Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 69.9% → 99.4%

Time: 8.2s

Alternatives: 12

Speedup: 7.0×

• 47.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 69.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.4% accurate, 0.5× speedup

Math

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

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))
        (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))))))
   (if (<= t_3 INFINITY)
     t_3
     (*
      (pow x1 4.0)
      (+ 6.0 (/ (* x2 (+ 8.0 (* 12.0 (/ 1.0 x1)))) (* x1 x1)))))))

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 tmp;
	if (t_3 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = pow(x1, 4.0) * (6.0 + ((x2 * (8.0 + (12.0 * (1.0 / x1)))) / (x1 * x1)));
	}
	return tmp;
}

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 tmp;
	if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = Math.pow(x1, 4.0) * (6.0 + ((x2 * (8.0 + (12.0 * (1.0 / x1)))) / (x1 * x1)));
	}
	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)))
	tmp = 0
	if t_3 <= math.inf:
		tmp = t_3
	else:
		tmp = math.pow(x1, 4.0) * (6.0 + ((x2 * (8.0 + (12.0 * (1.0 / x1)))) / (x1 * x1)))
	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))))
	tmp = 0.0
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(x2 * Float64(8.0 + Float64(12.0 * Float64(1.0 / x1)))) / Float64(x1 * x1))));
	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)));
	tmp = 0.0;
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = (x1 ^ 4.0) * (6.0 + ((x2 * (8.0 + (12.0 * (1.0 / x1)))) / (x1 * x1)));
	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]}, If[LessEqual[t$95$3, Infinity], t$95$3, N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(x2 * N[(8.0 + N[(12.0 * N[(1.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.9%

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

   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 69.9%

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

   2. Taylor expanded in x1 around -inf

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x2 around inf

      \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{\color{blue}{{x1}^{2}}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{\color{blue}{2}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{2}}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{2}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{2}}\right) \]

      6. pow2 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{x1 \cdot x1}\right) \]

      7. lift-*.f64 48.1

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{x1 \cdot x1}\right) \]

   7. Applied rewrites 48.1%

      \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{\color{blue}{x1 \cdot x1}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 95.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := x1 \cdot \left(\left(-\left(1 - 2 \cdot \left(1 - -3 \cdot \mathsf{fma}\left(2, x2, -3\right)\right)\right)\right) + x1 \cdot \left(9 + \mathsf{fma}\left(4, \mathsf{fma}\left(2, x2, -3\right), x1 \cdot \mathsf{fma}\left(6, x1, -3\right)\right)\right)\right)\\ t_3 := 2 \cdot \frac{x2}{1 + x1 \cdot x1}\\ \mathbf{if}\;x1 \leq -1600000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x1 \leq 310:\\ \;\;\;\;x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_0 + t\_1 \cdot t\_3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* (* 3.0 x1) x1))
        (t_2
         (*
          x1
          (+
           (- (- 1.0 (* 2.0 (- 1.0 (* -3.0 (fma 2.0 x2 -3.0))))))
           (*
            x1
            (+ 9.0 (fma 4.0 (fma 2.0 x2 -3.0) (* x1 (fma 6.0 x1 -3.0))))))))
        (t_3 (* 2.0 (/ x2 (+ 1.0 (* x1 x1))))))
   (if (<= x1 -1600000.0)
     t_2
     (if (<= x1 310.0)
       (+
        x1
        (+
         (+
          (+
           (+
            (*
             (+
              (* (* (* 2.0 x1) t_3) (- t_3 3.0))
              (* (* x1 x1) (- (* 4.0 t_3) 6.0)))
             t_0)
            (* t_1 t_3))
           (* (* x1 x1) x1))
          x1)
         (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))
       t_2))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = x1 * (-(1.0 - (2.0 * (1.0 - (-3.0 * fma(2.0, x2, -3.0))))) + (x1 * (9.0 + fma(4.0, fma(2.0, x2, -3.0), (x1 * fma(6.0, x1, -3.0))))));
	double t_3 = 2.0 * (x2 / (1.0 + (x1 * x1)));
	double tmp;
	if (x1 <= -1600000.0) {
		tmp = t_2;
	} else if (x1 <= 310.0) {
		tmp = x1 + (((((((((2.0 * x1) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_0) + (t_1 * t_3)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	} else {
		tmp = t_2;
	}
	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(x1 * Float64(Float64(-Float64(1.0 - Float64(2.0 * Float64(1.0 - Float64(-3.0 * fma(2.0, x2, -3.0)))))) + Float64(x1 * Float64(9.0 + fma(4.0, fma(2.0, x2, -3.0), Float64(x1 * fma(6.0, x1, -3.0)))))))
	t_3 = Float64(2.0 * Float64(x2 / Float64(1.0 + Float64(x1 * x1))))
	tmp = 0.0
	if (x1 <= -1600000.0)
		tmp = t_2;
	elseif (x1 <= 310.0)
		tmp = 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_0) + 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_0))));
	else
		tmp = t_2;
	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[(x1 * N[((-N[(1.0 - N[(2.0 * N[(1.0 - N[(-3.0 * N[(2.0 * x2 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(x1 * N[(9.0 + N[(4.0 * N[(2.0 * x2 + -3.0), $MachinePrecision] + N[(x1 * N[(6.0 * x1 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(x2 / N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1600000.0], t$95$2, If[LessEqual[x1, 310.0], 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$0), $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$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.6e6 or 310 < x1

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around -inf

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

   7. Applied rewrites 14.1%

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

   8. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   10. Applied rewrites 50.0%

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

   if -1.6e6 < x1 < 310

   1. Initial program 69.9%

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

   2. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 69.1

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 69.1

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 54.5

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

   10. Applied rewrites 54.5%

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

   11. Taylor expanded in x2 around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 54.5

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

   13. Applied rewrites 54.5%

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

Alternative 3: 95.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(\left(-\left(1 - 2 \cdot \left(1 - -3 \cdot \mathsf{fma}\left(2, x2, -3\right)\right)\right)\right) + x1 \cdot \left(9 + \mathsf{fma}\left(4, \mathsf{fma}\left(2, x2, -3\right), x1 \cdot \mathsf{fma}\left(6, x1, -3\right)\right)\right)\right)\\ \mathbf{if}\;x1 \leq -1450000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 128:\\ \;\;\;\;x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          x1
          (+
           (- (- 1.0 (* 2.0 (- 1.0 (* -3.0 (fma 2.0 x2 -3.0))))))
           (*
            x1
            (+ 9.0 (fma 4.0 (fma 2.0 x2 -3.0) (* x1 (fma 6.0 x1 -3.0)))))))))
   (if (<= x1 -1450000.0)
     t_0
     (if (<= x1 128.0)
       (+
        x1
        (+
         (+ (* x2 (fma -12.0 x1 (* 8.0 (* x1 x2)))) x1)
         (* 3.0 (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) 1.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (-(1.0 - (2.0 * (1.0 - (-3.0 * fma(2.0, x2, -3.0))))) + (x1 * (9.0 + fma(4.0, fma(2.0, x2, -3.0), (x1 * fma(6.0, x1, -3.0))))));
	double tmp;
	if (x1 <= -1450000.0) {
		tmp = t_0;
	} else if (x1 <= 128.0) {
		tmp = x1 + (((x2 * fma(-12.0, x1, (8.0 * (x1 * x2)))) + x1) + (3.0 * (((((3.0 * x1) * x1) - (2.0 * x2)) - x1) / 1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(Float64(-Float64(1.0 - Float64(2.0 * Float64(1.0 - Float64(-3.0 * fma(2.0, x2, -3.0)))))) + Float64(x1 * Float64(9.0 + fma(4.0, fma(2.0, x2, -3.0), Float64(x1 * fma(6.0, x1, -3.0)))))))
	tmp = 0.0
	if (x1 <= -1450000.0)
		tmp = t_0;
	elseif (x1 <= 128.0)
		tmp = Float64(x1 + Float64(Float64(Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2)))) + x1) + Float64(3.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(2.0 * x2)) - x1) / 1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[((-N[(1.0 - N[(2.0 * N[(1.0 - N[(-3.0 * N[(2.0 * x2 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(x1 * N[(9.0 + N[(4.0 * N[(2.0 * x2 + -3.0), $MachinePrecision] + N[(x1 * N[(6.0 * x1 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1450000.0], t$95$0, If[LessEqual[x1, 128.0], N[(x1 + N[(N[(N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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] / 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.45e6 or 128 < x1

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around -inf

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

   7. Applied rewrites 14.1%

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

   8. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   10. Applied rewrites 50.0%

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

   if -1.45e6 < x1 < 128

   1. Initial program 69.9%

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

   2. 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. lift-*.f64 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 69.6%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \left(-12 \cdot x1 + \color{blue}{8 \cdot \left(x1 \cdot x2\right)}\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      4. lift-*.f64 75.2

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

   10. Applied rewrites 75.2%

       \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{x1 \cdot x1}\right)\\ \mathbf{if}\;x1 \leq -1750000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 5.9 \cdot 10^{+18}:\\ \;\;\;\;x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (pow x1 4.0)
          (+ 6.0 (/ (* x2 (+ 8.0 (* 12.0 (/ 1.0 x1)))) (* x1 x1))))))
   (if (<= x1 -1750000.0)
     t_0
     (if (<= x1 5.9e+18)
       (+
        x1
        (+
         (+ (* x2 (fma -12.0 x1 (* 8.0 (* x1 x2)))) x1)
         (* 3.0 (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) 1.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = pow(x1, 4.0) * (6.0 + ((x2 * (8.0 + (12.0 * (1.0 / x1)))) / (x1 * x1)));
	double tmp;
	if (x1 <= -1750000.0) {
		tmp = t_0;
	} else if (x1 <= 5.9e+18) {
		tmp = x1 + (((x2 * fma(-12.0, x1, (8.0 * (x1 * x2)))) + x1) + (3.0 * (((((3.0 * x1) * x1) - (2.0 * x2)) - x1) / 1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(x2 * Float64(8.0 + Float64(12.0 * Float64(1.0 / x1)))) / Float64(x1 * x1))))
	tmp = 0.0
	if (x1 <= -1750000.0)
		tmp = t_0;
	elseif (x1 <= 5.9e+18)
		tmp = Float64(x1 + Float64(Float64(Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2)))) + x1) + Float64(3.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(2.0 * x2)) - 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[(N[(x2 * N[(8.0 + N[(12.0 * N[(1.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1750000.0], t$95$0, If[LessEqual[x1, 5.9e+18], N[(x1 + N[(N[(N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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] / 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.75e6 or 5.9e18 < x1

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around -inf

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x2 around inf

      \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{\color{blue}{{x1}^{2}}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{\color{blue}{2}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{2}}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{2}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{2}}\right) \]

      6. pow2 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{x1 \cdot x1}\right) \]

      7. lift-*.f64 48.1

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{x1 \cdot x1}\right) \]

   7. Applied rewrites 48.1%

      \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{\color{blue}{x1 \cdot x1}}\right) \]

   if -1.75e6 < x1 < 5.9e18

   1. Initial program 69.9%

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

   2. 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. lift-*.f64 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 69.6%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \left(-12 \cdot x1 + \color{blue}{8 \cdot \left(x1 \cdot x2\right)}\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      4. lift-*.f64 75.2

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

   10. Applied rewrites 75.2%

       \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 95.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{x1 \cdot x1}\right)\\ \mathbf{if}\;x1 \leq -1750000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1400:\\ \;\;\;\;x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, -3 \cdot x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (pow x1 4.0)
          (+ 6.0 (/ (* x2 (+ 8.0 (* 12.0 (/ 1.0 x1)))) (* x1 x1))))))
   (if (<= x1 -1750000.0)
     t_0
     (if (<= x1 1400.0)
       (+
        x1
        (+
         (+ (* x2 (fma -12.0 x1 (* 8.0 (* x1 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 + ((x2 * (8.0 + (12.0 * (1.0 / x1)))) / (x1 * x1)));
	double tmp;
	if (x1 <= -1750000.0) {
		tmp = t_0;
	} else if (x1 <= 1400.0) {
		tmp = x1 + (((x2 * fma(-12.0, x1, (8.0 * (x1 * 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) * Float64(6.0 + Float64(Float64(x2 * Float64(8.0 + Float64(12.0 * Float64(1.0 / x1)))) / Float64(x1 * x1))))
	tmp = 0.0
	if (x1 <= -1750000.0)
		tmp = t_0;
	elseif (x1 <= 1400.0)
		tmp = Float64(x1 + Float64(Float64(Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * 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] * N[(6.0 + N[(N[(x2 * N[(8.0 + N[(12.0 * N[(1.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1750000.0], t$95$0, If[LessEqual[x1, 1400.0], N[(x1 + N[(N[(N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -1.75e6 or 1400 < x1

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around -inf

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x2 around inf

      \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{\color{blue}{{x1}^{2}}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{\color{blue}{2}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{2}}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{2}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{{x1}^{2}}\right) \]

      6. pow2 N/A

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{x1 \cdot x1}\right) \]

      7. lift-*.f64 48.1

         \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{x1 \cdot x1}\right) \]

   7. Applied rewrites 48.1%

      \[\leadsto {x1}^{4} \cdot \left(6 + \frac{x2 \cdot \left(8 + 12 \cdot \frac{1}{x1}\right)}{\color{blue}{x1 \cdot x1}}\right) \]

   if -1.75e6 < x1 < 1400

   1. Initial program 69.9%

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

   2. 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. lift-*.f64 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 69.6%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \left(-12 \cdot x1 + \color{blue}{8 \cdot \left(x1 \cdot x2\right)}\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      4. lift-*.f64 75.2

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

   10. Applied rewrites 75.2%

       \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

   11. Taylor expanded in x1 around 0

       \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \color{blue}{\left(-6 \cdot x2 + -3 \cdot x1\right)}\right) \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \mathsf{fma}\left(-6, \color{blue}{x2}, -3 \cdot x1\right)\right) \]

       2. lower-*.f64 60.7

          \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, -3 \cdot x1\right)\right) \]

   13. Applied rewrites 60.7%

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

Alternative 6: 93.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1600000:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \mathsf{fma}\left(6, x1, -3\right)\\ \mathbf{elif}\;x1 \leq 370:\\ \;\;\;\;x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, -3 \cdot x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1600000.0)
   (* (* (* x1 x1) x1) (fma 6.0 x1 -3.0))
   (if (<= x1 370.0)
     (+
      x1
      (+
       (+ (* x2 (fma -12.0 x1 (* 8.0 (* x1 x2)))) x1)
       (fma -6.0 x2 (* -3.0 x1))))
     (* (pow x1 4.0) (- 6.0 (/ 3.0 x1))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1600000.0) {
		tmp = ((x1 * x1) * x1) * fma(6.0, x1, -3.0);
	} else if (x1 <= 370.0) {
		tmp = x1 + (((x2 * fma(-12.0, x1, (8.0 * (x1 * x2)))) + x1) + fma(-6.0, x2, (-3.0 * x1)));
	} else {
		tmp = pow(x1, 4.0) * (6.0 - (3.0 / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1600000.0)
		tmp = Float64(Float64(Float64(x1 * x1) * x1) * fma(6.0, x1, -3.0));
	elseif (x1 <= 370.0)
		tmp = Float64(x1 + Float64(Float64(Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2)))) + x1) + fma(-6.0, x2, Float64(-3.0 * x1))));
	else
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 - Float64(3.0 / x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1600000.0], N[(N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision] * N[(6.0 * x1 + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 370.0], N[(x1 + N[(N[(N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2 + N[(-3.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.6e6

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. pow3 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 - 3\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 - 3\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 - 3\right) \]

      5. sub-flip N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 + \left(\mathsf{neg}\left(3\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 + -3\right) \]

      7. lower-fma.f64 46.4

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

   7. Applied rewrites 46.4%

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

   if -1.6e6 < x1 < 370

   1. Initial program 69.9%

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

   2. 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. lift-*.f64 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 69.6%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \left(-12 \cdot x1 + \color{blue}{8 \cdot \left(x1 \cdot x2\right)}\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      4. lift-*.f64 75.2

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

   10. Applied rewrites 75.2%

       \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

   11. Taylor expanded in x1 around 0

       \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \color{blue}{\left(-6 \cdot x2 + -3 \cdot x1\right)}\right) \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \mathsf{fma}\left(-6, \color{blue}{x2}, -3 \cdot x1\right)\right) \]

       2. lower-*.f64 60.7

          \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, -3 \cdot x1\right)\right) \]

   13. Applied rewrites 60.7%

       \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \color{blue}{\mathsf{fma}\left(-6, x2, -3 \cdot x1\right)}\right) \]

   if 370 < x1

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 46.4

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

   4. Applied rewrites 46.4%

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 88.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1600000:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \mathsf{fma}\left(6, x1, -3\right)\\ \mathbf{elif}\;x1 \leq 370:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, x1 \cdot \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(2, x2, -3\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1600000.0)
   (* (* (* x1 x1) x1) (fma 6.0 x1 -3.0))
   (if (<= x1 370.0)
     (fma -6.0 x2 (* x1 (fma 4.0 (* x2 (fma 2.0 x2 -3.0)) -1.0)))
     (* (pow x1 4.0) (- 6.0 (/ 3.0 x1))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1600000.0) {
		tmp = ((x1 * x1) * x1) * fma(6.0, x1, -3.0);
	} else if (x1 <= 370.0) {
		tmp = fma(-6.0, x2, (x1 * fma(4.0, (x2 * fma(2.0, x2, -3.0)), -1.0)));
	} else {
		tmp = pow(x1, 4.0) * (6.0 - (3.0 / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1600000.0)
		tmp = Float64(Float64(Float64(x1 * x1) * x1) * fma(6.0, x1, -3.0));
	elseif (x1 <= 370.0)
		tmp = fma(-6.0, x2, Float64(x1 * fma(4.0, Float64(x2 * fma(2.0, x2, -3.0)), -1.0)));
	else
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 - Float64(3.0 / x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1600000.0], N[(N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision] * N[(6.0 * x1 + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 370.0], N[(-6.0 * x2 + N[(x1 * N[(4.0 * N[(x2 * N[(2.0 * x2 + -3.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.6e6

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. pow3 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 - 3\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 - 3\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 - 3\right) \]

      5. sub-flip N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 + \left(\mathsf{neg}\left(3\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 + -3\right) \]

      7. lower-fma.f64 46.4

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

   7. Applied rewrites 46.4%

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

   if -1.6e6 < x1 < 370

   1. Initial program 69.9%

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

   2. 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. lift-*.f64 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 69.6%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \left(-12 \cdot x1 + \color{blue}{8 \cdot \left(x1 \cdot x2\right)}\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      4. lift-*.f64 75.2

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

   10. Applied rewrites 75.2%

       \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

   11. Taylor expanded in x1 around 0

       \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   12. 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. sub-flip N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 N/A

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

   13. Applied rewrites 55.3%

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

   if 370 < x1

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 46.4

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

   4. Applied rewrites 46.4%

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 88.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \mathsf{fma}\left(6, x1, -3\right)\\ \mathbf{if}\;x1 \leq -1600000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 370:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, x1 \cdot \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(2, x2, -3\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* (* x1 x1) x1) (fma 6.0 x1 -3.0))))
   (if (<= x1 -1600000.0)
     t_0
     (if (<= x1 370.0)
       (fma -6.0 x2 (* x1 (fma 4.0 (* x2 (fma 2.0 x2 -3.0)) -1.0)))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = ((x1 * x1) * x1) * fma(6.0, x1, -3.0);
	double tmp;
	if (x1 <= -1600000.0) {
		tmp = t_0;
	} else if (x1 <= 370.0) {
		tmp = fma(-6.0, x2, (x1 * fma(4.0, (x2 * fma(2.0, x2, -3.0)), -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(x1 * x1) * x1) * fma(6.0, x1, -3.0))
	tmp = 0.0
	if (x1 <= -1600000.0)
		tmp = t_0;
	elseif (x1 <= 370.0)
		tmp = fma(-6.0, x2, Float64(x1 * fma(4.0, Float64(x2 * fma(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[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision] * N[(6.0 * x1 + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1600000.0], t$95$0, If[LessEqual[x1, 370.0], N[(-6.0 * x2 + N[(x1 * N[(4.0 * N[(x2 * N[(2.0 * x2 + -3.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.6e6 or 370 < x1

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. pow3 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 - 3\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 - 3\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 - 3\right) \]

      5. sub-flip N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 + \left(\mathsf{neg}\left(3\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 + -3\right) \]

      7. lower-fma.f64 46.4

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

   7. Applied rewrites 46.4%

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

   if -1.6e6 < x1 < 370

   1. Initial program 69.9%

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

   2. 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. lift-*.f64 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 69.6%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \left(-12 \cdot x1 + \color{blue}{8 \cdot \left(x1 \cdot x2\right)}\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

      4. lift-*.f64 75.2

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

   10. Applied rewrites 75.2%

       \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{1}\right) \]

   11. Taylor expanded in x1 around 0

       \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   12. 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. sub-flip N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 N/A

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

   13. Applied rewrites 55.3%

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

Alternative 9: 71.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \mathsf{fma}\left(6, x1, -3\right)\\ \mathbf{if}\;x1 \leq -1.16 \cdot 10^{-30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 3.5 \cdot 10^{-74}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 370:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* (* x1 x1) x1) (fma 6.0 x1 -3.0))))
   (if (<= x1 -1.16e-30)
     t_0
     (if (<= x1 3.5e-74)
       (* -6.0 x2)
       (if (<= x1 370.0) (* 8.0 (* x1 (* x2 x2))) t_0)))))

C

double code(double x1, double x2) {
	double t_0 = ((x1 * x1) * x1) * fma(6.0, x1, -3.0);
	double tmp;
	if (x1 <= -1.16e-30) {
		tmp = t_0;
	} else if (x1 <= 3.5e-74) {
		tmp = -6.0 * x2;
	} else if (x1 <= 370.0) {
		tmp = 8.0 * (x1 * (x2 * x2));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(x1 * x1) * x1) * fma(6.0, x1, -3.0))
	tmp = 0.0
	if (x1 <= -1.16e-30)
		tmp = t_0;
	elseif (x1 <= 3.5e-74)
		tmp = Float64(-6.0 * x2);
	elseif (x1 <= 370.0)
		tmp = Float64(8.0 * Float64(x1 * Float64(x2 * x2)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision] * N[(6.0 * x1 + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.16e-30], t$95$0, If[LessEqual[x1, 3.5e-74], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x1, 370.0], N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.16e-30 or 370 < x1

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. pow3 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 - 3\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 - 3\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 - 3\right) \]

      5. sub-flip N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 + \left(\mathsf{neg}\left(3\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot \left(6 \cdot x1 + -3\right) \]

      7. lower-fma.f64 46.4

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

   7. Applied rewrites 46.4%

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

   if -1.16e-30 < x1 < 3.50000000000000015e-74

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 26.2

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

   4. Applied rewrites 26.2%

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

   if 3.50000000000000015e-74 < x1 < 370

   1. Initial program 69.9%

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

   2. 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. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 17.8

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in x1 around 0

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

      1. pow2 N/A

         \[\leadsto 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right) \]

      3. lift-*.f64 22.7

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

   7. Applied rewrites 22.7%

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

Alternative 10: 55.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{if}\;x1 \leq -8.5 \cdot 10^{+61}:\\ \;\;\;\;-3 \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -4.2 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 3.5 \cdot 10^{-74}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 8.0 (* x1 (* x2 x2)))))
   (if (<= x1 -8.5e+61)
     (* -3.0 (* (* x1 x1) x1))
     (if (<= x1 -4.2e-20) t_0 (if (<= x1 3.5e-74) (* -6.0 x2) t_0)))))

C

double code(double x1, double x2) {
	double t_0 = 8.0 * (x1 * (x2 * x2));
	double tmp;
	if (x1 <= -8.5e+61) {
		tmp = -3.0 * ((x1 * x1) * x1);
	} else if (x1 <= -4.2e-20) {
		tmp = t_0;
	} else if (x1 <= 3.5e-74) {
		tmp = -6.0 * x2;
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = 8.0d0 * (x1 * (x2 * x2))
    if (x1 <= (-8.5d+61)) then
        tmp = (-3.0d0) * ((x1 * x1) * x1)
    else if (x1 <= (-4.2d-20)) then
        tmp = t_0
    else if (x1 <= 3.5d-74) then
        tmp = (-6.0d0) * x2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = 8.0 * (x1 * (x2 * x2));
	double tmp;
	if (x1 <= -8.5e+61) {
		tmp = -3.0 * ((x1 * x1) * x1);
	} else if (x1 <= -4.2e-20) {
		tmp = t_0;
	} else if (x1 <= 3.5e-74) {
		tmp = -6.0 * x2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = 8.0 * (x1 * (x2 * x2))
	tmp = 0
	if x1 <= -8.5e+61:
		tmp = -3.0 * ((x1 * x1) * x1)
	elif x1 <= -4.2e-20:
		tmp = t_0
	elif x1 <= 3.5e-74:
		tmp = -6.0 * x2
	else:
		tmp = t_0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(8.0 * Float64(x1 * Float64(x2 * x2)))
	tmp = 0.0
	if (x1 <= -8.5e+61)
		tmp = Float64(-3.0 * Float64(Float64(x1 * x1) * x1));
	elseif (x1 <= -4.2e-20)
		tmp = t_0;
	elseif (x1 <= 3.5e-74)
		tmp = Float64(-6.0 * x2);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = 8.0 * (x1 * (x2 * x2));
	tmp = 0.0;
	if (x1 <= -8.5e+61)
		tmp = -3.0 * ((x1 * x1) * x1);
	elseif (x1 <= -4.2e-20)
		tmp = t_0;
	elseif (x1 <= 3.5e-74)
		tmp = -6.0 * x2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -8.5e+61], N[(-3.0 * N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -4.2e-20], t$95$0, If[LessEqual[x1, 3.5e-74], N[(-6.0 * x2), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -8.50000000000000035e61

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

         \[\leadsto -3 \cdot {x1}^{\color{blue}{3}} \]

      2. pow3 N/A

         \[\leadsto -3 \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto -3 \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

      4. lift-*.f64 19.3

         \[\leadsto -3 \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

   7. Applied rewrites 19.3%

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

   if -8.50000000000000035e61 < x1 < -4.1999999999999998e-20 or 3.50000000000000015e-74 < x1

   1. Initial program 69.9%

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

   2. 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. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 17.8

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in x1 around 0

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

      1. pow2 N/A

         \[\leadsto 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right) \]

      3. lift-*.f64 22.7

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

   7. Applied rewrites 22.7%

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

   if -4.1999999999999998e-20 < x1 < 3.50000000000000015e-74

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 26.2

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

   4. Applied rewrites 26.2%

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

Alternative 11: 42.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.16 \cdot 10^{-30}:\\ \;\;\;\;-3 \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.16e-30) (* -3.0 (* (* x1 x1) x1)) (* -6.0 x2)))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.16e-30) {
		tmp = -3.0 * ((x1 * x1) * x1);
	} else {
		tmp = -6.0 * x2;
	}
	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) :: tmp
    if (x1 <= (-1.16d-30)) then
        tmp = (-3.0d0) * ((x1 * x1) * x1)
    else
        tmp = (-6.0d0) * x2
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.16e-30) {
		tmp = -3.0 * ((x1 * x1) * x1);
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if x1 <= -1.16e-30:
		tmp = -3.0 * ((x1 * x1) * x1)
	else:
		tmp = -6.0 * x2
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.16e-30)
		tmp = Float64(-3.0 * Float64(Float64(x1 * x1) * x1));
	else
		tmp = Float64(-6.0 * x2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if (x1 <= -1.16e-30)
		tmp = -3.0 * ((x1 * x1) * x1);
	else
		tmp = -6.0 * x2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.16e-30], N[(-3.0 * N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(-6.0 * x2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.16e-30

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

         \[\leadsto -3 \cdot {x1}^{\color{blue}{3}} \]

      2. pow3 N/A

         \[\leadsto -3 \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto -3 \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

      4. lift-*.f64 19.3

         \[\leadsto -3 \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

   7. Applied rewrites 19.3%

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

   if -1.16e-30 < x1

   1. Initial program 69.9%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 26.2

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

   4. Applied rewrites 26.2%

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

Alternative 12: 26.2% 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 69.9%

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

2. Taylor expanded in x1 around 0

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

   1. lower-*.f64 26.2

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

4. Applied rewrites 26.2%

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

Reproduce

herbie shell --seed 2025132 
(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))))))