Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.0% → 99.5%

Time: 21.9s

Alternatives: 21

Speedup: 8.3×

• 47.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x1 x2)
 :precision binary64
 (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
      (*
       (pow x1 4.0)
       (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1)))))))

C

double code(double x1, double x2) {
	double 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 + (pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	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 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := 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[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 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%

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

4. Final simplification 99.6%

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

Alternative 2: 58.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-175}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-95}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 6, -1\right)\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+71}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\\ \end{array} \end{array} \]

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 -5e+129)
     t_1
     (if (<= t_4 5e-175)
       (* x2 -6.0)
       (if (<= t_4 2e-95)
         (* x1 (fma x1 6.0 -1.0))
         (if (<= t_4 5e+71)
           (* x2 -6.0)
           (if (<= t_4 INFINITY) t_1 (fma x1 (fma 9.0 x1 -2.0) x1))))))))

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 <= -5e+129) {
		tmp = t_1;
	} else if (t_4 <= 5e-175) {
		tmp = x2 * -6.0;
	} else if (t_4 <= 2e-95) {
		tmp = x1 * fma(x1, 6.0, -1.0);
	} else if (t_4 <= 5e+71) {
		tmp = x2 * -6.0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(x1, fma(9.0, x1, -2.0), x1);
	}
	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 <= -5e+129)
		tmp = t_1;
	elseif (t_4 <= 5e-175)
		tmp = Float64(x2 * -6.0);
	elseif (t_4 <= 2e-95)
		tmp = Float64(x1 * fma(x1, 6.0, -1.0));
	elseif (t_4 <= 5e+71)
		tmp = Float64(x2 * -6.0);
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = fma(x1, fma(9.0, x1, -2.0), x1);
	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, -5e+129], t$95$1, If[LessEqual[t$95$4, 5e-175], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[t$95$4, 2e-95], N[(x1 * N[(x1 * 6.0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+71], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + x1), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.0000000000000003e129 or 4.99999999999999972e71 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 58.9%

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

   6. Taylor expanded in x2 around inf

      \[\leadsto \color{blue}{8 \cdot \left(x1 \cdot {x2}^{2}\right)} \]
   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 54.1

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

   8. Simplified 54.1%

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

   if -5.0000000000000003e129 < (+.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)))))) < 5e-175 or 1.99999999999999998e-95 < (+.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)))))) < 4.99999999999999972e71

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in 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 69.2

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

   8. Simplified 69.2%

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

   if 5e-175 < (+.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)))))) < 1.99999999999999998e-95

   1. Initial program 98.6%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x2 around inf

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{20 \cdot x2}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

      2. *-lowering-*.f64 100.0

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   8. Simplified 100.0%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   9. Taylor expanded in x2 around 0

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

       1. associate-+r+ N/A

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

       2. +-commutative N/A

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

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

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. distribute-lft-out N/A

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

       4. associate-+l- N/A

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

       5. metadata-eval N/A

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

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

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 89.0

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

   14. Simplified 89.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 52.9%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 81.5

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

   8. Simplified 81.5%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{+129}:\\ \;\;\;\;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 5 \cdot 10^{-175}:\\ \;\;\;\;x2 \cdot -6\\ \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 2 \cdot 10^{-95}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 6, -1\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 5 \cdot 10^{+71}:\\ \;\;\;\;x2 \cdot -6\\ \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:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := \mathsf{fma}\left(x2, x2 \cdot \left(x1 \cdot 8\right), x1\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+71}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x2 \cdot -6\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (fma x2 (* x2 (* x1 8.0)) 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+129)
     t_1
     (if (<= t_4 5e+71)
       (+ x1 (fma x1 (fma 9.0 x1 -2.0) (* x2 -6.0)))
       (if (<= t_4 INFINITY)
         t_1
         (* x1 (fma (* x1 x1) (fma x1 6.0 -3.0) 1.0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = fma(x2, (x2 * (x1 * 8.0)), x1);
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((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+129) {
		tmp = t_1;
	} else if (t_4 <= 5e+71) {
		tmp = x1 + fma(x1, fma(9.0, x1, -2.0), (x2 * -6.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 * fma((x1 * x1), fma(x1, 6.0, -3.0), 1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = fma(x2, Float64(x2 * Float64(x1 * 8.0)), x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(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+129)
		tmp = t_1;
	elseif (t_4 <= 5e+71)
		tmp = Float64(x1 + fma(x1, fma(9.0, x1, -2.0), Float64(x2 * -6.0)));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 * fma(Float64(x1 * x1), fma(x1, 6.0, -3.0), 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[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(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+129], t$95$1, If[LessEqual[t$95$4, 5e+71], N[(x1 + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * 6.0 + -3.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.0000000000000003e129 or 4.99999999999999972e71 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 58.2

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

   5. Simplified 58.2%

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

   6. Taylor expanded in x1 around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 59.1

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

   8. Simplified 59.1%

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

   if -5.0000000000000003e129 < (+.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)))))) < 4.99999999999999972e71

   1. Initial program 99.2%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 96.5%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 95.1

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

   8. Simplified 95.1%

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

   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 x1 around inf

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval 97.5

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

   5. Simplified 97.5%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 97.5

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

   8. Simplified 97.5%

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

4. Final simplification 82.9%

   \[\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^{+129}:\\ \;\;\;\;\mathsf{fma}\left(x2, x2 \cdot \left(x1 \cdot 8\right), x1\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+71}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -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, x2 \cdot \left(x1 \cdot 8\right), x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := \mathsf{fma}\left(x2, x2 \cdot \left(x1 \cdot 8\right), x1\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+71}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x2 \cdot -6\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (fma x2 (* x2 (* x1 8.0)) 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+129)
     t_1
     (if (<= t_4 5e+71)
       (+ x1 (fma x1 (fma 9.0 x1 -2.0) (* x2 -6.0)))
       (if (<= t_4 INFINITY) t_1 (fma x1 (fma 9.0 x1 -2.0) x1))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = fma(x2, (x2 * (x1 * 8.0)), x1);
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((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+129) {
		tmp = t_1;
	} else if (t_4 <= 5e+71) {
		tmp = x1 + fma(x1, fma(9.0, x1, -2.0), (x2 * -6.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(x1, fma(9.0, x1, -2.0), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = fma(x2, Float64(x2 * Float64(x1 * 8.0)), x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(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+129)
		tmp = t_1;
	elseif (t_4 <= 5e+71)
		tmp = Float64(x1 + fma(x1, fma(9.0, x1, -2.0), Float64(x2 * -6.0)));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = fma(x1, fma(9.0, x1, -2.0), x1);
	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[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(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+129], t$95$1, If[LessEqual[t$95$4, 5e+71], N[(x1 + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + x1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.0000000000000003e129 or 4.99999999999999972e71 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 58.2

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

   5. Simplified 58.2%

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

   6. Taylor expanded in x1 around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 59.1

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

   8. Simplified 59.1%

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

   if -5.0000000000000003e129 < (+.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)))))) < 4.99999999999999972e71

   1. Initial program 99.2%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 96.5%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 95.1

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

   8. Simplified 95.1%

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

   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 x1 around 0

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

   5. Simplified 52.9%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 81.5

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

   8. Simplified 81.5%

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

4. Final simplification 77.8%

   \[\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^{+129}:\\ \;\;\;\;\mathsf{fma}\left(x2, x2 \cdot \left(x1 \cdot 8\right), x1\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+71}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -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, x2 \cdot \left(x1 \cdot 8\right), x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := \mathsf{fma}\left(x2, x2 \cdot \left(x1 \cdot 8\right), x1\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(x1, \mathsf{fma}\left(6, x1, -2\right), x1\right)\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (fma x2 (* x2 (* x1 8.0)) 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+129)
     t_1
     (if (<= t_4 5e+71)
       (fma x2 -6.0 (fma x1 (fma 6.0 x1 -2.0) x1))
       (if (<= t_4 INFINITY) t_1 (fma x1 (fma 9.0 x1 -2.0) x1))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = fma(x2, (x2 * (x1 * 8.0)), x1);
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((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+129) {
		tmp = t_1;
	} else if (t_4 <= 5e+71) {
		tmp = fma(x2, -6.0, fma(x1, fma(6.0, x1, -2.0), x1));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(x1, fma(9.0, x1, -2.0), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = fma(x2, Float64(x2 * Float64(x1 * 8.0)), x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(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+129)
		tmp = t_1;
	elseif (t_4 <= 5e+71)
		tmp = fma(x2, -6.0, fma(x1, fma(6.0, x1, -2.0), x1));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = fma(x1, fma(9.0, x1, -2.0), x1);
	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[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(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+129], t$95$1, If[LessEqual[t$95$4, 5e+71], N[(x2 * -6.0 + N[(x1 * N[(6.0 * x1 + -2.0), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + x1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.0000000000000003e129 or 4.99999999999999972e71 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 58.2

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

   5. Simplified 58.2%

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

   6. Taylor expanded in x1 around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 59.1

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

   8. Simplified 59.1%

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

   if -5.0000000000000003e129 < (+.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)))))) < 4.99999999999999972e71

   1. Initial program 99.2%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 96.5%

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

   6. Taylor expanded in x2 around inf

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{20 \cdot x2}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

      2. *-lowering-*.f64 95.3

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   8. Simplified 95.3%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   9. Taylor expanded in x2 around 0

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

       1. associate-+r+ N/A

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

       2. +-commutative N/A

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

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

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

   11. Simplified 95.4%

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

   12. Taylor expanded in x1 around 0

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

   14. Simplified 94.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 52.9%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 81.5

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

   8. Simplified 81.5%

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

4. Final simplification 77.5%

   \[\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^{+129}:\\ \;\;\;\;\mathsf{fma}\left(x2, x2 \cdot \left(x1 \cdot 8\right), x1\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(x1, \mathsf{fma}\left(6, x1, -2\right), x1\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 \infty:\\ \;\;\;\;\mathsf{fma}\left(x2, x2 \cdot \left(x1 \cdot 8\right), x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 x1)))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_1 (* 2.0 x2)) x1) t_2))
        (t_4
         (*
          t_2
          (+
           (* (* (* x1 2.0) t_3) (- t_3 3.0))
           (* (* x1 x1) (- (* t_3 4.0) 6.0)))))
        (t_5 (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_2))))
   (if (<= (+ x1 (+ (+ x1 (+ (+ t_4 (* t_1 t_3)) t_0)) t_5)) INFINITY)
     (+ x1 (+ t_5 (+ x1 (+ t_0 (+ t_4 (* 3.0 t_1))))))
     (+
      x1
      (*
       (pow x1 4.0)
       (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1)))))))

C

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

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * x1))
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0))))
	t_5 = Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_2))
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(x1 + Float64(Float64(t_4 + Float64(t_1 * t_3)) + t_0)) + t_5)) <= Inf)
		tmp = Float64(x1 + Float64(t_5 + Float64(x1 + Float64(t_0 + Float64(t_4 + Float64(3.0 * t_1))))));
	else
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around inf

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

   5. Simplified 98.2%

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

   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%

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

4. Final simplification 98.8%

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

Alternative 7: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{if}\;x1 \leq -1.8 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;x1 + x2 \cdot \left(\frac{x2 \cdot \left(x1 \cdot 8\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - \mathsf{fma}\left(-6, \frac{x1 \cdot x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(-4, \frac{x1 \cdot \left(\frac{\left(x1 \cdot x1\right) \cdot 6}{\mathsf{fma}\left(x1, x1, 1\right)} - \mathsf{fma}\left(2, \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \frac{\left(x1 \cdot x1\right) \cdot -8}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \frac{6}{\mathsf{fma}\left(x1, x1, 1\right)}\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 1.65 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(x1 \cdot 8, x2, -6\right)\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          x1
          (*
           (pow x1 4.0)
           (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))))
   (if (<= x1 -1.8e+43)
     t_0
     (if (<= x1 -5.8e-7)
       (+
        x1
        (*
         x2
         (-
          (/ (* x2 (* x1 8.0)) (fma x1 x1 1.0))
          (fma
           -6.0
           (/ (* x1 x1) (fma x1 x1 1.0))
           (fma
            (fma x1 x1 1.0)
            (fma
             -4.0
             (/
              (*
               x1
               (-
                (/ (* (* x1 x1) 6.0) (fma x1 x1 1.0))
                (fma 2.0 (/ x1 (fma x1 x1 1.0)) 3.0)))
              (fma x1 x1 1.0))
             (/ (* (* x1 x1) -8.0) (fma x1 x1 1.0)))
            (/ 6.0 (fma x1 x1 1.0)))))))
       (if (<= x1 1.65e+19)
         (fma
          x2
          (fma x1 (fma 12.0 x1 -12.0) (fma (* x1 8.0) x2 -6.0))
          (fma x1 (fma 9.0 x1 -2.0) x1))
         t_0)))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1)));
	double tmp;
	if (x1 <= -1.8e+43) {
		tmp = t_0;
	} else if (x1 <= -5.8e-7) {
		tmp = x1 + (x2 * (((x2 * (x1 * 8.0)) / fma(x1, x1, 1.0)) - fma(-6.0, ((x1 * x1) / fma(x1, x1, 1.0)), fma(fma(x1, x1, 1.0), fma(-4.0, ((x1 * ((((x1 * x1) * 6.0) / fma(x1, x1, 1.0)) - fma(2.0, (x1 / fma(x1, x1, 1.0)), 3.0))) / fma(x1, x1, 1.0)), (((x1 * x1) * -8.0) / fma(x1, x1, 1.0))), (6.0 / fma(x1, x1, 1.0))))));
	} else if (x1 <= 1.65e+19) {
		tmp = fma(x2, fma(x1, fma(12.0, x1, -12.0), fma((x1 * 8.0), x2, -6.0)), fma(x1, fma(9.0, x1, -2.0), x1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1))))
	tmp = 0.0
	if (x1 <= -1.8e+43)
		tmp = t_0;
	elseif (x1 <= -5.8e-7)
		tmp = Float64(x1 + Float64(x2 * Float64(Float64(Float64(x2 * Float64(x1 * 8.0)) / fma(x1, x1, 1.0)) - fma(-6.0, Float64(Float64(x1 * x1) / fma(x1, x1, 1.0)), fma(fma(x1, x1, 1.0), fma(-4.0, Float64(Float64(x1 * Float64(Float64(Float64(Float64(x1 * x1) * 6.0) / fma(x1, x1, 1.0)) - fma(2.0, Float64(x1 / fma(x1, x1, 1.0)), 3.0))) / fma(x1, x1, 1.0)), Float64(Float64(Float64(x1 * x1) * -8.0) / fma(x1, x1, 1.0))), Float64(6.0 / fma(x1, x1, 1.0)))))));
	elseif (x1 <= 1.65e+19)
		tmp = fma(x2, fma(x1, fma(12.0, x1, -12.0), fma(Float64(x1 * 8.0), x2, -6.0)), fma(x1, fma(9.0, x1, -2.0), x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.8e+43], t$95$0, If[LessEqual[x1, -5.8e-7], N[(x1 + N[(x2 * N[(N[(N[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-6.0 * N[(N[(x1 * x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(-4.0 * N[(N[(x1 * N[(N[(N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * x1), $MachinePrecision] * -8.0), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1.65e+19], N[(x2 * N[(x1 * N[(12.0 * x1 + -12.0), $MachinePrecision] + N[(N[(x1 * 8.0), $MachinePrecision] * x2 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.80000000000000005e43 or 1.65e19 < x1

   1. Initial program 29.4%

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

   3. Taylor expanded in x1 around -inf

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

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

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

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

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

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

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

   5. Simplified 97.4%

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

   if -1.80000000000000005e43 < x1 < -5.7999999999999995e-7

   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 x2 around -inf

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

   5. Simplified 90.7%

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

   6. Taylor expanded in x2 around 0

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

   8. Simplified 90.9%

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

   if -5.7999999999999995e-7 < x1 < 1.65e19

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 90.6%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

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

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

   8. Simplified 97.9%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.8 \cdot 10^{+43}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;x1 + x2 \cdot \left(\frac{x2 \cdot \left(x1 \cdot 8\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - \mathsf{fma}\left(-6, \frac{x1 \cdot x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(-4, \frac{x1 \cdot \left(\frac{\left(x1 \cdot x1\right) \cdot 6}{\mathsf{fma}\left(x1, x1, 1\right)} - \mathsf{fma}\left(2, \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \frac{\left(x1 \cdot x1\right) \cdot -8}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \frac{6}{\mathsf{fma}\left(x1, x1, 1\right)}\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 1.65 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(x1 \cdot 8, x2, -6\right)\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{if}\;x1 \leq -8.4 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot 8, x2 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right)\\ \mathbf{elif}\;x1 \leq 2.4 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(x1 \cdot 8, x2, -6\right)\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          x1
          (*
           (pow x1 4.0)
           (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))))
   (if (<= x1 -8.4e+42)
     t_0
     (if (<= x1 -5.8e-7)
       (fma (* x2 8.0) (* x2 (/ x1 (fma x1 x1 1.0))) x1)
       (if (<= x1 2.4e+21)
         (fma
          x2
          (fma x1 (fma 12.0 x1 -12.0) (fma (* x1 8.0) x2 -6.0))
          (fma x1 (fma 9.0 x1 -2.0) x1))
         t_0)))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1)));
	double tmp;
	if (x1 <= -8.4e+42) {
		tmp = t_0;
	} else if (x1 <= -5.8e-7) {
		tmp = fma((x2 * 8.0), (x2 * (x1 / fma(x1, x1, 1.0))), x1);
	} else if (x1 <= 2.4e+21) {
		tmp = fma(x2, fma(x1, fma(12.0, x1, -12.0), fma((x1 * 8.0), x2, -6.0)), fma(x1, fma(9.0, x1, -2.0), x1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1))))
	tmp = 0.0
	if (x1 <= -8.4e+42)
		tmp = t_0;
	elseif (x1 <= -5.8e-7)
		tmp = fma(Float64(x2 * 8.0), Float64(x2 * Float64(x1 / fma(x1, x1, 1.0))), x1);
	elseif (x1 <= 2.4e+21)
		tmp = fma(x2, fma(x1, fma(12.0, x1, -12.0), fma(Float64(x1 * 8.0), x2, -6.0)), fma(x1, fma(9.0, x1, -2.0), x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -8.4e+42], t$95$0, If[LessEqual[x1, -5.8e-7], N[(N[(x2 * 8.0), $MachinePrecision] * N[(x2 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 2.4e+21], N[(x2 * N[(x1 * N[(12.0 * x1 + -12.0), $MachinePrecision] + N[(N[(x1 * 8.0), $MachinePrecision] * x2 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -8.39999999999999982e42 or 2.4e21 < x1

   1. Initial program 29.4%

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

   3. Taylor expanded in x1 around -inf

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

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

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

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

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

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

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

   5. Simplified 97.4%

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

   if -8.39999999999999982e42 < x1 < -5.7999999999999995e-7

   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 x2 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 82.4

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

   5. Simplified 82.4%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. accelerator-lowering-fma.f64 90.8

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

   7. Applied egg-rr 90.8%

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

   if -5.7999999999999995e-7 < x1 < 2.4e21

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 90.6%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

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

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

   8. Simplified 97.9%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -8.4 \cdot 10^{+42}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot 8, x2 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right)\\ \mathbf{elif}\;x1 \leq 2.4 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(x1 \cdot 8, x2, -6\right)\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \mathbf{if}\;x1 \leq -3.9 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot 8, x2 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right)\\ \mathbf{elif}\;x1 \leq 1.8 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(x1 \cdot 8, x2, -6\right)\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (fma (* x1 x1) (fma x1 6.0 -3.0) 1.0))))
   (if (<= x1 -3.9e+42)
     t_0
     (if (<= x1 -5.8e-7)
       (fma (* x2 8.0) (* x2 (/ x1 (fma x1 x1 1.0))) x1)
       (if (<= x1 1.8e+54)
         (fma
          x2
          (fma x1 (fma 12.0 x1 -12.0) (fma (* x1 8.0) x2 -6.0))
          (fma x1 (fma 9.0 x1 -2.0) x1))
         t_0)))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma((x1 * x1), fma(x1, 6.0, -3.0), 1.0);
	double tmp;
	if (x1 <= -3.9e+42) {
		tmp = t_0;
	} else if (x1 <= -5.8e-7) {
		tmp = fma((x2 * 8.0), (x2 * (x1 / fma(x1, x1, 1.0))), x1);
	} else if (x1 <= 1.8e+54) {
		tmp = fma(x2, fma(x1, fma(12.0, x1, -12.0), fma((x1 * 8.0), x2, -6.0)), fma(x1, fma(9.0, x1, -2.0), x1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(Float64(x1 * x1), fma(x1, 6.0, -3.0), 1.0))
	tmp = 0.0
	if (x1 <= -3.9e+42)
		tmp = t_0;
	elseif (x1 <= -5.8e-7)
		tmp = fma(Float64(x2 * 8.0), Float64(x2 * Float64(x1 / fma(x1, x1, 1.0))), x1);
	elseif (x1 <= 1.8e+54)
		tmp = fma(x2, fma(x1, fma(12.0, x1, -12.0), fma(Float64(x1 * 8.0), x2, -6.0)), fma(x1, fma(9.0, x1, -2.0), x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * 6.0 + -3.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -3.9e+42], t$95$0, If[LessEqual[x1, -5.8e-7], N[(N[(x2 * 8.0), $MachinePrecision] * N[(x2 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1.8e+54], N[(x2 * N[(x1 * N[(12.0 * x1 + -12.0), $MachinePrecision] + N[(N[(x1 * 8.0), $MachinePrecision] * x2 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.8999999999999997e42 or 1.8000000000000001e54 < x1

   1. Initial program 24.9%

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

   3. Taylor expanded in x1 around inf

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval 93.5

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

   5. Simplified 93.5%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 93.5

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

   8. Simplified 93.5%

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

   if -3.8999999999999997e42 < x1 < -5.7999999999999995e-7

   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 x2 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 82.4

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

   5. Simplified 82.4%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. accelerator-lowering-fma.f64 90.8

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

   7. Applied egg-rr 90.8%

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

   if -5.7999999999999995e-7 < x1 < 1.8000000000000001e54

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 88.4%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

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

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

   8. Simplified 95.4%

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

Alternative 10: 91.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \mathbf{if}\;x1 \leq -8.4 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot 8, x2 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right)\\ \mathbf{elif}\;x1 \leq 1.8 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, \mathsf{fma}\left(12, x1, -12\right)\right), -6\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(6, x1, -2\right), x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (fma (* x1 x1) (fma x1 6.0 -3.0) 1.0))))
   (if (<= x1 -8.4e+42)
     t_0
     (if (<= x1 -5.8e-7)
       (fma (* x2 8.0) (* x2 (/ x1 (fma x1 x1 1.0))) x1)
       (if (<= x1 1.8e+54)
         (fma
          x2
          (fma x1 (fma x2 8.0 (fma 12.0 x1 -12.0)) -6.0)
          (fma x1 (fma 6.0 x1 -2.0) x1))
         t_0)))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma((x1 * x1), fma(x1, 6.0, -3.0), 1.0);
	double tmp;
	if (x1 <= -8.4e+42) {
		tmp = t_0;
	} else if (x1 <= -5.8e-7) {
		tmp = fma((x2 * 8.0), (x2 * (x1 / fma(x1, x1, 1.0))), x1);
	} else if (x1 <= 1.8e+54) {
		tmp = fma(x2, fma(x1, fma(x2, 8.0, fma(12.0, x1, -12.0)), -6.0), fma(x1, fma(6.0, x1, -2.0), x1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(Float64(x1 * x1), fma(x1, 6.0, -3.0), 1.0))
	tmp = 0.0
	if (x1 <= -8.4e+42)
		tmp = t_0;
	elseif (x1 <= -5.8e-7)
		tmp = fma(Float64(x2 * 8.0), Float64(x2 * Float64(x1 / fma(x1, x1, 1.0))), x1);
	elseif (x1 <= 1.8e+54)
		tmp = fma(x2, fma(x1, fma(x2, 8.0, fma(12.0, x1, -12.0)), -6.0), fma(x1, fma(6.0, x1, -2.0), x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * 6.0 + -3.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -8.4e+42], t$95$0, If[LessEqual[x1, -5.8e-7], N[(N[(x2 * 8.0), $MachinePrecision] * N[(x2 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1.8e+54], N[(x2 * N[(x1 * N[(x2 * 8.0 + N[(12.0 * x1 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(6.0 * x1 + -2.0), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -8.39999999999999982e42 or 1.8000000000000001e54 < x1

   1. Initial program 24.9%

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

   3. Taylor expanded in x1 around inf

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval 93.5

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

   5. Simplified 93.5%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 93.5

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

   8. Simplified 93.5%

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

   if -8.39999999999999982e42 < x1 < -5.7999999999999995e-7

   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 x2 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 82.4

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

   5. Simplified 82.4%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. accelerator-lowering-fma.f64 90.8

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

   7. Applied egg-rr 90.8%

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

   if -5.7999999999999995e-7 < x1 < 1.8000000000000001e54

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 88.4%

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

   6. Taylor expanded in x2 around inf

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{20 \cdot x2}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

      2. *-lowering-*.f64 87.6

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   8. Simplified 87.6%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   9. Taylor expanded in x2 around 0

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

       1. associate-+r+ N/A

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

       2. +-commutative N/A

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

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

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

   11. Simplified 94.6%

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

Alternative 11: 62.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.6 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(x1, \left(x1 \cdot x1\right) \cdot -3, x1\right)\\ \mathbf{elif}\;x1 \leq -7.4 \cdot 10^{-28}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 \leq -1.7 \cdot 10^{-137}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 6, -1\right)\\ \mathbf{elif}\;x1 \leq 2.95 \cdot 10^{-127}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 \leq 5.4 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(x2, x2 \cdot \left(x1 \cdot 8\right), x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.6e+73)
   (fma x1 (* (* x1 x1) -3.0) x1)
   (if (<= x1 -7.4e-28)
     (* 8.0 (* x1 (* x2 x2)))
     (if (<= x1 -1.7e-137)
       (* x1 (fma x1 6.0 -1.0))
       (if (<= x1 2.95e-127)
         (* x2 -6.0)
         (if (<= x1 5.4e+149)
           (fma x2 (* x2 (* x1 8.0)) x1)
           (fma x1 (fma 9.0 x1 -2.0) x1)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.6e+73) {
		tmp = fma(x1, ((x1 * x1) * -3.0), x1);
	} else if (x1 <= -7.4e-28) {
		tmp = 8.0 * (x1 * (x2 * x2));
	} else if (x1 <= -1.7e-137) {
		tmp = x1 * fma(x1, 6.0, -1.0);
	} else if (x1 <= 2.95e-127) {
		tmp = x2 * -6.0;
	} else if (x1 <= 5.4e+149) {
		tmp = fma(x2, (x2 * (x1 * 8.0)), x1);
	} else {
		tmp = fma(x1, fma(9.0, x1, -2.0), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.6e+73)
		tmp = fma(x1, Float64(Float64(x1 * x1) * -3.0), x1);
	elseif (x1 <= -7.4e-28)
		tmp = Float64(8.0 * Float64(x1 * Float64(x2 * x2)));
	elseif (x1 <= -1.7e-137)
		tmp = Float64(x1 * fma(x1, 6.0, -1.0));
	elseif (x1 <= 2.95e-127)
		tmp = Float64(x2 * -6.0);
	elseif (x1 <= 5.4e+149)
		tmp = fma(x2, Float64(x2 * Float64(x1 * 8.0)), x1);
	else
		tmp = fma(x1, fma(9.0, x1, -2.0), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.6e+73], N[(x1 * N[(N[(x1 * x1), $MachinePrecision] * -3.0), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -7.4e-28], N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.7e-137], N[(x1 * N[(x1 * 6.0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.95e-127], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[x1, 5.4e+149], N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + x1), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x1 < -2.6000000000000001e73

   1. Initial program 9.4%

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

   3. Taylor expanded in x1 around inf

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 91.3

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

   8. Simplified 91.3%

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

   if -2.6000000000000001e73 < x1 < -7.40000000000000039e-28

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 51.8%

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

   6. Taylor expanded in x2 around inf

      \[\leadsto \color{blue}{8 \cdot \left(x1 \cdot {x2}^{2}\right)} \]
   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 46.9

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

   8. Simplified 46.9%

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

   if -7.40000000000000039e-28 < x1 < -1.70000000000000007e-137

   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

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

   5. Simplified 96.3%

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

   6. Taylor expanded in x2 around inf

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{20 \cdot x2}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

      2. *-lowering-*.f64 96.3

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   8. Simplified 96.3%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   9. Taylor expanded in x2 around 0

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

       1. associate-+r+ N/A

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

       2. +-commutative N/A

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

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

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. distribute-lft-out N/A

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

       4. associate-+l- N/A

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

       5. metadata-eval N/A

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

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

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 56.0

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

   14. Simplified 56.0%

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

   if -1.70000000000000007e-137 < x1 < 2.9499999999999999e-127

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 90.6%

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

   6. Taylor expanded in 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 80.0

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

   8. Simplified 80.0%

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

   if 2.9499999999999999e-127 < x1 < 5.4000000000000002e149

   1. Initial program 96.1%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in x1 around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 49.9

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

   8. Simplified 49.9%

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

   if 5.4000000000000002e149 < x1

   1. Initial program 3.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 78.5%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 97.2

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

   8. Simplified 97.2%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.6 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(x1, \left(x1 \cdot x1\right) \cdot -3, x1\right)\\ \mathbf{elif}\;x1 \leq -7.4 \cdot 10^{-28}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 \leq -1.7 \cdot 10^{-137}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 6, -1\right)\\ \mathbf{elif}\;x1 \leq 2.95 \cdot 10^{-127}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 \leq 5.4 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(x2, x2 \cdot \left(x1 \cdot 8\right), x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.1% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{if}\;x1 \leq -4 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(x1, \left(x1 \cdot x1\right) \cdot -3, x1\right)\\ \mathbf{elif}\;x1 \leq -1.9 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -1.5 \cdot 10^{-137}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 6, -1\right)\\ \mathbf{elif}\;x1 \leq 2.75 \cdot 10^{-127}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 \leq 5.4 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 8.0 (* x1 (* x2 x2)))))
   (if (<= x1 -4e+75)
     (fma x1 (* (* x1 x1) -3.0) x1)
     (if (<= x1 -1.9e-27)
       t_0
       (if (<= x1 -1.5e-137)
         (* x1 (fma x1 6.0 -1.0))
         (if (<= x1 2.75e-127)
           (* x2 -6.0)
           (if (<= x1 5.4e+149) t_0 (fma x1 (fma 9.0 x1 -2.0) x1))))))))

C

double code(double x1, double x2) {
	double t_0 = 8.0 * (x1 * (x2 * x2));
	double tmp;
	if (x1 <= -4e+75) {
		tmp = fma(x1, ((x1 * x1) * -3.0), x1);
	} else if (x1 <= -1.9e-27) {
		tmp = t_0;
	} else if (x1 <= -1.5e-137) {
		tmp = x1 * fma(x1, 6.0, -1.0);
	} else if (x1 <= 2.75e-127) {
		tmp = x2 * -6.0;
	} else if (x1 <= 5.4e+149) {
		tmp = t_0;
	} else {
		tmp = fma(x1, fma(9.0, x1, -2.0), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(8.0 * Float64(x1 * Float64(x2 * x2)))
	tmp = 0.0
	if (x1 <= -4e+75)
		tmp = fma(x1, Float64(Float64(x1 * x1) * -3.0), x1);
	elseif (x1 <= -1.9e-27)
		tmp = t_0;
	elseif (x1 <= -1.5e-137)
		tmp = Float64(x1 * fma(x1, 6.0, -1.0));
	elseif (x1 <= 2.75e-127)
		tmp = Float64(x2 * -6.0);
	elseif (x1 <= 5.4e+149)
		tmp = t_0;
	else
		tmp = fma(x1, fma(9.0, x1, -2.0), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -4e+75], N[(x1 * N[(N[(x1 * x1), $MachinePrecision] * -3.0), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -1.9e-27], t$95$0, If[LessEqual[x1, -1.5e-137], N[(x1 * N[(x1 * 6.0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.75e-127], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[x1, 5.4e+149], t$95$0, N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + x1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -3.99999999999999971e75

   1. Initial program 9.4%

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

   3. Taylor expanded in x1 around inf

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 91.3

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

   8. Simplified 91.3%

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

   if -3.99999999999999971e75 < x1 < -1.9e-27 or 2.75000000000000018e-127 < x1 < 5.4000000000000002e149

   1. Initial program 97.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 63.7%

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

   6. Taylor expanded in x2 around inf

      \[\leadsto \color{blue}{8 \cdot \left(x1 \cdot {x2}^{2}\right)} \]
   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 45.8

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

   8. Simplified 45.8%

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

   if -1.9e-27 < x1 < -1.4999999999999999e-137

   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

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

   5. Simplified 96.3%

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

   6. Taylor expanded in x2 around inf

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{20 \cdot x2}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

      2. *-lowering-*.f64 96.3

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   8. Simplified 96.3%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   9. Taylor expanded in x2 around 0

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

       1. associate-+r+ N/A

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

       2. +-commutative N/A

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

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

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. distribute-lft-out N/A

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

       4. associate-+l- N/A

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

       5. metadata-eval N/A

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

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

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 56.0

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

   14. Simplified 56.0%

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

   if -1.4999999999999999e-137 < x1 < 2.75000000000000018e-127

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 90.6%

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

   6. Taylor expanded in 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 80.0

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

   8. Simplified 80.0%

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

   if 5.4000000000000002e149 < x1

   1. Initial program 3.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 78.5%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 97.2

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

   8. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 13: 91.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \mathbf{if}\;x1 \leq -4.4 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot 8, x2 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right)\\ \mathbf{elif}\;x1 \leq 1.8 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, \mathsf{fma}\left(12, x1, -12\right)\right), -6\right), -x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (fma (* x1 x1) (fma x1 6.0 -3.0) 1.0))))
   (if (<= x1 -4.4e+42)
     t_0
     (if (<= x1 -5.8e-7)
       (fma (* x2 8.0) (* x2 (/ x1 (fma x1 x1 1.0))) x1)
       (if (<= x1 1.8e+54)
         (fma x2 (fma x1 (fma x2 8.0 (fma 12.0 x1 -12.0)) -6.0) (- x1))
         t_0)))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma((x1 * x1), fma(x1, 6.0, -3.0), 1.0);
	double tmp;
	if (x1 <= -4.4e+42) {
		tmp = t_0;
	} else if (x1 <= -5.8e-7) {
		tmp = fma((x2 * 8.0), (x2 * (x1 / fma(x1, x1, 1.0))), x1);
	} else if (x1 <= 1.8e+54) {
		tmp = fma(x2, fma(x1, fma(x2, 8.0, fma(12.0, x1, -12.0)), -6.0), -x1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(Float64(x1 * x1), fma(x1, 6.0, -3.0), 1.0))
	tmp = 0.0
	if (x1 <= -4.4e+42)
		tmp = t_0;
	elseif (x1 <= -5.8e-7)
		tmp = fma(Float64(x2 * 8.0), Float64(x2 * Float64(x1 / fma(x1, x1, 1.0))), x1);
	elseif (x1 <= 1.8e+54)
		tmp = fma(x2, fma(x1, fma(x2, 8.0, fma(12.0, x1, -12.0)), -6.0), Float64(-x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * 6.0 + -3.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -4.4e+42], t$95$0, If[LessEqual[x1, -5.8e-7], N[(N[(x2 * 8.0), $MachinePrecision] * N[(x2 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1.8e+54], N[(x2 * N[(x1 * N[(x2 * 8.0 + N[(12.0 * x1 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] + (-x1)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -4.4000000000000003e42 or 1.8000000000000001e54 < x1

   1. Initial program 24.9%

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

   3. Taylor expanded in x1 around inf

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval 93.5

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

   5. Simplified 93.5%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 93.5

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

   8. Simplified 93.5%

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

   if -4.4000000000000003e42 < x1 < -5.7999999999999995e-7

   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 x2 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 82.4

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

   5. Simplified 82.4%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. accelerator-lowering-fma.f64 90.8

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

   7. Applied egg-rr 90.8%

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

   if -5.7999999999999995e-7 < x1 < 1.8000000000000001e54

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 88.4%

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

   6. Taylor expanded in x2 around inf

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{20 \cdot x2}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

      2. *-lowering-*.f64 87.6

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   8. Simplified 87.6%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   9. Taylor expanded in x2 around 0

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

       1. associate-+r+ N/A

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

       2. +-commutative N/A

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

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

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

   11. Simplified 94.6%

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

   12. Taylor expanded in x1 around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 94.5

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

   14. Simplified 94.5%

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

Alternative 14: 91.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \mathbf{if}\;x1 \leq -5.4 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.8 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, \mathsf{fma}\left(12, x1, -12\right)\right), -6\right), -x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (fma (* x1 x1) (fma x1 6.0 -3.0) 1.0))))
   (if (<= x1 -5.4e+42)
     t_0
     (if (<= x1 1.8e+54)
       (fma x2 (fma x1 (fma x2 8.0 (fma 12.0 x1 -12.0)) -6.0) (- x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma((x1 * x1), fma(x1, 6.0, -3.0), 1.0);
	double tmp;
	if (x1 <= -5.4e+42) {
		tmp = t_0;
	} else if (x1 <= 1.8e+54) {
		tmp = fma(x2, fma(x1, fma(x2, 8.0, fma(12.0, x1, -12.0)), -6.0), -x1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(Float64(x1 * x1), fma(x1, 6.0, -3.0), 1.0))
	tmp = 0.0
	if (x1 <= -5.4e+42)
		tmp = t_0;
	elseif (x1 <= 1.8e+54)
		tmp = fma(x2, fma(x1, fma(x2, 8.0, fma(12.0, x1, -12.0)), -6.0), Float64(-x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * 6.0 + -3.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -5.4e+42], t$95$0, If[LessEqual[x1, 1.8e+54], N[(x2 * N[(x1 * N[(x2 * 8.0 + N[(12.0 * x1 + -12.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] + (-x1)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -5.4000000000000001e42 or 1.8000000000000001e54 < x1

   1. Initial program 24.9%

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

   3. Taylor expanded in x1 around inf

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval 93.5

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

   5. Simplified 93.5%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 93.5

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

   8. Simplified 93.5%

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

   if -5.4000000000000001e42 < x1 < 1.8000000000000001e54

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 85.3%

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

   6. Taylor expanded in x2 around inf

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{20 \cdot x2}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

      2. *-lowering-*.f64 84.6

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   8. Simplified 84.6%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   9. Taylor expanded in x2 around 0

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

       1. associate-+r+ N/A

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

       2. +-commutative N/A

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

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

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

   11. Simplified 91.1%

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

   12. Taylor expanded in x1 around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 91.0

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

   14. Simplified 91.0%

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

Alternative 15: 85.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \mathbf{if}\;x1 \leq -3.9 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.8 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -1\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (fma (* x1 x1) (fma x1 6.0 -3.0) 1.0))))
   (if (<= x1 -3.9e+42)
     t_0
     (if (<= x1 1.8e+54)
       (fma x1 (fma x2 (fma x2 8.0 -12.0) -1.0) (* x2 -6.0))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma((x1 * x1), fma(x1, 6.0, -3.0), 1.0);
	double tmp;
	if (x1 <= -3.9e+42) {
		tmp = t_0;
	} else if (x1 <= 1.8e+54) {
		tmp = fma(x1, fma(x2, fma(x2, 8.0, -12.0), -1.0), (x2 * -6.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(Float64(x1 * x1), fma(x1, 6.0, -3.0), 1.0))
	tmp = 0.0
	if (x1 <= -3.9e+42)
		tmp = t_0;
	elseif (x1 <= 1.8e+54)
		tmp = fma(x1, fma(x2, fma(x2, 8.0, -12.0), -1.0), Float64(x2 * -6.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * 6.0 + -3.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -3.9e+42], t$95$0, If[LessEqual[x1, 1.8e+54], N[(x1 * N[(x2 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.8999999999999997e42 or 1.8000000000000001e54 < x1

   1. Initial program 24.9%

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

   3. Taylor expanded in x1 around inf

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval 93.5

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

   5. Simplified 93.5%

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

   6. Taylor expanded in x1 around 0

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

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

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

      2. +-commutative N/A

         \[\leadsto x1 \cdot \color{blue}{\left({x1}^{2} \cdot \left(6 \cdot x1 - 3\right) + 1\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left({x1}^{2}, 6 \cdot x1 - 3, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto x1 \cdot \mathsf{fma}\left(\color{blue}{x1 \cdot x1}, 6 \cdot x1 - 3, 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto x1 \cdot \mathsf{fma}\left(\color{blue}{x1 \cdot x1}, 6 \cdot x1 - 3, 1\right) \]

      6. sub-neg N/A

         \[\leadsto x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \color{blue}{6 \cdot x1 + \left(\mathsf{neg}\left(3\right)\right)}, 1\right) \]

      7. *-commutative N/A

         \[\leadsto x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \color{blue}{x1 \cdot 6} + \left(\mathsf{neg}\left(3\right)\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto x1 \cdot \mathsf{fma}\left(x1 \cdot x1, x1 \cdot 6 + \color{blue}{-3}, 1\right) \]

      9. accelerator-lowering-fma.f64 93.5

         \[\leadsto x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \color{blue}{\mathsf{fma}\left(x1, 6, -3\right)}, 1\right) \]

   8. Simplified 93.5%

      \[\leadsto \color{blue}{x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)} \]

   if -3.8999999999999997e42 < x1 < 1.8000000000000001e54

   1. Initial program 99.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Simplified 85.3%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around inf

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{20 \cdot x2}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

      2. *-lowering-*.f64 84.6

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   8. Simplified 84.6%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   9. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 + \left(x1 \cdot \left(6 \cdot x1 - 2\right) + x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)\right)} \]
   10. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(x1 + x1 \cdot \left(6 \cdot x1 - 2\right)\right) + x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right) + \left(x1 + x1 \cdot \left(6 \cdot x1 - 2\right)\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6, x1 + x1 \cdot \left(6 \cdot x1 - 2\right)\right)} \]

   11. Simplified 91.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, \mathsf{fma}\left(12, x1, -12\right)\right), -6\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(6, x1, -2\right), x1\right)\right)} \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 1\right)} \]
   13. 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. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x2, 8, \mathsf{neg}\left(12\right)\right)}, -1\right), -6 \cdot x2\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, \color{blue}{-12}\right), -1\right), -6 \cdot x2\right) \]

       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 84.5

           \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -1\right), \color{blue}{x2 \cdot -6}\right) \]

   14. Simplified 84.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -1\right), x2 \cdot -6\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 55.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.05 \cdot 10^{-137}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\\ \mathbf{elif}\;x1 \leq 3.55 \cdot 10^{-124}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.05e-137)
   (+ x1 (* x1 (fma 9.0 x1 -2.0)))
   (if (<= x1 3.55e-124) (* x2 -6.0) (fma x1 (fma 9.0 x1 -2.0) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.05e-137) {
		tmp = x1 + (x1 * fma(9.0, x1, -2.0));
	} else if (x1 <= 3.55e-124) {
		tmp = x2 * -6.0;
	} else {
		tmp = fma(x1, fma(9.0, x1, -2.0), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.05e-137)
		tmp = Float64(x1 + Float64(x1 * fma(9.0, x1, -2.0)));
	elseif (x1 <= 3.55e-124)
		tmp = Float64(x2 * -6.0);
	else
		tmp = fma(x1, fma(9.0, x1, -2.0), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.05e-137], N[(x1 + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 3.55e-124], N[(x2 * -6.0), $MachinePrecision], N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.0499999999999999e-137

   1. Initial program 52.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Simplified 53.2%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \color{blue}{x1 \cdot \left(9 \cdot x1 - 2\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x1 + \color{blue}{x1 \cdot \left(9 \cdot x1 - 2\right)} \]

      2. sub-neg N/A

         \[\leadsto x1 + x1 \cdot \color{blue}{\left(9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto x1 + x1 \cdot \left(9 \cdot x1 + \color{blue}{-2}\right) \]

      4. accelerator-lowering-fma.f64 49.7

         \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(9, x1, -2\right)} \]

   8. Simplified 49.7%

      \[\leadsto x1 + \color{blue}{x1 \cdot \mathsf{fma}\left(9, x1, -2\right)} \]

   if -2.0499999999999999e-137 < x1 < 3.55000000000000019e-124

   1. Initial program 99.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Simplified 90.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in 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 79.1

         \[\leadsto \color{blue}{x2 \cdot -6} \]

   8. Simplified 79.1%

      \[\leadsto \color{blue}{x2 \cdot -6} \]

   if 3.55000000000000019e-124 < x1

   1. Initial program 62.6%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Simplified 71.4%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 + x1 \cdot \left(9 \cdot x1 - 2\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x1 \cdot \left(9 \cdot x1 - 2\right) + x1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x1, 9 \cdot x1 - 2, x1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)}, x1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x1, 9 \cdot x1 + \color{blue}{-2}, x1\right) \]

      5. accelerator-lowering-fma.f64 48.0

         \[\leadsto \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x1\right) \]

   8. Simplified 48.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 54.9% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)\\ \mathbf{if}\;x1 \leq -1.5 \cdot 10^{-137}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.45 \cdot 10^{-124}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma x1 (fma 9.0 x1 -2.0) x1)))
   (if (<= x1 -1.5e-137) t_0 (if (<= x1 1.45e-124) (* x2 -6.0) t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma(x1, fma(9.0, x1, -2.0), x1);
	double tmp;
	if (x1 <= -1.5e-137) {
		tmp = t_0;
	} else if (x1 <= 1.45e-124) {
		tmp = x2 * -6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(x1, fma(9.0, x1, -2.0), x1)
	tmp = 0.0
	if (x1 <= -1.5e-137)
		tmp = t_0;
	elseif (x1 <= 1.45e-124)
		tmp = Float64(x2 * -6.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -1.5e-137], t$95$0, If[LessEqual[x1, 1.45e-124], N[(x2 * -6.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.4999999999999999e-137 or 1.4500000000000001e-124 < x1

   1. Initial program 57.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Simplified 61.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 + x1 \cdot \left(9 \cdot x1 - 2\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x1 \cdot \left(9 \cdot x1 - 2\right) + x1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x1, 9 \cdot x1 - 2, x1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)}, x1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x1, 9 \cdot x1 + \color{blue}{-2}, x1\right) \]

      5. accelerator-lowering-fma.f64 48.9

         \[\leadsto \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x1\right) \]

   8. Simplified 48.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x1\right)} \]

   if -1.4999999999999999e-137 < x1 < 1.4500000000000001e-124

   1. Initial program 99.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Simplified 90.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in 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 79.1

         \[\leadsto \color{blue}{x2 \cdot -6} \]

   8. Simplified 79.1%

      \[\leadsto \color{blue}{x2 \cdot -6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 54.6% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(x1, 6, -1\right)\\ \mathbf{if}\;x1 \leq -3.3 \cdot 10^{-140}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{-124}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (fma x1 6.0 -1.0))))
   (if (<= x1 -3.3e-140) t_0 (if (<= x1 4.8e-124) (* x2 -6.0) t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma(x1, 6.0, -1.0);
	double tmp;
	if (x1 <= -3.3e-140) {
		tmp = t_0;
	} else if (x1 <= 4.8e-124) {
		tmp = x2 * -6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(x1, 6.0, -1.0))
	tmp = 0.0
	if (x1 <= -3.3e-140)
		tmp = t_0;
	elseif (x1 <= 4.8e-124)
		tmp = Float64(x2 * -6.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 6.0 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -3.3e-140], t$95$0, If[LessEqual[x1, 4.8e-124], N[(x2 * -6.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.29999999999999987e-140 or 4.79999999999999985e-124 < x1

   1. Initial program 57.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Simplified 61.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around inf

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{20 \cdot x2}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

      2. *-lowering-*.f64 61.1

         \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   8. Simplified 61.1%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \color{blue}{x2 \cdot 20}\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right) \]

   9. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 + \left(x1 \cdot \left(6 \cdot x1 - 2\right) + x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)\right)} \]
   10. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(x1 + x1 \cdot \left(6 \cdot x1 - 2\right)\right) + x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right) + \left(x1 + x1 \cdot \left(6 \cdot x1 - 2\right)\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6, x1 + x1 \cdot \left(6 \cdot x1 - 2\right)\right)} \]

   11. Simplified 52.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, \mathsf{fma}\left(12, x1, -12\right)\right), -6\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(6, x1, -2\right), x1\right)\right)} \]

   12. Taylor expanded in x2 around 0

       \[\leadsto \color{blue}{x1 + x1 \cdot \left(6 \cdot x1 - 2\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{x1 \cdot \left(6 \cdot x1 - 2\right) + x1} \]

       2. *-rgt-identity N/A

          \[\leadsto x1 \cdot \left(6 \cdot x1 - 2\right) + \color{blue}{x1 \cdot 1} \]

       3. distribute-lft-out N/A

          \[\leadsto \color{blue}{x1 \cdot \left(\left(6 \cdot x1 - 2\right) + 1\right)} \]

       4. associate-+l- N/A

          \[\leadsto x1 \cdot \color{blue}{\left(6 \cdot x1 - \left(2 - 1\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto x1 \cdot \left(6 \cdot x1 - \color{blue}{1}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x1 \cdot \left(6 \cdot x1 - 1\right)} \]

       7. sub-neg N/A

          \[\leadsto x1 \cdot \color{blue}{\left(6 \cdot x1 + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto x1 \cdot \left(\color{blue}{x1 \cdot 6} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto x1 \cdot \left(x1 \cdot 6 + \color{blue}{-1}\right) \]

       10. accelerator-lowering-fma.f64 48.3

           \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 6, -1\right)} \]

   14. Simplified 48.3%

       \[\leadsto \color{blue}{x1 \cdot \mathsf{fma}\left(x1, 6, -1\right)} \]

   if -3.29999999999999987e-140 < x1 < 4.79999999999999985e-124

   1. Initial program 99.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Simplified 90.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in 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 79.1

         \[\leadsto \color{blue}{x2 \cdot -6} \]

   8. Simplified 79.1%

      \[\leadsto \color{blue}{x2 \cdot -6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 25.6% accurate, 33.1× speedup

Math

\[\begin{array}{l} \\ x1 + x2 \cdot -6 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (+ x1 (* x2 -6.0)))

C

double code(double x1, double x2) {
	return x1 + (x2 * -6.0);
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = x1 + (x2 * (-6.0d0))
end function

Java

public static double code(double x1, double x2) {
	return x1 + (x2 * -6.0);
}

Python

def code(x1, x2):
	return x1 + (x2 * -6.0)

Julia

function code(x1, x2)
	return Float64(x1 + Float64(x2 * -6.0))
end

MATLAB

function tmp = code(x1, x2)
	tmp = x1 + (x2 * -6.0);
end

Wolfram

code[x1_, x2_] := N[(x1 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.0%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
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 24.5

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

5. Simplified 24.5%

   \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]
6. Add Preprocessing

Alternative 20: 25.4% accurate, 49.7× speedup

Math

\[\begin{array}{l} \\ x2 \cdot -6 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (* x2 -6.0))

C

double code(double x1, double x2) {
	return x2 * -6.0;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = x2 * (-6.0d0)
end function

Java

public static double code(double x1, double x2) {
	return x2 * -6.0;
}

Python

def code(x1, x2):
	return x2 * -6.0

Julia

function code(x1, x2)
	return Float64(x2 * -6.0)
end

MATLAB

function tmp = code(x1, x2)
	tmp = x2 * -6.0;
end

Wolfram

code[x1_, x2_] := N[(x2 * -6.0), $MachinePrecision]

Derivation

1. Initial program 68.0%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x1 around 0

   \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

5. Simplified 69.2%

   \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

6. Taylor expanded in 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 24.5

      \[\leadsto \color{blue}{x2 \cdot -6} \]

8. Simplified 24.5%

   \[\leadsto \color{blue}{x2 \cdot -6} \]
9. Add Preprocessing

Alternative 21: 3.2% accurate, 298.0× speedup

Math

\[\begin{array}{l} \\ x1 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 x1)

C

double code(double x1, double x2) {
	return x1;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = x1
end function

Java

public static double code(double x1, double x2) {
	return x1;
}

Python

def code(x1, x2):
	return x1

Julia

function code(x1, x2)
	return x1
end

MATLAB

function tmp = code(x1, x2)
	tmp = x1;
end

Wolfram

code[x1_, x2_] := x1

Derivation

1. Initial program 68.0%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
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 24.5

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

5. Simplified 24.5%

   \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

6. Taylor expanded in x1 around inf

   \[\leadsto \color{blue}{x1} \]
7. Step-by-step derivation

8. Simplified 3.1%

   \[\leadsto \color{blue}{x1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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))))))