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (- 0.0 x1))

C

double code(double x1, double x2) {
	return 0.0 - x1;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = 0.0d0 - x1
end function

Java

public static double code(double x1, double x2) {
	return 0.0 - x1;
}

Python

def code(x1, x2):
	return 0.0 - x1

Julia

function code(x1, x2)
	return Float64(0.0 - x1)
end

MATLAB

function tmp = code(x1, x2)
	tmp = 0.0 - x1;
end

Wolfram

code[x1_, x2_] := N[(0.0 - x1), $MachinePrecision]

Derivation

1. Initial program 71.2%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x1 around 0

   \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2\right) + -6 \cdot x2\right)} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, 4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 2, -6 \cdot x2\right)} \]

   3. sub-neg N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -6 \cdot x2\right) \]

   4. *-commutative N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) \cdot 4} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

   5. associate-*l* N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{x2 \cdot \left(\left(2 \cdot x2 - 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

   6. *-commutative N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \color{blue}{\left(4 \cdot \left(2 \cdot x2 - 3\right)\right)} + \left(\mathsf{neg}\left(2\right)\right), -6 \cdot x2\right) \]

   7. metadata-eval N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, x2 \cdot \left(4 \cdot \left(2 \cdot x2 - 3\right)\right) + \color{blue}{-2}, -6 \cdot x2\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x2, 4 \cdot \left(2 \cdot x2 - 3\right), -2\right)}, -6 \cdot x2\right) \]

   9. sub-neg N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 4 \cdot \color{blue}{\left(2 \cdot x2 + \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

   10. distribute-lft-in N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{4 \cdot \left(2 \cdot x2\right) + 4 \cdot \left(\mathsf{neg}\left(3\right)\right)}, -2\right), -6 \cdot x2\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(4, 2 \cdot x2, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}, -2\right), -6 \cdot x2\right) \]

   12. *-commutative N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, \color{blue}{x2 \cdot 2}, 4 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), -2\right), -6 \cdot x2\right) \]

   14. metadata-eval N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, 4 \cdot \color{blue}{-3}\right), -2\right), -6 \cdot x2\right) \]

   15. metadata-eval N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, \color{blue}{-12}\right), -2\right), -6 \cdot x2\right) \]

   16. *-commutative N/A

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

   17. *-lowering-*.f64 54.5

       \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), \color{blue}{x2 \cdot -6}\right) \]

5. Simplified 54.5%

   \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right), x2 \cdot -6\right)} \]

6. Taylor expanded in x2 around 0

   \[\leadsto \color{blue}{x1 + -2 \cdot x1} \]
7. Step-by-step derivation

   1. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(-2 + 1\right) \cdot x1} \]

   2. metadata-eval N/A

      \[\leadsto \color{blue}{-1} \cdot x1 \]

   3. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

   4. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - x1} \]

   5. --lowering--.f64 12.9

      \[\leadsto \color{blue}{0 - x1} \]

8. Simplified 12.9%

   \[\leadsto \color{blue}{0 - x1} \]
9. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x1\right)} \]

   2. neg-lowering-neg.f64 12.9

      \[\leadsto \color{blue}{-x1} \]

10. Applied egg-rr 12.9%

    \[\leadsto \color{blue}{-x1} \]

11. Final simplification 12.9%

    \[\leadsto 0 - x1 \]
12. Add Preprocessing

Alternative 24: 3.3% 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 71.2%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
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 31.4

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

5. Simplified 31.4%

   \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

6. Taylor expanded in x1 around inf

   \[\leadsto \color{blue}{x1} \]
7. Step-by-step derivation

8. Simplified 3.4%

   \[\leadsto \color{blue}{x1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(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))))))