Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 69.8% → 99.6%

Time: 8.8s

Alternatives: 21

Speedup: 6.0×

• 46.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_0)
	t_3 = Float64(Float64(fma(Float64(3.0 * x1), x1, Float64(x2 + x2)) - x1) / fma(x1, x1, 1.0))
	tmp = 0.0
	if (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_0) + Float64(t_1 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0)))) <= Inf)
		tmp = fma(Float64(fma(-2.0, x2, t_1) - x1), Float64(Float64(1.0 / fma(x1, x1, 1.0)) * 3.0), Float64(fma(fma(fma(t_3, 4.0, -6.0), Float64(x1 * x1), Float64(Float64(t_3 - 3.0) * Float64(t_3 * Float64(x1 + x1)))), fma(x1, x1, 1.0), fma(Float64(t_3 * Float64(3.0 * x1)), x1, Float64(fma(x1, x1, 1.0) * x1))) + x1));
	else
		tmp = Float64((x1 ^ 4.0) * 6.0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1 + N[(x2 + x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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$0), $MachinePrecision] + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(-2.0 * x2 + t$95$1), $MachinePrecision] - x1), $MachinePrecision] * N[(N[(1.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[(N[(t$95$3 * 4.0 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$3 - 3.0), $MachinePrecision] * N[(t$95$3 * N[(x1 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(t$95$3 * N[(3.0 * x1), $MachinePrecision]), $MachinePrecision] * x1 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $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 69.8%

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

   3. Applied rewrites 70.0%

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

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

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

      \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

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))
        (t_3 (/ (- (fma (* 3.0 x1) x1 (+ x2 x2)) x1) (fma x1 x1 1.0))))
   (if (<=
        (+
         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))))
        INFINITY)
     (+
      x1
      (fma
       (* x1 x1)
       x1
       (+
        (fma
         (fma (fma t_3 4.0 -6.0) (* x1 x1) (* (- t_3 3.0) (* t_3 (+ x1 x1))))
         (fma x1 x1 1.0)
         (* t_3 t_0))
        (fma (/ (- (fma -2.0 x2 t_0) x1) (fma x1 x1 1.0)) 3.0 x1))))
     (* (pow x1 4.0) 6.0))))

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

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)
	t_3 = Float64(Float64(fma(Float64(3.0 * x1), x1, Float64(x2 + x2)) - x1) / fma(x1, x1, 1.0))
	tmp = 0.0
	if (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)))) <= Inf)
		tmp = Float64(x1 + fma(Float64(x1 * x1), x1, Float64(fma(fma(fma(t_3, 4.0, -6.0), Float64(x1 * x1), Float64(Float64(t_3 - 3.0) * Float64(t_3 * Float64(x1 + x1)))), fma(x1, x1, 1.0), Float64(t_3 * t_0)) + fma(Float64(Float64(fma(-2.0, x2, t_0) - x1) / fma(x1, x1, 1.0)), 3.0, x1))));
	else
		tmp = Float64((x1 ^ 4.0) * 6.0);
	end
	return tmp
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]}, Block[{t$95$3 = N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1 + N[(x2 + x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(t$95$3 * 4.0 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$3 - 3.0), $MachinePrecision] * N[(t$95$3 * N[(x1 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $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 69.8%

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

   3. Applied rewrites 74.6%

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

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

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

      \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

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))
        (t_3 (/ (- (fma (* 3.0 x1) x1 (+ x2 x2)) x1) (fma x1 x1 1.0))))
   (if (<=
        (+
         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))))
        INFINITY)
     (+
      (fma
       (fma (fma t_3 4.0 -6.0) (* x1 x1) (* (- t_3 3.0) (* t_3 (+ x1 x1))))
       (fma x1 x1 1.0)
       (fma (* t_3 (* 3.0 x1)) x1 (* (fma x1 x1 1.0) x1)))
      (fma (/ (- (fma -2.0 x2 t_0) x1) (fma x1 x1 1.0)) 3.0 x1))
     (* (pow x1 4.0) 6.0))))

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

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)
	t_3 = Float64(Float64(fma(Float64(3.0 * x1), x1, Float64(x2 + x2)) - x1) / fma(x1, x1, 1.0))
	tmp = 0.0
	if (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)))) <= Inf)
		tmp = Float64(fma(fma(fma(t_3, 4.0, -6.0), Float64(x1 * x1), Float64(Float64(t_3 - 3.0) * Float64(t_3 * Float64(x1 + x1)))), fma(x1, x1, 1.0), fma(Float64(t_3 * Float64(3.0 * x1)), x1, Float64(fma(x1, x1, 1.0) * x1))) + fma(Float64(Float64(fma(-2.0, x2, t_0) - x1) / fma(x1, x1, 1.0)), 3.0, x1));
	else
		tmp = Float64((x1 ^ 4.0) * 6.0);
	end
	return tmp
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]}, Block[{t$95$3 = N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1 + N[(x2 + x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(N[(N[(t$95$3 * 4.0 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$3 - 3.0), $MachinePrecision] * N[(t$95$3 * N[(x1 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(t$95$3 * N[(3.0 * x1), $MachinePrecision]), $MachinePrecision] * x1 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $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 69.8%

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

   3. Applied rewrites 70.0%

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

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

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

      \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

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

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))
        (t_3 (- (fma (* 3.0 x1) x1 (+ x2 x2)) x1))
        (t_4 (/ t_3 (fma x1 x1 1.0))))
   (if (<=
        (+
         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))))
        INFINITY)
     (fma
      (fma x1 x1 1.0)
      x1
      (fma
       (* t_4 3.0)
       (* x1 x1)
       (fma
        (fma
         (fma t_3 (/ 4.0 (fma x1 x1 1.0)) -6.0)
         (* x1 x1)
         (* (* t_3 (/ (+ x1 x1) (fma x1 x1 1.0))) (- t_4 3.0)))
        (fma x1 x1 1.0)
        (fma (- (fma -2.0 x2 t_0) x1) (/ 3.0 (fma x1 x1 1.0)) x1))))
     (* (pow x1 4.0) 6.0))))

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

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)
	t_3 = Float64(fma(Float64(3.0 * x1), x1, Float64(x2 + x2)) - x1)
	t_4 = Float64(t_3 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (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)))) <= Inf)
		tmp = fma(fma(x1, x1, 1.0), x1, fma(Float64(t_4 * 3.0), Float64(x1 * x1), fma(fma(fma(t_3, Float64(4.0 / fma(x1, x1, 1.0)), -6.0), Float64(x1 * x1), Float64(Float64(t_3 * Float64(Float64(x1 + x1) / fma(x1, x1, 1.0))) * Float64(t_4 - 3.0))), fma(x1, x1, 1.0), fma(Float64(fma(-2.0, x2, t_0) - x1), Float64(3.0 / fma(x1, x1, 1.0)), x1))));
	else
		tmp = Float64((x1 ^ 4.0) * 6.0);
	end
	return tmp
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]}, Block[{t$95$3 = N[(N[(N[(3.0 * x1), $MachinePrecision] * x1 + N[(x2 + x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(t$95$4 * 3.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(N[(t$95$3 * N[(4.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$3 * N[(N[(x1 + x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] * N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $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 69.8%

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

   3. Applied rewrites 74.6%

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

   5. Applied rewrites 74.6%

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

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

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

      \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ t_4 := \mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1\\ t_5 := \frac{t\_4}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;t\_3 \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \mathsf{fma}\left(t\_5 \cdot 3, x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_4, 4, -6\right), x1 \cdot x1, \left(t\_4 \cdot \frac{x1 + x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(t\_5 - 3\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, t\_0\right) - x1, \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right)\right)\right)\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_5, 4, -6\right), x1 \cdot x1, \mathsf{fma}\left(t\_4, \frac{1}{\mathsf{fma}\left(x1, x1, 1\right)}, -3\right) \cdot \left(t\_5 \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(x1, x1, 1\right), t\_5 \cdot t\_0\right) + \mathsf{fma}\left(-2 \cdot x2, 3, x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot 6\\ \end{array} \end{array} \]

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))
        (t_3
         (+
          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)))))
        (t_4 (- (fma (* 3.0 x1) x1 (+ x2 x2)) x1))
        (t_5 (/ t_4 (fma x1 x1 1.0))))
   (if (<= t_3 0.4)
     (fma
      (fma x1 x1 1.0)
      x1
      (fma
       (* t_5 3.0)
       (* x1 x1)
       (fma
        (fma
         (fma t_4 4.0 -6.0)
         (* x1 x1)
         (* (* t_4 (/ (+ x1 x1) (fma x1 x1 1.0))) (- t_5 3.0)))
        (fma x1 x1 1.0)
        (fma (- (fma -2.0 x2 t_0) x1) (/ 3.0 (fma x1 x1 1.0)) x1))))
     (if (<= t_3 INFINITY)
       (+
        x1
        (fma
         (* x1 x1)
         x1
         (+
          (fma
           (fma
            (fma t_5 4.0 -6.0)
            (* x1 x1)
            (* (fma t_4 (/ 1.0 (fma x1 x1 1.0)) -3.0) (* t_5 (+ x1 x1))))
           (fma x1 x1 1.0)
           (* t_5 t_0))
          (fma (* -2.0 x2) 3.0 x1))))
       (* (pow x1 4.0) 6.0)))))

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;
	double t_3 = 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)));
	double t_4 = fma((3.0 * x1), x1, (x2 + x2)) - x1;
	double t_5 = t_4 / fma(x1, x1, 1.0);
	double tmp;
	if (t_3 <= 0.4) {
		tmp = fma(fma(x1, x1, 1.0), x1, fma((t_5 * 3.0), (x1 * x1), fma(fma(fma(t_4, 4.0, -6.0), (x1 * x1), ((t_4 * ((x1 + x1) / fma(x1, x1, 1.0))) * (t_5 - 3.0))), fma(x1, x1, 1.0), fma((fma(-2.0, x2, t_0) - x1), (3.0 / fma(x1, x1, 1.0)), x1))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = x1 + fma((x1 * x1), x1, (fma(fma(fma(t_5, 4.0, -6.0), (x1 * x1), (fma(t_4, (1.0 / fma(x1, x1, 1.0)), -3.0) * (t_5 * (x1 + x1)))), fma(x1, x1, 1.0), (t_5 * t_0)) + fma((-2.0 * x2), 3.0, x1)));
	} else {
		tmp = pow(x1, 4.0) * 6.0;
	}
	return tmp;
}

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)
	t_3 = 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))))
	t_4 = Float64(fma(Float64(3.0 * x1), x1, Float64(x2 + x2)) - x1)
	t_5 = Float64(t_4 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (t_3 <= 0.4)
		tmp = fma(fma(x1, x1, 1.0), x1, fma(Float64(t_5 * 3.0), Float64(x1 * x1), fma(fma(fma(t_4, 4.0, -6.0), Float64(x1 * x1), Float64(Float64(t_4 * Float64(Float64(x1 + x1) / fma(x1, x1, 1.0))) * Float64(t_5 - 3.0))), fma(x1, x1, 1.0), fma(Float64(fma(-2.0, x2, t_0) - x1), Float64(3.0 / fma(x1, x1, 1.0)), x1))));
	elseif (t_3 <= Inf)
		tmp = Float64(x1 + fma(Float64(x1 * x1), x1, Float64(fma(fma(fma(t_5, 4.0, -6.0), Float64(x1 * x1), Float64(fma(t_4, Float64(1.0 / fma(x1, x1, 1.0)), -3.0) * Float64(t_5 * Float64(x1 + x1)))), fma(x1, x1, 1.0), Float64(t_5 * t_0)) + fma(Float64(-2.0 * x2), 3.0, x1))));
	else
		tmp = Float64((x1 ^ 4.0) * 6.0);
	end
	return tmp
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]}, Block[{t$95$3 = 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]}, Block[{t$95$4 = N[(N[(N[(3.0 * x1), $MachinePrecision] * x1 + N[(x2 + x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.4], N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(t$95$5 * 3.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(N[(t$95$4 * 4.0 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$4 * N[(N[(x1 + x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$5 - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] * N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(t$95$5 * 4.0 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$4 * N[(1.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision] * N[(t$95$5 * N[(x1 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(t$95$5 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * x2), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $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)))))) < 0.40000000000000002

   1. Initial program 69.8%

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

   3. Applied rewrites 74.6%

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 65.9%

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

   if 0.40000000000000002 < (+.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 69.8%

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

   3. Applied rewrites 74.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x1 around 0

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

      1. lower-*.f64 61.9

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

   8. Applied rewrites 61.9%

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

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

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

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

Alternative 6: 98.4% accurate, 0.5× speedup

Math

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

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))
        (t_3 (- (fma (* 3.0 x1) x1 (+ x2 x2)) x1)))
   (if (<=
        (+
         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))))
        INFINITY)
     (+
      (fma
       (fma
        (fma t_3 (/ 4.0 (fma x1 x1 1.0)) -6.0)
        (* x1 x1)
        (*
         (* t_3 (/ (+ x1 x1) (fma x1 x1 1.0)))
         (- (/ t_3 (fma x1 x1 1.0)) 3.0)))
       (fma x1 x1 1.0)
       (fma
        x1
        (fma (/ (* t_3 x1) (fma x1 x1 1.0)) 3.0 (* x1 x1))
        (fma (- (fma -2.0 x2 t_0) x1) (/ 3.0 (fma x1 x1 1.0)) x1)))
      x1)
     (* (pow x1 4.0) 6.0))))

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

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)
	t_3 = Float64(fma(Float64(3.0 * x1), x1, Float64(x2 + x2)) - x1)
	tmp = 0.0
	if (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)))) <= Inf)
		tmp = Float64(fma(fma(fma(t_3, Float64(4.0 / fma(x1, x1, 1.0)), -6.0), Float64(x1 * x1), Float64(Float64(t_3 * Float64(Float64(x1 + x1) / fma(x1, x1, 1.0))) * Float64(Float64(t_3 / fma(x1, x1, 1.0)) - 3.0))), fma(x1, x1, 1.0), fma(x1, fma(Float64(Float64(t_3 * x1) / fma(x1, x1, 1.0)), 3.0, Float64(x1 * x1)), fma(Float64(fma(-2.0, x2, t_0) - x1), Float64(3.0 / fma(x1, x1, 1.0)), x1))) + x1);
	else
		tmp = Float64((x1 ^ 4.0) * 6.0);
	end
	return tmp
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]}, Block[{t$95$3 = N[(N[(N[(3.0 * x1), $MachinePrecision] * x1 + N[(x2 + x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(N[(N[(t$95$3 * N[(4.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$3 * N[(N[(x1 + x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(x1 * N[(N[(N[(t$95$3 * x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] * N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $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 69.8%

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

   3. Applied rewrites 74.6%

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

   5. Applied rewrites 74.0%

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

   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 69.8%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

      \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_2 + t\_1 \cdot t\_3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ t_5 := \frac{t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_6 := \mathsf{fma}\left(-1, x1, 2 \cdot x2\right)\\ \mathbf{if}\;t\_4 \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \mathsf{fma}\left(t\_6 \cdot 3, x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \frac{4}{\mathsf{fma}\left(x1, x1, 1\right)}, -6\right), x1 \cdot x1, \left(t\_0 \cdot \frac{x1 + x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(t\_6 - 3\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, t\_1\right) - x1, \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, x1\right)\right)\right)\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_5, 4, -6\right), x1 \cdot x1, \mathsf{fma}\left(t\_0, \frac{1}{\mathsf{fma}\left(x1, x1, 1\right)}, -3\right) \cdot \left(t\_5 \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(x1, x1, 1\right), t\_5 \cdot t\_1\right) + \mathsf{fma}\left(-2 \cdot x2, 3, x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (fma (* 3.0 x1) x1 (+ x2 x2)) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_1 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* 4.0 t_3) 6.0)))
               t_2)
              (* t_1 t_3))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_2)))))
        (t_5 (/ t_0 (fma x1 x1 1.0)))
        (t_6 (fma -1.0 x1 (* 2.0 x2))))
   (if (<= t_4 0.4)
     (fma
      (fma x1 x1 1.0)
      x1
      (fma
       (* t_6 3.0)
       (* x1 x1)
       (fma
        (fma
         (fma t_0 (/ 4.0 (fma x1 x1 1.0)) -6.0)
         (* x1 x1)
         (* (* t_0 (/ (+ x1 x1) (fma x1 x1 1.0))) (- t_6 3.0)))
        (fma x1 x1 1.0)
        (fma (- (fma -2.0 x2 t_1) x1) (/ 3.0 (fma x1 x1 1.0)) x1))))
     (if (<= t_4 INFINITY)
       (+
        x1
        (fma
         (* x1 x1)
         x1
         (+
          (fma
           (fma
            (fma t_5 4.0 -6.0)
            (* x1 x1)
            (* (fma t_0 (/ 1.0 (fma x1 x1 1.0)) -3.0) (* t_5 (+ x1 x1))))
           (fma x1 x1 1.0)
           (* t_5 t_1))
          (fma (* -2.0 x2) 3.0 x1))))
       (* (pow x1 4.0) 6.0)))))

C

double code(double x1, double x2) {
	double t_0 = fma((3.0 * x1), x1, (x2 + x2)) - x1;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_1 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + (((((((((2.0 * x1) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_2) + (t_1 * t_3)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)));
	double t_5 = t_0 / fma(x1, x1, 1.0);
	double t_6 = fma(-1.0, x1, (2.0 * x2));
	double tmp;
	if (t_4 <= 0.4) {
		tmp = fma(fma(x1, x1, 1.0), x1, fma((t_6 * 3.0), (x1 * x1), fma(fma(fma(t_0, (4.0 / fma(x1, x1, 1.0)), -6.0), (x1 * x1), ((t_0 * ((x1 + x1) / fma(x1, x1, 1.0))) * (t_6 - 3.0))), fma(x1, x1, 1.0), fma((fma(-2.0, x2, t_1) - x1), (3.0 / fma(x1, x1, 1.0)), x1))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = x1 + fma((x1 * x1), x1, (fma(fma(fma(t_5, 4.0, -6.0), (x1 * x1), (fma(t_0, (1.0 / fma(x1, x1, 1.0)), -3.0) * (t_5 * (x1 + x1)))), fma(x1, x1, 1.0), (t_5 * t_1)) + fma((-2.0 * x2), 3.0, x1)));
	} else {
		tmp = pow(x1, 4.0) * 6.0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(Float64(3.0 * x1), x1, Float64(x2 + x2)) - x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	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(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))) * t_2) + Float64(t_1 * t_3)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_2))))
	t_5 = Float64(t_0 / fma(x1, x1, 1.0))
	t_6 = fma(-1.0, x1, Float64(2.0 * x2))
	tmp = 0.0
	if (t_4 <= 0.4)
		tmp = fma(fma(x1, x1, 1.0), x1, fma(Float64(t_6 * 3.0), Float64(x1 * x1), fma(fma(fma(t_0, Float64(4.0 / fma(x1, x1, 1.0)), -6.0), Float64(x1 * x1), Float64(Float64(t_0 * Float64(Float64(x1 + x1) / fma(x1, x1, 1.0))) * Float64(t_6 - 3.0))), fma(x1, x1, 1.0), fma(Float64(fma(-2.0, x2, t_1) - x1), Float64(3.0 / fma(x1, x1, 1.0)), x1))));
	elseif (t_4 <= Inf)
		tmp = Float64(x1 + fma(Float64(x1 * x1), x1, Float64(fma(fma(fma(t_5, 4.0, -6.0), Float64(x1 * x1), Float64(fma(t_0, Float64(1.0 / fma(x1, x1, 1.0)), -3.0) * Float64(t_5 * Float64(x1 + x1)))), fma(x1, x1, 1.0), Float64(t_5 * t_1)) + fma(Float64(-2.0 * x2), 3.0, x1))));
	else
		tmp = Float64((x1 ^ 4.0) * 6.0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(3.0 * x1), $MachinePrecision] * x1 + N[(x2 + x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $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[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $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]}, Block[{t$95$5 = N[(t$95$0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(-1.0 * x1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.4], N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(t$95$6 * 3.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(N[(t$95$0 * N[(4.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$0 * N[(N[(x1 + x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$6 - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(N[(-2.0 * x2 + t$95$1), $MachinePrecision] - x1), $MachinePrecision] * N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(t$95$5 * 4.0 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$0 * N[(1.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision] * N[(t$95$5 * N[(x1 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(t$95$5 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * x2), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $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)))))) < 0.40000000000000002

   1. Initial program 69.8%

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

   3. Applied rewrites 74.6%

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 73.2

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in x1 around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-*.f64 58.9

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

   11. Applied rewrites 58.9%

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

   if 0.40000000000000002 < (+.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 69.8%

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

   3. Applied rewrites 74.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x1 around 0

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

      1. lower-*.f64 61.9

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

   8. Applied rewrites 61.9%

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

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

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

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

Alternative 8: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-1, x1, 2 \cdot x2\right)\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ \mathbf{if}\;x1 \leq -3.6 \cdot 10^{+19}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)\\ \mathbf{elif}\;x1 \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 4, -6\right), x1 \cdot x1, \left(t\_0 - 3\right) \cdot \left(t\_0 \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(x1, x1, 1\right), t\_0 \cdot t\_1\right) + \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, t\_1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\left(\left(-\frac{\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, -\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 3, 1\right), -2, 2\right)}{x1}\right) + 9}{x1}\right) + 3}{x1}\right) + 6\right) \cdot {x1}^{4}}{x1}\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma -1.0 x1 (* 2.0 x2))) (t_1 (* (* 3.0 x1) x1)))
   (if (<= x1 -3.6e+19)
     (*
      (pow x1 4.0)
      (+
       6.0
       (*
        -1.0
        (/ (+ 3.0 (* -1.0 (/ (+ 9.0 (* 4.0 (- (* 2.0 x2) 3.0))) x1))) x1))))
     (if (<= x1 8.5e+18)
       (+
        x1
        (fma
         (* x1 x1)
         x1
         (+
          (fma
           (fma (fma t_0 4.0 -6.0) (* x1 x1) (* (- t_0 3.0) (* t_0 (+ x1 x1))))
           (fma x1 x1 1.0)
           (* t_0 t_1))
          (fma (/ (- (fma -2.0 x2 t_1) x1) (fma x1 x1 1.0)) 3.0 x1))))
       (*
        (+
         1.0
         (/
          (*
           (+
            (-
             (/
              (+
               (-
                (/
                 (+
                  (fma
                   (fma 2.0 x2 -3.0)
                   4.0
                   (- (/ (fma (fma (fma 2.0 x2 -3.0) 3.0 1.0) -2.0 2.0) x1)))
                  9.0)
                 x1))
               3.0)
              x1))
            6.0)
           (pow x1 4.0))
          x1))
        x1)))))

C

double code(double x1, double x2) {
	double t_0 = fma(-1.0, x1, (2.0 * x2));
	double t_1 = (3.0 * x1) * x1;
	double tmp;
	if (x1 <= -3.6e+19) {
		tmp = pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * ((9.0 + (4.0 * ((2.0 * x2) - 3.0))) / x1))) / x1)));
	} else if (x1 <= 8.5e+18) {
		tmp = x1 + fma((x1 * x1), x1, (fma(fma(fma(t_0, 4.0, -6.0), (x1 * x1), ((t_0 - 3.0) * (t_0 * (x1 + x1)))), fma(x1, x1, 1.0), (t_0 * t_1)) + fma(((fma(-2.0, x2, t_1) - x1) / fma(x1, x1, 1.0)), 3.0, x1)));
	} else {
		tmp = (1.0 + (((-((-((fma(fma(2.0, x2, -3.0), 4.0, -(fma(fma(fma(2.0, x2, -3.0), 3.0, 1.0), -2.0, 2.0) / x1)) + 9.0) / x1) + 3.0) / x1) + 6.0) * pow(x1, 4.0)) / x1)) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(-1.0, x1, Float64(2.0 * x2))
	t_1 = Float64(Float64(3.0 * x1) * x1)
	tmp = 0.0
	if (x1 <= -3.6e+19)
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 + Float64(-1.0 * Float64(Float64(3.0 + Float64(-1.0 * Float64(Float64(9.0 + Float64(4.0 * Float64(Float64(2.0 * x2) - 3.0))) / x1))) / x1))));
	elseif (x1 <= 8.5e+18)
		tmp = Float64(x1 + fma(Float64(x1 * x1), x1, Float64(fma(fma(fma(t_0, 4.0, -6.0), Float64(x1 * x1), Float64(Float64(t_0 - 3.0) * Float64(t_0 * Float64(x1 + x1)))), fma(x1, x1, 1.0), Float64(t_0 * t_1)) + fma(Float64(Float64(fma(-2.0, x2, t_1) - x1) / fma(x1, x1, 1.0)), 3.0, x1))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(fma(fma(2.0, x2, -3.0), 4.0, Float64(-Float64(fma(fma(fma(2.0, x2, -3.0), 3.0, 1.0), -2.0, 2.0) / x1))) + 9.0) / x1)) + 3.0) / x1)) + 6.0) * (x1 ^ 4.0)) / x1)) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(-1.0 * x1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, If[LessEqual[x1, -3.6e+19], N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(-1.0 * N[(N[(3.0 + N[(-1.0 * N[(N[(9.0 + N[(4.0 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8.5e+18], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(t$95$0 * 4.0 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$0 - 3.0), $MachinePrecision] * N[(t$95$0 * N[(x1 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-2.0 * x2 + t$95$1), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[((-N[(N[((-N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + (-N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 3.0 + 1.0), $MachinePrecision] * -2.0 + 2.0), $MachinePrecision] / x1), $MachinePrecision])), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision]) + 3.0), $MachinePrecision] / x1), $MachinePrecision]) + 6.0), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.6e19

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

   4. Applied rewrites 47.9%

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

   if -3.6e19 < x1 < 8.5e18

   1. Initial program 69.8%

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

   3. Applied rewrites 74.6%

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

   4. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 60.4

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

   6. Applied rewrites 60.4%

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

   7. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 59.3

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

   9. Applied rewrites 59.3%

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

   10. Taylor expanded in x1 around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-*.f64 58.6

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

   12. Applied rewrites 58.6%

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

   13. Taylor expanded in x1 around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-*.f64 54.4

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

   15. Applied rewrites 54.4%

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

   if 8.5e18 < x1

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 48.5%

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

   6. Applied rewrites 48.5%

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

Alternative 9: 94.9% accurate, 0.6× speedup

Math

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

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))
        (t_3 (- (fma (* 3.0 x1) x1 (+ x2 x2)) x1)))
   (if (<=
        (+
         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))))
        INFINITY)
     (fma
      (fma x1 x1 1.0)
      x1
      (fma
       (* (* 2.0 x2) 3.0)
       (* x1 x1)
       (fma
        (fma
         (fma t_3 (/ 4.0 (fma x1 x1 1.0)) -6.0)
         (* x1 x1)
         (* (* t_3 (/ (+ x1 x1) (fma x1 x1 1.0))) (- (* 2.0 x2) 3.0)))
        (fma x1 x1 1.0)
        (fma (- (fma -2.0 x2 t_0) x1) (/ 3.0 (fma x1 x1 1.0)) x1))))
     (* (pow x1 4.0) 6.0))))

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

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)
	t_3 = Float64(fma(Float64(3.0 * x1), x1, Float64(x2 + x2)) - x1)
	tmp = 0.0
	if (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)))) <= Inf)
		tmp = fma(fma(x1, x1, 1.0), x1, fma(Float64(Float64(2.0 * x2) * 3.0), Float64(x1 * x1), fma(fma(fma(t_3, Float64(4.0 / fma(x1, x1, 1.0)), -6.0), Float64(x1 * x1), Float64(Float64(t_3 * Float64(Float64(x1 + x1) / fma(x1, x1, 1.0))) * Float64(Float64(2.0 * x2) - 3.0))), fma(x1, x1, 1.0), fma(Float64(fma(-2.0, x2, t_0) - x1), Float64(3.0 / fma(x1, x1, 1.0)), x1))));
	else
		tmp = Float64((x1 ^ 4.0) * 6.0);
	end
	return tmp
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]}, Block[{t$95$3 = N[(N[(N[(3.0 * x1), $MachinePrecision] * x1 + N[(x2 + x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(N[(2.0 * x2), $MachinePrecision] * 3.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(N[(t$95$3 * N[(4.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$3 * N[(N[(x1 + x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] * N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $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 69.8%

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

   3. Applied rewrites 74.6%

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x1 around 0

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

      1. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in x1 around 0

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

       1. lower-*.f64 69.1

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

   11. Applied rewrites 69.1%

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

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

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

      \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 94.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.1 \cdot 10^{+24}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)\\ \mathbf{elif}\;x1 \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;x1 + \left(\left(\left(\left(4 \cdot x1\right) \cdot \mathsf{fma}\left(2, x2, -3\right)\right) \cdot x2 + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\left(\left(-\frac{\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, -\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 3, 1\right), -2, 2\right)}{x1}\right) + 9}{x1}\right) + 3}{x1}\right) + 6\right) \cdot {x1}^{4}}{x1}\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.1e+24)
   (*
    (pow x1 4.0)
    (+
     6.0
     (*
      -1.0
      (/ (+ 3.0 (* -1.0 (/ (+ 9.0 (* 4.0 (- (* 2.0 x2) 3.0))) x1))) x1))))
   (if (<= x1 8.5e+18)
     (+
      x1
      (+
       (+ (* (* (* 4.0 x1) (fma 2.0 x2 -3.0)) x2) x1)
       (* 3.0 (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))))
     (*
      (+
       1.0
       (/
        (*
         (+
          (-
           (/
            (+
             (-
              (/
               (+
                (fma
                 (fma 2.0 x2 -3.0)
                 4.0
                 (- (/ (fma (fma (fma 2.0 x2 -3.0) 3.0 1.0) -2.0 2.0) x1)))
                9.0)
               x1))
             3.0)
            x1))
          6.0)
         (pow x1 4.0))
        x1))
      x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.1e+24) {
		tmp = pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * ((9.0 + (4.0 * ((2.0 * x2) - 3.0))) / x1))) / x1)));
	} else if (x1 <= 8.5e+18) {
		tmp = x1 + (((((4.0 * x1) * fma(2.0, x2, -3.0)) * x2) + x1) + (3.0 * (((((3.0 * x1) * x1) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))));
	} else {
		tmp = (1.0 + (((-((-((fma(fma(2.0, x2, -3.0), 4.0, -(fma(fma(fma(2.0, x2, -3.0), 3.0, 1.0), -2.0, 2.0) / x1)) + 9.0) / x1) + 3.0) / x1) + 6.0) * pow(x1, 4.0)) / x1)) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.1e+24)
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 + Float64(-1.0 * Float64(Float64(3.0 + Float64(-1.0 * Float64(Float64(9.0 + Float64(4.0 * Float64(Float64(2.0 * x2) - 3.0))) / x1))) / x1))));
	elseif (x1 <= 8.5e+18)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(4.0 * x1) * fma(2.0, x2, -3.0)) * x2) + x1) + Float64(3.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0)))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(fma(fma(2.0, x2, -3.0), 4.0, Float64(-Float64(fma(fma(fma(2.0, x2, -3.0), 3.0, 1.0), -2.0, 2.0) / x1))) + 9.0) / x1)) + 3.0) / x1)) + 6.0) * (x1 ^ 4.0)) / x1)) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.1e+24], N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(-1.0 * N[(N[(3.0 + N[(-1.0 * N[(N[(9.0 + N[(4.0 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8.5e+18], N[(x1 + N[(N[(N[(N[(N[(4.0 * x1), $MachinePrecision] * N[(2.0 * x2 + -3.0), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[((-N[(N[((-N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + (-N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 3.0 + 1.0), $MachinePrecision] * -2.0 + 2.0), $MachinePrecision] / x1), $MachinePrecision])), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision]) + 3.0), $MachinePrecision] / x1), $MachinePrecision]) + 6.0), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.10000000000000001e24

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

   4. Applied rewrites 47.9%

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

   if -1.10000000000000001e24 < x1 < 8.5e18

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 48.9

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

   4. Applied rewrites 48.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 54.7

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 54.7

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

   6. Applied rewrites 54.7%

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

   if 8.5e18 < x1

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 48.5%

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

   6. Applied rewrites 48.5%

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

Alternative 11: 94.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)\\ \mathbf{if}\;x1 \leq -1.1 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;x1 + \left(\left(\left(\left(4 \cdot x1\right) \cdot \mathsf{fma}\left(2, x2, -3\right)\right) \cdot x2 + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (pow x1 4.0)
          (+
           6.0
           (*
            -1.0
            (/
             (+ 3.0 (* -1.0 (/ (+ 9.0 (* 4.0 (- (* 2.0 x2) 3.0))) x1)))
             x1))))))
   (if (<= x1 -1.1e+24)
     t_0
     (if (<= x1 8.5e+18)
       (+
        x1
        (+
         (+ (* (* (* 4.0 x1) (fma 2.0 x2 -3.0)) x2) x1)
         (*
          3.0
          (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * ((9.0 + (4.0 * ((2.0 * x2) - 3.0))) / x1))) / x1)));
	double tmp;
	if (x1 <= -1.1e+24) {
		tmp = t_0;
	} else if (x1 <= 8.5e+18) {
		tmp = x1 + (((((4.0 * x1) * fma(2.0, x2, -3.0)) * x2) + x1) + (3.0 * (((((3.0 * x1) * x1) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64((x1 ^ 4.0) * Float64(6.0 + Float64(-1.0 * Float64(Float64(3.0 + Float64(-1.0 * Float64(Float64(9.0 + Float64(4.0 * Float64(Float64(2.0 * x2) - 3.0))) / x1))) / x1))))
	tmp = 0.0
	if (x1 <= -1.1e+24)
		tmp = t_0;
	elseif (x1 <= 8.5e+18)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(4.0 * x1) * fma(2.0, x2, -3.0)) * x2) + x1) + Float64(3.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0)))));
	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[(-1.0 * N[(N[(3.0 + N[(-1.0 * N[(N[(9.0 + N[(4.0 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.1e+24], t$95$0, If[LessEqual[x1, 8.5e+18], N[(x1 + N[(N[(N[(N[(N[(4.0 * x1), $MachinePrecision] * N[(2.0 * x2 + -3.0), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.10000000000000001e24 or 8.5e18 < x1

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

   4. Applied rewrites 47.9%

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

   if -1.10000000000000001e24 < x1 < 8.5e18

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 48.9

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

   4. Applied rewrites 48.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 54.7

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 54.7

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

   6. Applied rewrites 54.7%

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

Alternative 12: 92.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot 6\\ \mathbf{if}\;x1 \leq -3.7 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;x1 + \left(\left(\left(\left(4 \cdot x1\right) \cdot \mathsf{fma}\left(2, x2, -3\right)\right) \cdot x2 + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (pow x1 4.0) 6.0)))
   (if (<= x1 -3.7e+27)
     t_0
     (if (<= x1 8.5e+18)
       (+
        x1
        (+
         (+ (* (* (* 4.0 x1) (fma 2.0 x2 -3.0)) x2) x1)
         (*
          3.0
          (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = pow(x1, 4.0) * 6.0;
	double tmp;
	if (x1 <= -3.7e+27) {
		tmp = t_0;
	} else if (x1 <= 8.5e+18) {
		tmp = x1 + (((((4.0 * x1) * fma(2.0, x2, -3.0)) * x2) + x1) + (3.0 * (((((3.0 * x1) * x1) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64((x1 ^ 4.0) * 6.0)
	tmp = 0.0
	if (x1 <= -3.7e+27)
		tmp = t_0;
	elseif (x1 <= 8.5e+18)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(4.0 * x1) * fma(2.0, x2, -3.0)) * x2) + x1) + Float64(3.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[x1, -3.7e+27], t$95$0, If[LessEqual[x1, 8.5e+18], N[(x1 + N[(N[(N[(N[(N[(4.0 * x1), $MachinePrecision] * N[(2.0 * x2 + -3.0), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.70000000000000002e27 or 8.5e18 < x1

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

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

   if -3.70000000000000002e27 < x1 < 8.5e18

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 48.9

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

   4. Applied rewrites 48.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 54.7

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 54.7

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

   6. Applied rewrites 54.7%

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

Alternative 13: 92.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot 6\\ \mathbf{if}\;x1 \leq -3.7 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;x1 + \left(\left(\left(\left(4 \cdot x1\right) \cdot \mathsf{fma}\left(2, x2, -3\right)\right) \cdot x2 + x1\right) + \mathsf{fma}\left(-6, x2, -3 \cdot x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (pow x1 4.0) 6.0)))
   (if (<= x1 -3.7e+27)
     t_0
     (if (<= x1 8.5e+18)
       (+
        x1
        (+
         (+ (* (* (* 4.0 x1) (fma 2.0 x2 -3.0)) x2) x1)
         (fma -6.0 x2 (* -3.0 x1))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = pow(x1, 4.0) * 6.0;
	double tmp;
	if (x1 <= -3.7e+27) {
		tmp = t_0;
	} else if (x1 <= 8.5e+18) {
		tmp = x1 + (((((4.0 * x1) * fma(2.0, x2, -3.0)) * x2) + x1) + fma(-6.0, x2, (-3.0 * x1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64((x1 ^ 4.0) * 6.0)
	tmp = 0.0
	if (x1 <= -3.7e+27)
		tmp = t_0;
	elseif (x1 <= 8.5e+18)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(4.0 * x1) * fma(2.0, x2, -3.0)) * x2) + x1) + fma(-6.0, x2, Float64(-3.0 * x1))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[x1, -3.7e+27], t$95$0, If[LessEqual[x1, 8.5e+18], N[(x1 + N[(N[(N[(N[(N[(4.0 * x1), $MachinePrecision] * N[(2.0 * x2 + -3.0), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2 + N[(-3.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.70000000000000002e27 or 8.5e18 < x1

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

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

   if -3.70000000000000002e27 < x1 < 8.5e18

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 48.9

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

   4. Applied rewrites 48.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 54.7

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

      10. lift--.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 54.7

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

   6. Applied rewrites 54.7%

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

   7. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 60.6

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

   9. Applied rewrites 60.6%

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

Alternative 14: 87.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot 6\\ \mathbf{if}\;x1 \leq -3.7 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (pow x1 4.0) 6.0)))
   (if (<= x1 -3.7e+27)
     t_0
     (if (<= x1 8.5e+18)
       (fma -6.0 x2 (* x1 (- (* 4.0 (* x2 (- (* 2.0 x2) 3.0))) 1.0)))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = pow(x1, 4.0) * 6.0;
	double tmp;
	if (x1 <= -3.7e+27) {
		tmp = t_0;
	} else if (x1 <= 8.5e+18) {
		tmp = fma(-6.0, x2, (x1 * ((4.0 * (x2 * ((2.0 * x2) - 3.0))) - 1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64((x1 ^ 4.0) * 6.0)
	tmp = 0.0
	if (x1 <= -3.7e+27)
		tmp = t_0;
	elseif (x1 <= 8.5e+18)
		tmp = fma(-6.0, x2, Float64(x1 * Float64(Float64(4.0 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0))) - 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[x1, -3.7e+27], t$95$0, If[LessEqual[x1, 8.5e+18], N[(-6.0 * x2 + N[(x1 * N[(N[(4.0 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.70000000000000002e27 or 8.5e18 < x1

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

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

   if -3.70000000000000002e27 < x1 < 8.5e18

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 55.2%

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

Alternative 15: 70.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot 6\\ t_1 := \frac{x2 \cdot \left(x2 \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot 8\\ \mathbf{if}\;x1 \leq -7.4 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq -1.3 \cdot 10^{-163}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot -2\right)\\ \mathbf{elif}\;x1 \leq 2.8 \cdot 10^{-210}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 1.66 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (pow x1 4.0) 6.0))
        (t_1 (* (/ (* x2 (* x2 x1)) (fma x1 x1 1.0)) 8.0)))
   (if (<= x1 -7.4e+28)
     t_0
     (if (<= x1 -1.15e-24)
       t_1
       (if (<= x1 -1.3e-163)
         (+ x1 (fma (* x1 x1) x1 (* x1 -2.0)))
         (if (<= x1 2.8e-210) (* -6.0 x2) (if (<= x1 1.66e+21) t_1 t_0)))))))

C

double code(double x1, double x2) {
	double t_0 = pow(x1, 4.0) * 6.0;
	double t_1 = ((x2 * (x2 * x1)) / fma(x1, x1, 1.0)) * 8.0;
	double tmp;
	if (x1 <= -7.4e+28) {
		tmp = t_0;
	} else if (x1 <= -1.15e-24) {
		tmp = t_1;
	} else if (x1 <= -1.3e-163) {
		tmp = x1 + fma((x1 * x1), x1, (x1 * -2.0));
	} else if (x1 <= 2.8e-210) {
		tmp = -6.0 * x2;
	} else if (x1 <= 1.66e+21) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64((x1 ^ 4.0) * 6.0)
	t_1 = Float64(Float64(Float64(x2 * Float64(x2 * x1)) / fma(x1, x1, 1.0)) * 8.0)
	tmp = 0.0
	if (x1 <= -7.4e+28)
		tmp = t_0;
	elseif (x1 <= -1.15e-24)
		tmp = t_1;
	elseif (x1 <= -1.3e-163)
		tmp = Float64(x1 + fma(Float64(x1 * x1), x1, Float64(x1 * -2.0)));
	elseif (x1 <= 2.8e-210)
		tmp = Float64(-6.0 * x2);
	elseif (x1 <= 1.66e+21)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x2 * N[(x2 * x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]}, If[LessEqual[x1, -7.4e+28], t$95$0, If[LessEqual[x1, -1.15e-24], t$95$1, If[LessEqual[x1, -1.3e-163], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.8e-210], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x1, 1.66e+21], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -7.3999999999999998e28 or 1.66e21 < x1

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

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

   if -7.3999999999999998e28 < x1 < -1.1500000000000001e-24 or 2.8e-210 < x1 < 1.66e21

   1. Initial program 69.8%

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

   2. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \]

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 17.3

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

   4. Applied rewrites 17.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.3

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

      4. lift-*.f64 N/A

         \[\leadsto \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \cdot 8 \]

      5. *-commutative N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      6. lower-*.f64 17.3

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      8. unpow2 N/A

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

      9. lower-*.f64 17.3

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

      10. lift-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 17.3

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

   6. Applied rewrites 17.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 19.8

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

   8. Applied rewrites 19.8%

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

   if -1.1500000000000001e-24 < x1 < -1.30000000000000001e-163

   1. Initial program 69.8%

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

   3. Applied rewrites 74.6%

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

   4. Taylor expanded in x1 around 0

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

   6. Applied rewrites 41.6%

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

   7. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - \color{blue}{2}\right)\right) \]

      2. lower--.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      5. lower-*.f64 17.9

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

   9. Applied rewrites 17.9%

      \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)}\right) \]

   10. Taylor expanded in x1 around 0

       \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot -2\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.8%

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

   if -1.30000000000000001e-163 < x1 < 2.8e-210

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 26.3

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

   4. Applied rewrites 26.3%

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

Alternative 16: 49.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot -2\right)\\ \mathbf{if}\;x1 \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \frac{x2 \cdot x2}{\mathsf{fma}\left(x1, x1, 1\right)}\right)\\ \mathbf{elif}\;x1 \leq -1.3 \cdot 10^{-163}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.8 \cdot 10^{-210}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 1.9 \cdot 10^{+97}:\\ \;\;\;\;\frac{x2 \cdot \left(x2 \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot 8\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (fma (* x1 x1) x1 (* x1 -2.0)))))
   (if (<= x1 -1.15e-24)
     (* 8.0 (* x1 (/ (* x2 x2) (fma x1 x1 1.0))))
     (if (<= x1 -1.3e-163)
       t_0
       (if (<= x1 2.8e-210)
         (* -6.0 x2)
         (if (<= x1 1.9e+97)
           (* (/ (* x2 (* x2 x1)) (fma x1 x1 1.0)) 8.0)
           t_0))))))

C

double code(double x1, double x2) {
	double t_0 = x1 + fma((x1 * x1), x1, (x1 * -2.0));
	double tmp;
	if (x1 <= -1.15e-24) {
		tmp = 8.0 * (x1 * ((x2 * x2) / fma(x1, x1, 1.0)));
	} else if (x1 <= -1.3e-163) {
		tmp = t_0;
	} else if (x1 <= 2.8e-210) {
		tmp = -6.0 * x2;
	} else if (x1 <= 1.9e+97) {
		tmp = ((x2 * (x2 * x1)) / fma(x1, x1, 1.0)) * 8.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + fma(Float64(x1 * x1), x1, Float64(x1 * -2.0)))
	tmp = 0.0
	if (x1 <= -1.15e-24)
		tmp = Float64(8.0 * Float64(x1 * Float64(Float64(x2 * x2) / fma(x1, x1, 1.0))));
	elseif (x1 <= -1.3e-163)
		tmp = t_0;
	elseif (x1 <= 2.8e-210)
		tmp = Float64(-6.0 * x2);
	elseif (x1 <= 1.9e+97)
		tmp = Float64(Float64(Float64(x2 * Float64(x2 * x1)) / fma(x1, x1, 1.0)) * 8.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.15e-24], N[(8.0 * N[(x1 * N[(N[(x2 * x2), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.3e-163], t$95$0, If[LessEqual[x1, 2.8e-210], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x1, 1.9e+97], N[(N[(N[(x2 * N[(x2 * x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.1500000000000001e-24

   1. Initial program 69.8%

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

   2. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \]

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 17.3

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

   4. Applied rewrites 17.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lower-/.f64 17.0

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 17.0

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

   6. Applied rewrites 17.0%

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

   if -1.1500000000000001e-24 < x1 < -1.30000000000000001e-163 or 1.90000000000000018e97 < x1

   1. Initial program 69.8%

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

   3. Applied rewrites 74.6%

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

   4. Taylor expanded in x1 around 0

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

   6. Applied rewrites 41.6%

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

   7. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - \color{blue}{2}\right)\right) \]

      2. lower--.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      5. lower-*.f64 17.9

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

   9. Applied rewrites 17.9%

      \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)}\right) \]

   10. Taylor expanded in x1 around 0

       \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot -2\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.8%

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

   if -1.30000000000000001e-163 < x1 < 2.8e-210

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 26.3

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

   4. Applied rewrites 26.3%

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

   if 2.8e-210 < x1 < 1.90000000000000018e97

   1. Initial program 69.8%

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

   2. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \]

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 17.3

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

   4. Applied rewrites 17.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.3

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

      4. lift-*.f64 N/A

         \[\leadsto \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \cdot 8 \]

      5. *-commutative N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      6. lower-*.f64 17.3

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      8. unpow2 N/A

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

      9. lower-*.f64 17.3

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

      10. lift-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 17.3

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

   6. Applied rewrites 17.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 19.8

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

   8. Applied rewrites 19.8%

      \[\leadsto \frac{x2 \cdot \left(x2 \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot 8 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 48.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot -2\right)\\ \mathbf{if}\;x1 \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot 8\\ \mathbf{elif}\;x1 \leq -1.3 \cdot 10^{-163}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.8 \cdot 10^{-210}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 1.9 \cdot 10^{+97}:\\ \;\;\;\;\frac{x2 \cdot \left(x2 \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot 8\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (fma (* x1 x1) x1 (* x1 -2.0)))))
   (if (<= x1 -1.15e-24)
     (* (* (* x2 x2) (/ x1 (fma x1 x1 1.0))) 8.0)
     (if (<= x1 -1.3e-163)
       t_0
       (if (<= x1 2.8e-210)
         (* -6.0 x2)
         (if (<= x1 1.9e+97)
           (* (/ (* x2 (* x2 x1)) (fma x1 x1 1.0)) 8.0)
           t_0))))))

C

double code(double x1, double x2) {
	double t_0 = x1 + fma((x1 * x1), x1, (x1 * -2.0));
	double tmp;
	if (x1 <= -1.15e-24) {
		tmp = ((x2 * x2) * (x1 / fma(x1, x1, 1.0))) * 8.0;
	} else if (x1 <= -1.3e-163) {
		tmp = t_0;
	} else if (x1 <= 2.8e-210) {
		tmp = -6.0 * x2;
	} else if (x1 <= 1.9e+97) {
		tmp = ((x2 * (x2 * x1)) / fma(x1, x1, 1.0)) * 8.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + fma(Float64(x1 * x1), x1, Float64(x1 * -2.0)))
	tmp = 0.0
	if (x1 <= -1.15e-24)
		tmp = Float64(Float64(Float64(x2 * x2) * Float64(x1 / fma(x1, x1, 1.0))) * 8.0);
	elseif (x1 <= -1.3e-163)
		tmp = t_0;
	elseif (x1 <= 2.8e-210)
		tmp = Float64(-6.0 * x2);
	elseif (x1 <= 1.9e+97)
		tmp = Float64(Float64(Float64(x2 * Float64(x2 * x1)) / fma(x1, x1, 1.0)) * 8.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.15e-24], N[(N[(N[(x2 * x2), $MachinePrecision] * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision], If[LessEqual[x1, -1.3e-163], t$95$0, If[LessEqual[x1, 2.8e-210], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x1, 1.9e+97], N[(N[(N[(x2 * N[(x2 * x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.1500000000000001e-24

   1. Initial program 69.8%

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

   2. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \]

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 17.3

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

   4. Applied rewrites 17.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.3

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

      4. lift-*.f64 N/A

         \[\leadsto \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \cdot 8 \]

      5. *-commutative N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      6. lower-*.f64 17.3

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      8. unpow2 N/A

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

      9. lower-*.f64 17.3

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

      10. lift-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 17.3

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

   6. Applied rewrites 17.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-/.f64 17.0

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

   8. Applied rewrites 17.0%

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

   if -1.1500000000000001e-24 < x1 < -1.30000000000000001e-163 or 1.90000000000000018e97 < x1

   1. Initial program 69.8%

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

   3. Applied rewrites 74.6%

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

   4. Taylor expanded in x1 around 0

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

   6. Applied rewrites 41.6%

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

   7. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - \color{blue}{2}\right)\right) \]

      2. lower--.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      5. lower-*.f64 17.9

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

   9. Applied rewrites 17.9%

      \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)}\right) \]

   10. Taylor expanded in x1 around 0

       \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot -2\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.8%

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

   if -1.30000000000000001e-163 < x1 < 2.8e-210

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 26.3

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

   4. Applied rewrites 26.3%

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

   if 2.8e-210 < x1 < 1.90000000000000018e97

   1. Initial program 69.8%

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

   2. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \]

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 17.3

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

   4. Applied rewrites 17.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.3

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

      4. lift-*.f64 N/A

         \[\leadsto \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \cdot 8 \]

      5. *-commutative N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      6. lower-*.f64 17.3

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      8. unpow2 N/A

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

      9. lower-*.f64 17.3

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

      10. lift-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 17.3

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

   6. Applied rewrites 17.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 19.8

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

   8. Applied rewrites 19.8%

      \[\leadsto \frac{x2 \cdot \left(x2 \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot 8 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 18: 47.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot -2\right)\\ t_1 := \left(\left(x2 \cdot x2\right) \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot 8\\ \mathbf{if}\;x1 \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq -1.3 \cdot 10^{-163}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 10^{-114}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 9.2 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (fma (* x1 x1) x1 (* x1 -2.0))))
        (t_1 (* (* (* x2 x2) (/ x1 (fma x1 x1 1.0))) 8.0)))
   (if (<= x1 -1.15e-24)
     t_1
     (if (<= x1 -1.3e-163)
       t_0
       (if (<= x1 1e-114) (* -6.0 x2) (if (<= x1 9.2e+96) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = x1 + fma((x1 * x1), x1, (x1 * -2.0));
	double t_1 = ((x2 * x2) * (x1 / fma(x1, x1, 1.0))) * 8.0;
	double tmp;
	if (x1 <= -1.15e-24) {
		tmp = t_1;
	} else if (x1 <= -1.3e-163) {
		tmp = t_0;
	} else if (x1 <= 1e-114) {
		tmp = -6.0 * x2;
	} else if (x1 <= 9.2e+96) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + fma(Float64(x1 * x1), x1, Float64(x1 * -2.0)))
	t_1 = Float64(Float64(Float64(x2 * x2) * Float64(x1 / fma(x1, x1, 1.0))) * 8.0)
	tmp = 0.0
	if (x1 <= -1.15e-24)
		tmp = t_1;
	elseif (x1 <= -1.3e-163)
		tmp = t_0;
	elseif (x1 <= 1e-114)
		tmp = Float64(-6.0 * x2);
	elseif (x1 <= 9.2e+96)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x2 * x2), $MachinePrecision] * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]}, If[LessEqual[x1, -1.15e-24], t$95$1, If[LessEqual[x1, -1.3e-163], t$95$0, If[LessEqual[x1, 1e-114], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x1, 9.2e+96], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.1500000000000001e-24 or 1.0000000000000001e-114 < x1 < 9.2000000000000006e96

   1. Initial program 69.8%

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

   2. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \]

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 17.3

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

   4. Applied rewrites 17.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.3

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

      4. lift-*.f64 N/A

         \[\leadsto \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \cdot 8 \]

      5. *-commutative N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      6. lower-*.f64 17.3

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      8. unpow2 N/A

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

      9. lower-*.f64 17.3

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

      10. lift-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 17.3

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

   6. Applied rewrites 17.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-/.f64 17.0

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

   8. Applied rewrites 17.0%

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

   if -1.1500000000000001e-24 < x1 < -1.30000000000000001e-163 or 9.2000000000000006e96 < x1

   1. Initial program 69.8%

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

   3. Applied rewrites 74.6%

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

   4. Taylor expanded in x1 around 0

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

   6. Applied rewrites 41.6%

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

   7. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - \color{blue}{2}\right)\right) \]

      2. lower--.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      5. lower-*.f64 17.9

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

   9. Applied rewrites 17.9%

      \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)}\right) \]

   10. Taylor expanded in x1 around 0

       \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot -2\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.8%

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

   if -1.30000000000000001e-163 < x1 < 1.0000000000000001e-114

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 26.3

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

   4. Applied rewrites 26.3%

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

Alternative 19: 47.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot -2\right)\\ t_1 := \frac{\left(x2 \cdot x2\right) \cdot x1}{1} \cdot 8\\ \mathbf{if}\;x1 \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq -1.3 \cdot 10^{-163}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 10^{-114}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 1.9 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (fma (* x1 x1) x1 (* x1 -2.0))))
        (t_1 (* (/ (* (* x2 x2) x1) 1.0) 8.0)))
   (if (<= x1 -1.15e-24)
     t_1
     (if (<= x1 -1.3e-163)
       t_0
       (if (<= x1 1e-114) (* -6.0 x2) (if (<= x1 1.9e+97) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = x1 + fma((x1 * x1), x1, (x1 * -2.0));
	double t_1 = (((x2 * x2) * x1) / 1.0) * 8.0;
	double tmp;
	if (x1 <= -1.15e-24) {
		tmp = t_1;
	} else if (x1 <= -1.3e-163) {
		tmp = t_0;
	} else if (x1 <= 1e-114) {
		tmp = -6.0 * x2;
	} else if (x1 <= 1.9e+97) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + fma(Float64(x1 * x1), x1, Float64(x1 * -2.0)))
	t_1 = Float64(Float64(Float64(Float64(x2 * x2) * x1) / 1.0) * 8.0)
	tmp = 0.0
	if (x1 <= -1.15e-24)
		tmp = t_1;
	elseif (x1 <= -1.3e-163)
		tmp = t_0;
	elseif (x1 <= 1e-114)
		tmp = Float64(-6.0 * x2);
	elseif (x1 <= 1.9e+97)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] / 1.0), $MachinePrecision] * 8.0), $MachinePrecision]}, If[LessEqual[x1, -1.15e-24], t$95$1, If[LessEqual[x1, -1.3e-163], t$95$0, If[LessEqual[x1, 1e-114], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x1, 1.9e+97], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.1500000000000001e-24 or 1.0000000000000001e-114 < x1 < 1.90000000000000018e97

   1. Initial program 69.8%

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

   2. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \]

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 17.3

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

   4. Applied rewrites 17.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.3

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

      4. lift-*.f64 N/A

         \[\leadsto \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \cdot 8 \]

      5. *-commutative N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      6. lower-*.f64 17.3

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      8. unpow2 N/A

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

      9. lower-*.f64 17.3

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

      10. lift-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 17.3

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

   6. Applied rewrites 17.3%

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

   7. Taylor expanded in x1 around 0

      \[\leadsto \frac{\left(x2 \cdot x2\right) \cdot x1}{1} \cdot 8 \]
   8. Step-by-step derivation

   9. Applied rewrites 22.3%

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

   if -1.1500000000000001e-24 < x1 < -1.30000000000000001e-163 or 1.90000000000000018e97 < x1

   1. Initial program 69.8%

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

   3. Applied rewrites 74.6%

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

   4. Taylor expanded in x1 around 0

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

   6. Applied rewrites 41.6%

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

   7. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - \color{blue}{2}\right)\right) \]

      2. lower--.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

      5. lower-*.f64 17.9

         \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)\right) \]

   9. Applied rewrites 17.9%

      \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)}\right) \]

   10. Taylor expanded in x1 around 0

       \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot -2\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.8%

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

   if -1.30000000000000001e-163 < x1 < 1.0000000000000001e-114

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 26.3

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

   4. Applied rewrites 26.3%

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

Alternative 20: 42.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ t_4 := \frac{\left(x2 \cdot x2\right) \cdot x1}{1} \cdot 8\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+257}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 10^{+266}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

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))
        (t_3
         (+
          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)))))
        (t_4 (* (/ (* (* x2 x2) x1) 1.0) 8.0)))
   (if (<= t_3 -5e+257) t_4 (if (<= t_3 1e+266) (* -6.0 x2) t_4))))

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;
	double t_3 = 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)));
	double t_4 = (((x2 * x2) * x1) / 1.0) * 8.0;
	double tmp;
	if (t_3 <= -5e+257) {
		tmp = t_4;
	} else if (t_3 <= 1e+266) {
		tmp = -6.0 * x2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x1, x2)
use fmin_fmax_functions
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (3.0d0 * x1) * x1
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    t_3 = 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)))
    t_4 = (((x2 * x2) * x1) / 1.0d0) * 8.0d0
    if (t_3 <= (-5d+257)) then
        tmp = t_4
    else if (t_3 <= 1d+266) then
        tmp = (-6.0d0) * x2
    else
        tmp = t_4
    end if
    code = tmp
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;
	double t_3 = 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)));
	double t_4 = (((x2 * x2) * x1) / 1.0) * 8.0;
	double tmp;
	if (t_3 <= -5e+257) {
		tmp = t_4;
	} else if (t_3 <= 1e+266) {
		tmp = -6.0 * x2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

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
	t_3 = 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)))
	t_4 = (((x2 * x2) * x1) / 1.0) * 8.0
	tmp = 0
	if t_3 <= -5e+257:
		tmp = t_4
	elif t_3 <= 1e+266:
		tmp = -6.0 * x2
	else:
		tmp = t_4
	return tmp

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)
	t_3 = 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))))
	t_4 = Float64(Float64(Float64(Float64(x2 * x2) * x1) / 1.0) * 8.0)
	tmp = 0.0
	if (t_3 <= -5e+257)
		tmp = t_4;
	elseif (t_3 <= 1e+266)
		tmp = Float64(-6.0 * x2);
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = 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;
	t_3 = 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)));
	t_4 = (((x2 * x2) * x1) / 1.0) * 8.0;
	tmp = 0.0;
	if (t_3 <= -5e+257)
		tmp = t_4;
	elseif (t_3 <= 1e+266)
		tmp = -6.0 * x2;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
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]}, Block[{t$95$3 = 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]}, Block[{t$95$4 = N[(N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] / 1.0), $MachinePrecision] * 8.0), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+257], t$95$4, If[LessEqual[t$95$3, 1e+266], N[(-6.0 * x2), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.00000000000000028e257 or 1e266 < (+.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 69.8%

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

   2. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \]

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 17.3

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

   4. Applied rewrites 17.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.3

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

      4. lift-*.f64 N/A

         \[\leadsto \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \cdot 8 \]

      5. *-commutative N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      6. lower-*.f64 17.3

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      8. unpow2 N/A

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

      9. lower-*.f64 17.3

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

      10. lift-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 17.3

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

   6. Applied rewrites 17.3%

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

   7. Taylor expanded in x1 around 0

      \[\leadsto \frac{\left(x2 \cdot x2\right) \cdot x1}{1} \cdot 8 \]
   8. Step-by-step derivation

   9. Applied rewrites 22.3%

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

   if -5.00000000000000028e257 < (+.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)))))) < 1e266

   1. Initial program 69.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 26.3

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

   4. Applied rewrites 26.3%

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

Alternative 21: 26.3% accurate, 46.3× speedup

Math

\[\begin{array}{l} \\ -6 \cdot x2 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (* -6.0 x2))

C

double code(double x1, double x2) {
	return -6.0 * x2;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x1, x2)
use fmin_fmax_functions
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = (-6.0d0) * x2
end function

Java

public static double code(double x1, double x2) {
	return -6.0 * x2;
}

Python

def code(x1, x2):
	return -6.0 * x2

Julia

function code(x1, x2)
	return Float64(-6.0 * x2)
end

MATLAB

function tmp = code(x1, x2)
	tmp = -6.0 * x2;
end

Wolfram

code[x1_, x2_] := N[(-6.0 * x2), $MachinePrecision]

Derivation

1. Initial program 69.8%

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

2. Taylor expanded in x1 around 0

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

   1. lower-*.f64 26.3

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

4. Applied rewrites 26.3%

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

Reproduce

herbie shell --seed 2025154 
(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))))))