Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.2% → 98.9%

Time: 44.4s

Alternatives: 19

Speedup: 7.6×

• 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: 70.2% 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: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ t_1 := \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\\ t_2 := \frac{t\_1}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_3 := x1 \cdot \left(x1 \cdot 3\right)\\ \mathbf{if}\;x1 \leq -1.8 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.35 \cdot 10^{+54}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(t\_2, 4, -6\right), \frac{\left(-3 + t\_2\right) \cdot \left(\left(x1 \cdot 2\right) \cdot t\_1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot x1, t\_3 \cdot t\_2\right) + \mathsf{fma}\left(3, \frac{t\_3 - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (pow x1 4.0)
          (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))
        (t_1 (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))))
        (t_2 (/ t_1 (fma x1 x1 1.0)))
        (t_3 (* x1 (* x1 3.0))))
   (if (<= x1 -1.8e+44)
     t_0
     (if (<= x1 2.35e+54)
       (+
        x1
        (fma
         (fma
          x1
          (* x1 (fma t_2 4.0 -6.0))
          (/ (* (+ -3.0 t_2) (* (* x1 2.0) t_1)) (fma x1 x1 1.0)))
         (fma x1 x1 1.0)
         (+
          (fma x1 (* x1 x1) (* t_3 t_2))
          (fma 3.0 (/ (- t_3 (fma 2.0 x2 x1)) (fma x1 x1 1.0)) x1))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1));
	double t_1 = fma(2.0, x2, fma(x1, (x1 * 3.0), -x1));
	double t_2 = t_1 / fma(x1, x1, 1.0);
	double t_3 = x1 * (x1 * 3.0);
	double tmp;
	if (x1 <= -1.8e+44) {
		tmp = t_0;
	} else if (x1 <= 2.35e+54) {
		tmp = x1 + fma(fma(x1, (x1 * fma(t_2, 4.0, -6.0)), (((-3.0 + t_2) * ((x1 * 2.0) * t_1)) / fma(x1, x1, 1.0))), fma(x1, x1, 1.0), (fma(x1, (x1 * x1), (t_3 * t_2)) + fma(3.0, ((t_3 - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), x1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = 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)))
	t_1 = fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1)))
	t_2 = Float64(t_1 / fma(x1, x1, 1.0))
	t_3 = Float64(x1 * Float64(x1 * 3.0))
	tmp = 0.0
	if (x1 <= -1.8e+44)
		tmp = t_0;
	elseif (x1 <= 2.35e+54)
		tmp = Float64(x1 + fma(fma(x1, Float64(x1 * fma(t_2, 4.0, -6.0)), Float64(Float64(Float64(-3.0 + t_2) * Float64(Float64(x1 * 2.0) * t_1)) / fma(x1, x1, 1.0))), fma(x1, x1, 1.0), Float64(fma(x1, Float64(x1 * x1), Float64(t_3 * t_2)) + fma(3.0, Float64(Float64(t_3 - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), x1))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.8e+44], t$95$0, If[LessEqual[x1, 2.35e+54], N[(x1 + N[(N[(x1 * N[(x1 * N[(t$95$2 * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-3.0 + t$95$2), $MachinePrecision] * N[(N[(x1 * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(x1 * N[(x1 * x1), $MachinePrecision] + N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(t$95$3 - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.8e44 or 2.34999999999999996e54 < x1

   1. Initial program 31.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

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

   if -1.8e44 < x1 < 2.34999999999999996e54

   1. Initial program 99.3%

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

   4. Applied rewrites 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.8 \cdot 10^{+44}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 2.35 \cdot 10^{+54}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(-3 + \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot x1, \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(3, \frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot x1\right)\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \mathsf{fma}\left(x2, -6, 8 \cdot \left(x2 \cdot \left(x1 \cdot x2\right)\right)\right)\\ t_3 := x1 \cdot x1 + 1\\ t_4 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_3}\\ t_5 := x1 + \left(\left(x1 + \left(\left(t\_3 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_4\right) \cdot \left(t\_4 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_4 \cdot 4 - 6\right)\right) + t\_1 \cdot t\_4\right) + t\_0\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_3}\right)\\ \mathbf{if}\;t\_5 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \left(6 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 x1)))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (fma x2 -6.0 (* 8.0 (* x2 (* x1 x2)))))
        (t_3 (+ (* x1 x1) 1.0))
        (t_4 (/ (- (+ t_1 (* 2.0 x2)) x1) t_3))
        (t_5
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_3
               (+
                (* (* (* x1 2.0) t_4) (- t_4 3.0))
                (* (* x1 x1) (- (* t_4 4.0) 6.0))))
              (* t_1 t_4))
             t_0))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_3))))))
   (if (<= t_5 -500.0)
     t_2
     (if (<= t_5 1e-8)
       (fma x2 -6.0 (* x1 (fma 9.0 x1 -1.0)))
       (if (<= t_5 5e+289) t_2 (+ x1 (* x1 (* 6.0 t_0))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * x1);
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = fma(x2, -6.0, (8.0 * (x2 * (x1 * x2))));
	double t_3 = (x1 * x1) + 1.0;
	double t_4 = ((t_1 + (2.0 * x2)) - x1) / t_3;
	double t_5 = x1 + ((x1 + (((t_3 * ((((x1 * 2.0) * t_4) * (t_4 - 3.0)) + ((x1 * x1) * ((t_4 * 4.0) - 6.0)))) + (t_1 * t_4)) + t_0)) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_3)));
	double tmp;
	if (t_5 <= -500.0) {
		tmp = t_2;
	} else if (t_5 <= 1e-8) {
		tmp = fma(x2, -6.0, (x1 * fma(9.0, x1, -1.0)));
	} else if (t_5 <= 5e+289) {
		tmp = t_2;
	} else {
		tmp = x1 + (x1 * (6.0 * t_0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * x1))
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = fma(x2, -6.0, Float64(8.0 * Float64(x2 * Float64(x1 * x2))))
	t_3 = Float64(Float64(x1 * x1) + 1.0)
	t_4 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_3)
	t_5 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_3 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_4) * Float64(t_4 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_4 * 4.0) - 6.0)))) + Float64(t_1 * t_4)) + t_0)) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_3))))
	tmp = 0.0
	if (t_5 <= -500.0)
		tmp = t_2;
	elseif (t_5 <= 1e-8)
		tmp = fma(x2, -6.0, Float64(x1 * fma(9.0, x1, -1.0)));
	elseif (t_5 <= 5e+289)
		tmp = t_2;
	else
		tmp = Float64(x1 + Float64(x1 * Float64(6.0 * t_0)));
	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[(x2 * -6.0 + N[(8.0 * N[(x2 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$3 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(t$95$4 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$4 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -500.0], t$95$2, If[LessEqual[t$95$5, 1e-8], N[(x2 * -6.0 + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 5e+289], t$95$2, N[(x1 + N[(x1 * N[(6.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -500 or 1e-8 < (+.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)))))) < 5.00000000000000031e289

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), x2 \cdot 14\right) + -6\right), -1\right)\right)\right)} \]

   9. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(x2, -6, 8 \cdot \left(x1 \cdot {x2}^{2}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 78.5%

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

   if -500 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1e-8

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 41.1

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), x2 \cdot 14\right) + -6\right), -1\right)\right)\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 99.8%

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

   if 5.00000000000000031e289 < (+.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 32.1%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 88.5

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

   5. Applied rewrites 88.5%

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

   7. Applied rewrites 88.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -500:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, 8 \cdot \left(x2 \cdot \left(x1 \cdot x2\right)\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\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^{+289}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, 8 \cdot \left(x2 \cdot \left(x1 \cdot x2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \left(6 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := \mathsf{fma}\left(x2, -6, 8 \cdot \left(x2 \cdot \left(x1 \cdot x2\right)\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (fma x2 -6.0 (* 8.0 (* x2 (* x1 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 -500.0)
     t_1
     (if (<= t_4 1e-8)
       (fma x2 -6.0 (* x1 (fma 9.0 x1 -1.0)))
       (if (<= t_4 INFINITY) t_1 (+ x1 (* (* x1 x1) 9.0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = fma(x2, -6.0, (8.0 * (x2 * (x1 * 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 <= -500.0) {
		tmp = t_1;
	} else if (t_4 <= 1e-8) {
		tmp = fma(x2, -6.0, (x1 * fma(9.0, x1, -1.0)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = fma(x2, -6.0, Float64(8.0 * Float64(x2 * Float64(x1 * 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 <= -500.0)
		tmp = t_1;
	elseif (t_4 <= 1e-8)
		tmp = fma(x2, -6.0, Float64(x1 * fma(9.0, x1, -1.0)));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	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 * -6.0 + N[(8.0 * N[(x2 * N[(x1 * x2), $MachinePrecision]), $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, -500.0], t$95$1, If[LessEqual[t$95$4, 1e-8], N[(x2 * -6.0 + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -500 or 1e-8 < (+.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 99.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), x2 \cdot 14\right) + -6\right), -1\right)\right)\right)} \]

   9. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(x2, -6, 8 \cdot \left(x1 \cdot {x2}^{2}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 70.8%

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

   if -500 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1e-8

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 41.1

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), x2 \cdot 14\right) + -6\right), -1\right)\right)\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 72.3%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 88.7%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 88.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -500:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, 8 \cdot \left(x2 \cdot \left(x1 \cdot x2\right)\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\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, -6, 8 \cdot \left(x2 \cdot \left(x1 \cdot x2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 + 8 \cdot \left(x2 \cdot \left(x1 \cdot x2\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+197}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ x1 (* 8.0 (* x2 (* x1 x2)))))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_0 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -2e+197)
     t_1
     (if (<= t_4 5e+143)
       (fma x2 -6.0 (* x1 (fma 9.0 x1 -1.0)))
       (if (<= t_4 INFINITY) t_1 (+ x1 (* (* x1 x1) 9.0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x1 + (8.0 * (x2 * (x1 * x2)));
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_0 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -2e+197) {
		tmp = t_1;
	} else if (t_4 <= 5e+143) {
		tmp = fma(x2, -6.0, (x1 * fma(9.0, x1, -1.0)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x1 + Float64(8.0 * Float64(x2 * Float64(x1 * x2))))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_0 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -2e+197)
		tmp = t_1;
	elseif (t_4 <= 5e+143)
		tmp = fma(x2, -6.0, Float64(x1 * fma(9.0, x1, -1.0)));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x1 + N[(8.0 * N[(x2 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+197], t$95$1, If[LessEqual[t$95$4, 5e+143], N[(x2 * -6.0 + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.9999999999999999e197 or 5.00000000000000012e143 < (+.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. Applied rewrites 45.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 65.6%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 55.3%

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

   if -1.9999999999999999e197 < (+.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)))))) < 5.00000000000000012e143

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), x2 \cdot 14\right) + -6\right), -1\right)\right)\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 90.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 72.3%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 88.7%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 88.7%

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

4. Final simplification 77.6%

   \[\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 -2 \cdot 10^{+197}:\\ \;\;\;\;x1 + 8 \cdot \left(x2 \cdot \left(x1 \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^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\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:\\ \;\;\;\;x1 + 8 \cdot \left(x2 \cdot \left(x1 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 + 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ x1 (* 8.0 (* x1 (* x2 x2)))))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_0 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -2e+307)
     t_1
     (if (<= t_4 5e+143)
       (fma x2 -6.0 (* x1 (fma 9.0 x1 -1.0)))
       (if (<= t_4 INFINITY) t_1 (+ x1 (* (* x1 x1) 9.0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x1 + (8.0 * (x1 * (x2 * x2)));
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_0 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -2e+307) {
		tmp = t_1;
	} else if (t_4 <= 5e+143) {
		tmp = fma(x2, -6.0, (x1 * fma(9.0, x1, -1.0)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x1 + Float64(8.0 * Float64(x1 * Float64(x2 * x2))))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_0 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -2e+307)
		tmp = t_1;
	elseif (t_4 <= 5e+143)
		tmp = fma(x2, -6.0, Float64(x1 * fma(9.0, x1, -1.0)));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x1 + N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+307], t$95$1, If[LessEqual[t$95$4, 5e+143], N[(x2 * -6.0 + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.99999999999999997e307 or 5.00000000000000012e143 < (+.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. Applied rewrites 47.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 46.9%

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

   if -1.99999999999999997e307 < (+.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)))))) < 5.00000000000000012e143

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), x2 \cdot 14\right) + -6\right), -1\right)\right)\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 85.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 72.3%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 88.7%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 88.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2 \cdot 10^{+307}:\\ \;\;\;\;x1 + 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^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\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:\\ \;\;\;\;x1 + 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -500:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;t\_3 \leq 10^{-66}:\\ \;\;\;\;x1 + x1 \cdot -2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+248}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

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 -500.0)
     (* x2 -6.0)
     (if (<= t_3 1e-66)
       (+ x1 (* x1 -2.0))
       (if (<= t_3 5e+248) (* x2 -6.0) (+ x1 (* (* x1 x1) 9.0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -500.0) {
		tmp = x2 * -6.0;
	} else if (t_3 <= 1e-66) {
		tmp = x1 + (x1 * -2.0);
	} else if (t_3 <= 5e+248) {
		tmp = x2 * -6.0;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = x1 * (x1 * 3.0d0)
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    t_3 = x1 + ((x1 + (((t_1 * ((((x1 * 2.0d0) * t_2) * (t_2 - 3.0d0)) + ((x1 * x1) * ((t_2 * 4.0d0) - 6.0d0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0d0 * (((t_0 - (2.0d0 * x2)) - x1) / t_1)))
    if (t_3 <= (-500.0d0)) then
        tmp = x2 * (-6.0d0)
    else if (t_3 <= 1d-66) then
        tmp = x1 + (x1 * (-2.0d0))
    else if (t_3 <= 5d+248) then
        tmp = x2 * (-6.0d0)
    else
        tmp = x1 + ((x1 * x1) * 9.0d0)
    end if
    code = tmp
end function

Java

public static 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 <= -500.0) {
		tmp = x2 * -6.0;
	} else if (t_3 <= 1e-66) {
		tmp = x1 + (x1 * -2.0);
	} else if (t_3 <= 5e+248) {
		tmp = x2 * -6.0;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Python

def code(x1, x2):
	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)))
	tmp = 0
	if t_3 <= -500.0:
		tmp = x2 * -6.0
	elif t_3 <= 1e-66:
		tmp = x1 + (x1 * -2.0)
	elif t_3 <= 5e+248:
		tmp = x2 * -6.0
	else:
		tmp = x1 + ((x1 * x1) * 9.0)
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= -500.0)
		tmp = Float64(x2 * -6.0);
	elseif (t_3 <= 1e-66)
		tmp = Float64(x1 + Float64(x1 * -2.0));
	elseif (t_3 <= 5e+248)
		tmp = Float64(x2 * -6.0);
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	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)));
	tmp = 0.0;
	if (t_3 <= -500.0)
		tmp = x2 * -6.0;
	elseif (t_3 <= 1e-66)
		tmp = x1 + (x1 * -2.0);
	elseif (t_3 <= 5e+248)
		tmp = x2 * -6.0;
	else
		tmp = x1 + ((x1 * x1) * 9.0);
	end
	tmp_2 = 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, -500.0], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[t$95$3, 1e-66], N[(x1 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+248], N[(x2 * -6.0), $MachinePrecision], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -500 or 9.9999999999999998e-67 < (+.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)))))) < 4.9999999999999996e248

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.6

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

   8. Applied rewrites 49.6%

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

   if -500 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 9.9999999999999998e-67

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 99.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 62.7%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 61.6%

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

   if 4.9999999999999996e248 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 38.8%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 60.8%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 56.0%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 56.2%

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

4. Final simplification 54.4%

   \[\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 -500:\\ \;\;\;\;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 10^{-66}:\\ \;\;\;\;x1 + x1 \cdot -2\\ \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^{+248}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 0.5× speedup

Math

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

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
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_1 t_3))
             t_0))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 INFINITY) t_4 (+ x1 (* x1 (* 6.0 t_0))))))

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

Java

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

Python

def code(x1, x2):
	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 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_1 * t_3)) + t_0)) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)))
	tmp = 0
	if t_4 <= math.inf:
		tmp = t_4
	else:
		tmp = x1 + (x1 * (6.0 * t_0))
	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(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_1 * t_3)) + t_0)) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(x1 + Float64(x1 * Float64(6.0 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	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 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_1 * t_3)) + t_0)) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)));
	tmp = 0.0;
	if (t_4 <= Inf)
		tmp = t_4;
	else
		tmp = x1 + (x1 * (6.0 * t_0));
	end
	tmp_2 = 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[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, Infinity], t$95$4, N[(x1 + N[(x1 * N[(6.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.6%

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

Alternative 8: 96.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq 0.000155:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{elif}\;x1 \leq 2.35 \cdot 10^{+54}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1
         (*
          (pow x1 4.0)
          (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2)))
   (if (<= x1 -3.6e+40)
     t_1
     (if (<= x1 0.000155)
       (+
        x1
        (fma x2 (fma x1 (fma x2 8.0 -12.0) -6.0) (* x1 (fma 9.0 x1 -2.0))))
       (if (<= x1 2.35e+54)
         (+
          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 3.0)))
         t_1)))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1));
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double tmp;
	if (x1 <= -3.6e+40) {
		tmp = t_1;
	} else if (x1 <= 0.000155) {
		tmp = x1 + fma(x2, fma(x1, fma(x2, 8.0, -12.0), -6.0), (x1 * fma(9.0, x1, -2.0)));
	} else if (x1 <= 2.35e+54) {
		tmp = 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 * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = 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)))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	tmp = 0.0
	if (x1 <= -3.6e+40)
		tmp = t_1;
	elseif (x1 <= 0.000155)
		tmp = Float64(x1 + fma(x2, fma(x1, fma(x2, 8.0, -12.0), -6.0), Float64(x1 * fma(9.0, x1, -2.0))));
	elseif (x1 <= 2.35e+54)
		tmp = 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 * 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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]}, 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]}, If[LessEqual[x1, -3.6e+40], t$95$1, If[LessEqual[x1, 0.000155], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.35e+54], 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 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.59999999999999996e40 or 2.34999999999999996e54 < x1

   1. Initial program 31.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

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

   if -3.59999999999999996e40 < x1 < 1.55e-4

   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. Applied rewrites 84.2%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 98.2%

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

   if 1.55e-4 < x1 < 2.34999999999999996e54

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 0.000155:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{elif}\;x1 \leq 2.35 \cdot 10^{+54}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ t_1 := \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\\ t_2 := \frac{t\_1}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;x1 \leq -1.8 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.35 \cdot 10^{+54}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(t\_2, 4, -6\right), \frac{\left(-3 + t\_2\right) \cdot \left(\left(x1 \cdot 2\right) \cdot t\_1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot t\_2, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (pow x1 4.0)
          (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))
        (t_1 (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))))
        (t_2 (/ t_1 (fma x1 x1 1.0))))
   (if (<= x1 -1.8e+44)
     t_0
     (if (<= x1 2.35e+54)
       (+
        x1
        (fma
         (/ (- (* x1 (* x1 3.0)) (fma 2.0 x2 x1)) (fma x1 x1 1.0))
         3.0
         (fma
          (fma x1 x1 1.0)
          (fma
           x1
           (* x1 (fma t_2 4.0 -6.0))
           (/ (* (+ -3.0 t_2) (* (* x1 2.0) t_1)) (fma x1 x1 1.0)))
          (fma x1 (* (* x1 3.0) t_2) (fma x1 (* x1 x1) x1)))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1));
	double t_1 = fma(2.0, x2, fma(x1, (x1 * 3.0), -x1));
	double t_2 = t_1 / fma(x1, x1, 1.0);
	double tmp;
	if (x1 <= -1.8e+44) {
		tmp = t_0;
	} else if (x1 <= 2.35e+54) {
		tmp = x1 + fma((((x1 * (x1 * 3.0)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), fma(x1, (x1 * fma(t_2, 4.0, -6.0)), (((-3.0 + t_2) * ((x1 * 2.0) * t_1)) / fma(x1, x1, 1.0))), fma(x1, ((x1 * 3.0) * t_2), fma(x1, (x1 * x1), x1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = 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)))
	t_1 = fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1)))
	t_2 = Float64(t_1 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (x1 <= -1.8e+44)
		tmp = t_0;
	elseif (x1 <= 2.35e+54)
		tmp = Float64(x1 + fma(Float64(Float64(Float64(x1 * Float64(x1 * 3.0)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(t_2, 4.0, -6.0)), Float64(Float64(Float64(-3.0 + t_2) * Float64(Float64(x1 * 2.0) * t_1)) / fma(x1, x1, 1.0))), fma(x1, Float64(Float64(x1 * 3.0) * t_2), fma(x1, Float64(x1 * x1), x1)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.8e+44], t$95$0, If[LessEqual[x1, 2.35e+54], N[(x1 + N[(N[(N[(N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(t$95$2 * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-3.0 + t$95$2), $MachinePrecision] * N[(N[(x1 * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.8e44 or 2.34999999999999996e54 < x1

   1. Initial program 31.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

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

   if -1.8e44 < x1 < 2.34999999999999996e54

   1. Initial program 99.3%

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

   4. Applied rewrites 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.8 \cdot 10^{+44}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 2.35 \cdot 10^{+54}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(-3 + \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (pow x1 4.0)
          (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1)))))
   (if (<= x1 -3.6e+40)
     t_0
     (if (<= x1 2.9e+26)
       (+
        x1
        (fma x2 (fma x1 (fma x2 8.0 -12.0) -6.0) (* x1 (fma 9.0 x1 -2.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = 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 <= -3.6e+40) {
		tmp = t_0;
	} else if (x1 <= 2.9e+26) {
		tmp = x1 + fma(x2, fma(x1, fma(x2, 8.0, -12.0), -6.0), (x1 * fma(9.0, x1, -2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = 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 <= -3.6e+40)
		tmp = t_0;
	elseif (x1 <= 2.9e+26)
		tmp = Float64(x1 + fma(x2, fma(x1, fma(x2, 8.0, -12.0), -6.0), Float64(x1 * fma(9.0, x1, -2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -3.6e+40], t$95$0, If[LessEqual[x1, 2.9e+26], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.59999999999999996e40 or 2.9e26 < x1

   1. Initial program 35.5%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 98.3%

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

   if -3.59999999999999996e40 < x1 < 2.9e26

   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. Applied rewrites 82.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 96.0%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 96.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;x1 + 6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + 6 \cdot {x1}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -3.6e+40)
   (+ x1 (* 6.0 (* x1 (* x1 (* x1 x1)))))
   (if (<= x1 2.9e+26)
     (+ x1 (fma x2 (fma x1 (fma x2 8.0 -12.0) -6.0) (* x1 (fma 9.0 x1 -2.0))))
     (+ x1 (* 6.0 (pow x1 4.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -3.6e+40) {
		tmp = x1 + (6.0 * (x1 * (x1 * (x1 * x1))));
	} else if (x1 <= 2.9e+26) {
		tmp = x1 + fma(x2, fma(x1, fma(x2, 8.0, -12.0), -6.0), (x1 * fma(9.0, x1, -2.0)));
	} else {
		tmp = x1 + (6.0 * pow(x1, 4.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -3.6e+40)
		tmp = Float64(x1 + Float64(6.0 * Float64(x1 * Float64(x1 * Float64(x1 * x1)))));
	elseif (x1 <= 2.9e+26)
		tmp = Float64(x1 + fma(x2, fma(x1, fma(x2, 8.0, -12.0), -6.0), Float64(x1 * fma(9.0, x1, -2.0))));
	else
		tmp = Float64(x1 + Float64(6.0 * (x1 ^ 4.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -3.6e+40], N[(x1 + N[(6.0 * N[(x1 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.9e+26], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(6.0 * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.59999999999999996e40

   1. Initial program 16.0%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   7. Applied rewrites 98.0%

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

   if -3.59999999999999996e40 < x1 < 2.9e26

   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. Applied rewrites 82.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 96.0%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 96.0%

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

   if 2.9e26 < x1

   1. Initial program 50.6%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;x1 + 6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + 6 \cdot {x1}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;x1 + 6 \cdot \left(x1 \cdot t\_0\right)\\ \mathbf{elif}\;x1 \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \left(6 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 x1))))
   (if (<= x1 -3.6e+40)
     (+ x1 (* 6.0 (* x1 t_0)))
     (if (<= x1 2.9e+26)
       (+
        x1
        (fma x2 (fma x1 (fma x2 8.0 -12.0) -6.0) (* x1 (fma 9.0 x1 -2.0))))
       (+ x1 (* x1 (* 6.0 t_0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * x1);
	double tmp;
	if (x1 <= -3.6e+40) {
		tmp = x1 + (6.0 * (x1 * t_0));
	} else if (x1 <= 2.9e+26) {
		tmp = x1 + fma(x2, fma(x1, fma(x2, 8.0, -12.0), -6.0), (x1 * fma(9.0, x1, -2.0)));
	} else {
		tmp = x1 + (x1 * (6.0 * t_0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -3.6e+40)
		tmp = Float64(x1 + Float64(6.0 * Float64(x1 * t_0)));
	elseif (x1 <= 2.9e+26)
		tmp = Float64(x1 + fma(x2, fma(x1, fma(x2, 8.0, -12.0), -6.0), Float64(x1 * fma(9.0, x1, -2.0))));
	else
		tmp = Float64(x1 + Float64(x1 * Float64(6.0 * t_0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -3.6e+40], N[(x1 + N[(6.0 * N[(x1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.9e+26], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(x1 * N[(6.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.59999999999999996e40

   1. Initial program 16.0%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   7. Applied rewrites 98.0%

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

   if -3.59999999999999996e40 < x1 < 2.9e26

   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. Applied rewrites 82.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 96.0%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 96.0%

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

   if 2.9e26 < x1

   1. Initial program 50.6%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 88.2

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

   5. Applied rewrites 88.2%

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

   7. Applied rewrites 88.2%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;x1 + 6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \left(6 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 92.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;x1 + 6 \cdot \left(x1 \cdot t\_0\right)\\ \mathbf{elif}\;x1 \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right), x1 \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \left(6 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 x1))))
   (if (<= x1 -3.6e+40)
     (+ x1 (* 6.0 (* x1 t_0)))
     (if (<= x1 2.9e+26)
       (+ x1 (fma x2 (fma x1 (fma x2 8.0 -12.0) -6.0) (* x1 -2.0)))
       (+ x1 (* x1 (* 6.0 t_0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * x1);
	double tmp;
	if (x1 <= -3.6e+40) {
		tmp = x1 + (6.0 * (x1 * t_0));
	} else if (x1 <= 2.9e+26) {
		tmp = x1 + fma(x2, fma(x1, fma(x2, 8.0, -12.0), -6.0), (x1 * -2.0));
	} else {
		tmp = x1 + (x1 * (6.0 * t_0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -3.6e+40)
		tmp = Float64(x1 + Float64(6.0 * Float64(x1 * t_0)));
	elseif (x1 <= 2.9e+26)
		tmp = Float64(x1 + fma(x2, fma(x1, fma(x2, 8.0, -12.0), -6.0), Float64(x1 * -2.0)));
	else
		tmp = Float64(x1 + Float64(x1 * Float64(6.0 * t_0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -3.6e+40], N[(x1 + N[(6.0 * N[(x1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.9e+26], N[(x1 + N[(x2 * N[(x1 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(x1 * N[(6.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.59999999999999996e40

   1. Initial program 16.0%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   7. Applied rewrites 98.0%

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

   if -3.59999999999999996e40 < x1 < 2.9e26

   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. Applied rewrites 82.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 96.0%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 96.0%

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

   12. Taylor expanded in x1 around 0

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

   14. Applied rewrites 95.4%

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

   if 2.9e26 < x1

   1. Initial program 50.6%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 88.2

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

   5. Applied rewrites 88.2%

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

   7. Applied rewrites 88.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;x1 + 6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right), x1 \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \left(6 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + x2 \cdot -6\\ \mathbf{if}\;2 \cdot x2 \leq -1 \cdot 10^{-85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;2 \cdot x2 \leq 5 \cdot 10^{-110}:\\ \;\;\;\;x1 + x1 \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (* x2 -6.0))))
   (if (<= (* 2.0 x2) -1e-85)
     t_0
     (if (<= (* 2.0 x2) 5e-110) (+ x1 (* x1 -2.0)) t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (x2 * -6.0);
	double tmp;
	if ((2.0 * x2) <= -1e-85) {
		tmp = t_0;
	} else if ((2.0 * x2) <= 5e-110) {
		tmp = x1 + (x1 * -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x1 + (x2 * (-6.0d0))
    if ((2.0d0 * x2) <= (-1d-85)) then
        tmp = t_0
    else if ((2.0d0 * x2) <= 5d-110) then
        tmp = x1 + (x1 * (-2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = x1 + (x2 * -6.0);
	double tmp;
	if ((2.0 * x2) <= -1e-85) {
		tmp = t_0;
	} else if ((2.0 * x2) <= 5e-110) {
		tmp = x1 + (x1 * -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = x1 + (x2 * -6.0)
	tmp = 0
	if (2.0 * x2) <= -1e-85:
		tmp = t_0
	elif (2.0 * x2) <= 5e-110:
		tmp = x1 + (x1 * -2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64(x2 * -6.0))
	tmp = 0.0
	if (Float64(2.0 * x2) <= -1e-85)
		tmp = t_0;
	elseif (Float64(2.0 * x2) <= 5e-110)
		tmp = Float64(x1 + Float64(x1 * -2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 + (x2 * -6.0);
	tmp = 0.0;
	if ((2.0 * x2) <= -1e-85)
		tmp = t_0;
	elseif ((2.0 * x2) <= 5e-110)
		tmp = x1 + (x1 * -2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * x2), $MachinePrecision], -1e-85], t$95$0, If[LessEqual[N[(2.0 * x2), $MachinePrecision], 5e-110], N[(x1 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -9.9999999999999998e-86 or 5e-110 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 73.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.8

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

   5. Applied rewrites 30.8%

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

   if -9.9999999999999998e-86 < (*.f64 #s(literal 2 binary64) x2) < 5e-110

   1. Initial program 64.6%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 75.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 33.1%

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

Alternative 15: 31.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -1 \cdot 10^{-85}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;2 \cdot x2 \leq 5 \cdot 10^{-110}:\\ \;\;\;\;x1 + x1 \cdot -2\\ \mathbf{else}:\\ \;\;\;\;x2 \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* 2.0 x2) -1e-85)
   (* x2 -6.0)
   (if (<= (* 2.0 x2) 5e-110) (+ x1 (* x1 -2.0)) (* x2 -6.0))))

C

double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -1e-85) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 5e-110) {
		tmp = x1 + (x1 * -2.0);
	} else {
		tmp = x2 * -6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((2.0d0 * x2) <= (-1d-85)) then
        tmp = x2 * (-6.0d0)
    else if ((2.0d0 * x2) <= 5d-110) then
        tmp = x1 + (x1 * (-2.0d0))
    else
        tmp = x2 * (-6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -1e-85) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 5e-110) {
		tmp = x1 + (x1 * -2.0);
	} else {
		tmp = x2 * -6.0;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (2.0 * x2) <= -1e-85:
		tmp = x2 * -6.0
	elif (2.0 * x2) <= 5e-110:
		tmp = x1 + (x1 * -2.0)
	else:
		tmp = x2 * -6.0
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(2.0 * x2) <= -1e-85)
		tmp = Float64(x2 * -6.0);
	elseif (Float64(2.0 * x2) <= 5e-110)
		tmp = Float64(x1 + Float64(x1 * -2.0));
	else
		tmp = Float64(x2 * -6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((2.0 * x2) <= -1e-85)
		tmp = x2 * -6.0;
	elseif ((2.0 * x2) <= 5e-110)
		tmp = x1 + (x1 * -2.0);
	else
		tmp = x2 * -6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(2.0 * x2), $MachinePrecision], -1e-85], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[N[(2.0 * x2), $MachinePrecision], 5e-110], N[(x1 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision], N[(x2 * -6.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -9.9999999999999998e-86 or 5e-110 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 73.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.8

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.6

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

   8. Applied rewrites 30.6%

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

   if -9.9999999999999998e-86 < (*.f64 #s(literal 2 binary64) x2) < 5e-110

   1. Initial program 64.6%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 75.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 33.1%

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

Alternative 16: 58.3% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -7.2 \cdot 10^{-135}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -19, 9\right), -1\right)\\ \mathbf{elif}\;x1 \leq 1.52 \cdot 10^{-111}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -7.2e-135)
   (* x1 (fma x1 (fma x1 -19.0 9.0) -1.0))
   (if (<= x1 1.52e-111) (* x2 -6.0) (+ x1 (* x1 (fma 9.0 x1 -2.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -7.2e-135) {
		tmp = x1 * fma(x1, fma(x1, -19.0, 9.0), -1.0);
	} else if (x1 <= 1.52e-111) {
		tmp = x2 * -6.0;
	} else {
		tmp = x1 + (x1 * fma(9.0, x1, -2.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -7.2e-135)
		tmp = Float64(x1 * fma(x1, fma(x1, -19.0, 9.0), -1.0));
	elseif (x1 <= 1.52e-111)
		tmp = Float64(x2 * -6.0);
	else
		tmp = Float64(x1 + Float64(x1 * fma(9.0, x1, -2.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -7.2e-135], N[(x1 * N[(x1 * N[(x1 * -19.0 + 9.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1.52e-111], N[(x2 * -6.0), $MachinePrecision], N[(x1 + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -7.19999999999999955e-135

   1. Initial program 48.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 8.2

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

   5. Applied rewrites 8.2%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 9.3

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

   8. Applied rewrites 9.3%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 66.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(-2, \mathsf{fma}\left(x2, 2, -3\right), \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x1, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, -4, 6\right), 1 + \mathsf{fma}\left(x2, 6, -9\right)\right) + \left(x2 \cdot -2\right) \cdot \mathsf{fma}\left(x2, 2, -3\right), \mathsf{fma}\left(\mathsf{fma}\left(x2, 2, -3\right), x2 \cdot 4, -3\right)\right), \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + -6\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(x2, 2, -3\right), x2 \cdot 4, -1\right)\right), x2 \cdot -6\right)} \]

   12. Taylor expanded in x2 around 0

       \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -19 \cdot x1\right) - 1\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 67.5%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -19, 9\right), -1\right)} \]

   if -7.19999999999999955e-135 < x1 < 1.51999999999999998e-111

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

   if 1.51999999999999998e-111 < x1

   1. Initial program 67.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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. Applied rewrites 68.1%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 42.7%

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

Alternative 17: 54.4% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\\ \mathbf{if}\;x1 \leq -7.2 \cdot 10^{-135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.52 \cdot 10^{-111}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (* x1 (fma 9.0 x1 -2.0)))))
   (if (<= x1 -7.2e-135) t_0 (if (<= x1 1.52e-111) (* x2 -6.0) t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (x1 * fma(9.0, x1, -2.0));
	double tmp;
	if (x1 <= -7.2e-135) {
		tmp = t_0;
	} else if (x1 <= 1.52e-111) {
		tmp = x2 * -6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64(x1 * fma(9.0, x1, -2.0)))
	tmp = 0.0
	if (x1 <= -7.2e-135)
		tmp = t_0;
	elseif (x1 <= 1.52e-111)
		tmp = Float64(x2 * -6.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -7.2e-135], t$95$0, If[LessEqual[x1, 1.52e-111], N[(x2 * -6.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -7.19999999999999955e-135 or 1.51999999999999998e-111 < x1

   1. Initial program 58.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 67.0%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 49.2%

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

   if -7.19999999999999955e-135 < x1 < 1.51999999999999998e-111

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

Alternative 18: 67.1% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -3.8 \cdot 10^{+40}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -19, 9\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -3.8e+40)
   (* x1 (fma x1 (fma x1 -19.0 9.0) -1.0))
   (fma x2 -6.0 (* x1 (fma 9.0 x1 -1.0)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -3.8e+40) {
		tmp = x1 * fma(x1, fma(x1, -19.0, 9.0), -1.0);
	} else {
		tmp = fma(x2, -6.0, (x1 * fma(9.0, x1, -1.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -3.8e+40)
		tmp = Float64(x1 * fma(x1, fma(x1, -19.0, 9.0), -1.0));
	else
		tmp = fma(x2, -6.0, Float64(x1 * fma(9.0, x1, -1.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -3.8e+40], N[(x1 * N[(x1 * N[(x1 * -19.0 + 9.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(x2 * -6.0 + N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.80000000000000004e40

   1. Initial program 16.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.8

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

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 1.9

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

   8. Applied rewrites 1.9%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 65.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(-2, \mathsf{fma}\left(x2, 2, -3\right), \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x1, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, -4, 6\right), 1 + \mathsf{fma}\left(x2, 6, -9\right)\right) + \left(x2 \cdot -2\right) \cdot \mathsf{fma}\left(x2, 2, -3\right), \mathsf{fma}\left(\mathsf{fma}\left(x2, 2, -3\right), x2 \cdot 4, -3\right)\right), \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + -6\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(x2, 2, -3\right), x2 \cdot 4, -1\right)\right), x2 \cdot -6\right)} \]

   12. Taylor expanded in x2 around 0

       \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -19 \cdot x1\right) - 1\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 85.2%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -19, 9\right), -1\right)} \]

   if -3.80000000000000004e40 < x1

   1. Initial program 83.9%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), x2 \cdot 14\right) + -6\right), -1\right)\right)\right)} \]

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 60.4%

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

Alternative 19: 26.0% 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 70.7%

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

3. Taylor expanded in x1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.1

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

5. Applied rewrites 25.1%

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

6. Taylor expanded in x1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.1

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

8. Applied rewrites 25.1%

   \[\leadsto \color{blue}{x2 \cdot -6} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(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))))))