Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.3% → 99.5%

Time: 14.4s

Alternatives: 16

Speedup: 6.3×

• 46.3% 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: 70.3% 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: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-neg.f64 99.4

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

   8. Applied rewrites 99.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

Alternative 2: 61.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ t_4 := \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+304}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-26}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-93}:\\ \;\;\;\;-x1\\ \mathbf{elif}\;t\_3 \leq 10^{+229}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
               t_1)
              (* t_0 t_2))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1)))))
        (t_4 (* (* (* x2 x2) x1) 8.0)))
   (if (<= t_3 -5e+304)
     t_4
     (if (<= t_3 -2e-26)
       (* -6.0 x2)
       (if (<= t_3 5e-93)
         (- x1)
         (if (<= t_3 1e+229)
           (* -6.0 x2)
           (if (<= t_3 INFINITY) t_4 (* (* x1 x1) 9.0))))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double t_4 = ((x2 * x2) * x1) * 8.0;
	double tmp;
	if (t_3 <= -5e+304) {
		tmp = t_4;
	} else if (t_3 <= -2e-26) {
		tmp = -6.0 * x2;
	} else if (t_3 <= 5e-93) {
		tmp = -x1;
	} else if (t_3 <= 1e+229) {
		tmp = -6.0 * x2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = (x1 * x1) * 9.0;
	}
	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 t_4 = ((x2 * x2) * x1) * 8.0;
	double tmp;
	if (t_3 <= -5e+304) {
		tmp = t_4;
	} else if (t_3 <= -2e-26) {
		tmp = -6.0 * x2;
	} else if (t_3 <= 5e-93) {
		tmp = -x1;
	} else if (t_3 <= 1e+229) {
		tmp = -6.0 * x2;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_4;
	} else {
		tmp = (x1 * x1) * 9.0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (3.0 * x1) * x1
	t_1 = (x1 * x1) + 1.0
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1
	t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))
	t_4 = ((x2 * x2) * x1) * 8.0
	tmp = 0
	if t_3 <= -5e+304:
		tmp = t_4
	elif t_3 <= -2e-26:
		tmp = -6.0 * x2
	elif t_3 <= 5e-93:
		tmp = -x1
	elif t_3 <= 1e+229:
		tmp = -6.0 * x2
	elif t_3 <= math.inf:
		tmp = t_4
	else:
		tmp = (x1 * x1) * 9.0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	t_4 = Float64(Float64(Float64(x2 * x2) * x1) * 8.0)
	tmp = 0.0
	if (t_3 <= -5e+304)
		tmp = t_4;
	elseif (t_3 <= -2e-26)
		tmp = Float64(-6.0 * x2);
	elseif (t_3 <= 5e-93)
		tmp = Float64(-x1);
	elseif (t_3 <= 1e+229)
		tmp = Float64(-6.0 * x2);
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(Float64(x1 * x1) * 9.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (3.0 * x1) * x1;
	t_1 = (x1 * x1) + 1.0;
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	t_4 = ((x2 * x2) * x1) * 8.0;
	tmp = 0.0;
	if (t_3 <= -5e+304)
		tmp = t_4;
	elseif (t_3 <= -2e-26)
		tmp = -6.0 * x2;
	elseif (t_3 <= 5e-93)
		tmp = -x1;
	elseif (t_3 <= 1e+229)
		tmp = -6.0 * x2;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = (x1 * x1) * 9.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+304], t$95$4, If[LessEqual[t$95$3, -2e-26], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[t$95$3, 5e-93], (-x1), If[LessEqual[t$95$3, 1e+229], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -4.9999999999999997e304 or 9.9999999999999999e228 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 39.0%

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

   if -4.9999999999999997e304 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2.0000000000000001e-26 or 4.99999999999999994e-93 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 9.9999999999999999e228

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if -2.0000000000000001e-26 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.99999999999999994e-93

   1. Initial program 98.9%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 65.3%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto -1 \cdot x1 \]
   10. Step-by-step derivation

   11. Applied rewrites 65.3%

       \[\leadsto -x1 \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 86.4%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 86.4%

       \[\leadsto \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ t_4 := \mathsf{fma}\left(8 \cdot x1, x2, -\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, 12\right), x1, 6\right)\right) \cdot x2\\ \mathbf{if}\;t\_3 \leq -2000000000000:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(12, x2, 9\right), x1, \mathsf{fma}\left(-12, x2, -1\right)\right) \cdot x1\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
               t_1)
              (* t_0 t_2))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1)))))
        (t_4 (* (fma (* 8.0 x1) x2 (- (fma (fma -12.0 x1 12.0) x1 6.0))) x2)))
   (if (<= t_3 -2000000000000.0)
     t_4
     (if (<= t_3 5e+167)
       (fma x2 -6.0 (* (fma (fma 12.0 x2 9.0) x1 (fma -12.0 x2 -1.0)) x1))
       (if (<= t_3 INFINITY) t_4 (* (* x1 x1) 9.0))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double t_4 = fma((8.0 * x1), x2, -fma(fma(-12.0, x1, 12.0), x1, 6.0)) * x2;
	double tmp;
	if (t_3 <= -2000000000000.0) {
		tmp = t_4;
	} else if (t_3 <= 5e+167) {
		tmp = fma(x2, -6.0, (fma(fma(12.0, x2, 9.0), x1, fma(-12.0, x2, -1.0)) * x1));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = (x1 * x1) * 9.0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	t_4 = Float64(fma(Float64(8.0 * x1), x2, Float64(-fma(fma(-12.0, x1, 12.0), x1, 6.0))) * x2)
	tmp = 0.0
	if (t_3 <= -2000000000000.0)
		tmp = t_4;
	elseif (t_3 <= 5e+167)
		tmp = fma(x2, -6.0, Float64(fma(fma(12.0, x2, 9.0), x1, fma(-12.0, x2, -1.0)) * x1));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(Float64(x1 * x1) * 9.0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(8.0 * x1), $MachinePrecision] * x2 + (-N[(N[(-12.0 * x1 + 12.0), $MachinePrecision] * x1 + 6.0), $MachinePrecision])), $MachinePrecision] * x2), $MachinePrecision]}, If[LessEqual[t$95$3, -2000000000000.0], t$95$4, If[LessEqual[t$95$3, 5e+167], N[(x2 * -6.0 + N[(N[(N[(12.0 * x2 + 9.0), $MachinePrecision] * x1 + N[(-12.0 * x2 + -1.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e12 or 4.9999999999999997e167 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 48.5%

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

   6. Taylor expanded in x2 around -inf

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 63.3%

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

   if -2e12 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.9999999999999997e167

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right), \color{blue}{x2}, \left(9 \cdot x1 - 1\right) \cdot x1\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto -6 \cdot x2 + x1 \cdot \color{blue}{\left(\left(-12 \cdot x2 + x1 \cdot \left(9 + 12 \cdot x2\right)\right) - 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 86.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(12, x2, 9\right), x1, -12 \cdot x2\right) - 1, x1, -6 \cdot x2\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 87.0%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(12, x2, 9\right), x1, \mathsf{fma}\left(-12, x2, -1\right)\right) \cdot x1\right) \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 86.4%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 86.4%

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

Alternative 4: 51.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-26}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-93}:\\ \;\;\;\;-x1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+237}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

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 -2e-26)
     (* -6.0 x2)
     (if (<= t_3 5e-93)
       (- x1)
       (if (<= t_3 2e+237) (* -6.0 x2) (* (* x1 x1) 9.0))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -2e-26) {
		tmp = -6.0 * x2;
	} else if (t_3 <= 5e-93) {
		tmp = -x1;
	} else if (t_3 <= 2e+237) {
		tmp = -6.0 * x2;
	} else {
		tmp = (x1 * x1) * 9.0;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (3.0d0 * x1) * x1
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    t_3 = x1 + (((((((((2.0d0 * x1) * t_2) * (t_2 - 3.0d0)) + ((x1 * x1) * ((4.0d0 * t_2) - 6.0d0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0d0 * (((t_0 - (2.0d0 * x2)) - x1) / t_1)))
    if (t_3 <= (-2d-26)) then
        tmp = (-6.0d0) * x2
    else if (t_3 <= 5d-93) then
        tmp = -x1
    else if (t_3 <= 2d+237) then
        tmp = (-6.0d0) * x2
    else
        tmp = (x1 * x1) * 9.0d0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -2e-26) {
		tmp = -6.0 * x2;
	} else if (t_3 <= 5e-93) {
		tmp = -x1;
	} else if (t_3 <= 2e+237) {
		tmp = -6.0 * x2;
	} else {
		tmp = (x1 * x1) * 9.0;
	}
	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 <= -2e-26:
		tmp = -6.0 * x2
	elif t_3 <= 5e-93:
		tmp = -x1
	elif t_3 <= 2e+237:
		tmp = -6.0 * x2
	else:
		tmp = (x1 * x1) * 9.0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= -2e-26)
		tmp = Float64(-6.0 * x2);
	elseif (t_3 <= 5e-93)
		tmp = Float64(-x1);
	elseif (t_3 <= 2e+237)
		tmp = Float64(-6.0 * x2);
	else
		tmp = Float64(Float64(x1 * x1) * 9.0);
	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 <= -2e-26)
		tmp = -6.0 * x2;
	elseif (t_3 <= 5e-93)
		tmp = -x1;
	elseif (t_3 <= 2e+237)
		tmp = -6.0 * x2;
	else
		tmp = (x1 * x1) * 9.0;
	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, -2e-26], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[t$95$3, 5e-93], (-x1), If[LessEqual[t$95$3, 2e+237], N[(-6.0 * x2), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2.0000000000000001e-26 or 4.99999999999999994e-93 < (+.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)))))) < 1.99999999999999988e237

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 54.1

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

   5. Applied rewrites 54.1%

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

   if -2.0000000000000001e-26 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.99999999999999994e-93

   1. Initial program 98.9%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 65.3%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto -1 \cdot x1 \]
   10. Step-by-step derivation

   11. Applied rewrites 65.3%

       \[\leadsto -x1 \]

   if 1.99999999999999988e237 < (+.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 31.9%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 60.3%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 60.3%

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

Alternative 5: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \left(3 \cdot x1\right) \cdot x1\\ t_3 := \frac{\left(t\_2 + 2 \cdot x2\right) - x1}{t\_1}\\ t_4 := \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\\ t_5 := \left(2 \cdot x1\right) \cdot t\_3\\ t_6 := 3 \cdot \frac{\left(t\_2 - 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(t\_5 \cdot \left(t\_3 - 3\right) + t\_4\right) \cdot t\_1 + t\_2 \cdot t\_3\right) + t\_0\right) + x1\right) + t\_6\right) \leq \infty:\\ \;\;\;\;x1 + \left(\left(\left(\left(\left(t\_5 \cdot \mathsf{fma}\left(2, x2, -3\right) + t\_4\right) \cdot t\_1 + t\_2 \cdot 3\right) + t\_0\right) + x1\right) + t\_6\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (* (* 3.0 x1) x1))
        (t_3 (/ (- (+ t_2 (* 2.0 x2)) x1) t_1))
        (t_4 (* (* x1 x1) (- (* 4.0 t_3) 6.0)))
        (t_5 (* (* 2.0 x1) t_3))
        (t_6 (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_1))))
   (if (<=
        (+
         x1
         (+
          (+ (+ (+ (* (+ (* t_5 (- t_3 3.0)) t_4) t_1) (* t_2 t_3)) t_0) x1)
          t_6))
        INFINITY)
     (+
      x1
      (+
       (+ (+ (+ (* (+ (* t_5 (fma 2.0 x2 -3.0)) t_4) t_1) (* t_2 3.0)) t_0) x1)
       t_6))
     (*
      (- 6.0 (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) x1)) x1))
      (pow x1 4.0)))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = (3.0 * x1) * x1;
	double t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1;
	double t_4 = (x1 * x1) * ((4.0 * t_3) - 6.0);
	double t_5 = (2.0 * x1) * t_3;
	double t_6 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	double tmp;
	if ((x1 + (((((((t_5 * (t_3 - 3.0)) + t_4) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_6)) <= ((double) INFINITY)) {
		tmp = x1 + (((((((t_5 * fma(2.0, x2, -3.0)) + t_4) * t_1) + (t_2 * 3.0)) + t_0) + x1) + t_6);
	} else {
		tmp = (6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)) * pow(x1, 4.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(3.0 * x1) * x1)
	t_3 = Float64(Float64(Float64(t_2 + Float64(2.0 * x2)) - x1) / t_1)
	t_4 = Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))
	t_5 = Float64(Float64(2.0 * x1) * t_3)
	t_6 = Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_1))
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(t_5 * Float64(t_3 - 3.0)) + t_4) * t_1) + Float64(t_2 * t_3)) + t_0) + x1) + t_6)) <= Inf)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(t_5 * fma(2.0, x2, -3.0)) + t_4) * t_1) + Float64(t_2 * 3.0)) + t_0) + x1) + t_6));
	else
		tmp = Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)) * (x1 ^ 4.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$6 = N[(3.0 * N[(N[(N[(t$95$2 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(N[(N[(N[(N[(N[(t$95$5 * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision], Infinity], N[(x1 + N[(N[(N[(N[(N[(N[(N[(t$95$5 * N[(2.0 * x2 + -3.0), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$2 * 3.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 - N[(N[(3.0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x1 around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 95.6

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

   8. Applied rewrites 95.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

Alternative 6: 96.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -490000000000 \lor \neg \left(x1 \leq 1700\right):\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(12, x1, -12\right) \cdot x1\right), x2, \left(9 \cdot x1 - 1\right) \cdot x1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -490000000000.0) (not (<= x1 1700.0)))
   (*
    (- 6.0 (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) x1)) x1))
    (pow x1 4.0))
   (fma
    x2
    -6.0
    (fma
     (fma (* x1 x2) 8.0 (* (fma 12.0 x1 -12.0) x1))
     x2
     (* (- (* 9.0 x1) 1.0) x1)))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -490000000000.0) || !(x1 <= 1700.0)) {
		tmp = (6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)) * pow(x1, 4.0);
	} else {
		tmp = fma(x2, -6.0, fma(fma((x1 * x2), 8.0, (fma(12.0, x1, -12.0) * x1)), x2, (((9.0 * x1) - 1.0) * x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -490000000000.0) || !(x1 <= 1700.0))
		tmp = Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)) * (x1 ^ 4.0));
	else
		tmp = fma(x2, -6.0, fma(fma(Float64(x1 * x2), 8.0, Float64(fma(12.0, x1, -12.0) * x1)), x2, Float64(Float64(Float64(9.0 * x1) - 1.0) * x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -490000000000.0], N[Not[LessEqual[x1, 1700.0]], $MachinePrecision]], N[(N[(6.0 - N[(N[(3.0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], N[(x2 * -6.0 + N[(N[(N[(x1 * x2), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(N[(9.0 * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -4.9e11 or 1700 < x1

   1. Initial program 36.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

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

   if -4.9e11 < x1 < 1700

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 86.9%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(x2, -6, x1 \cdot \left(9 \cdot x1 - 1\right) + x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 98.3%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(12, x1, -12\right) \cdot x1\right), x2, \left(9 \cdot x1 - 1\right) \cdot x1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -490000000000 \lor \neg \left(x1 \leq 1700\right):\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(12, x1, -12\right) \cdot x1\right), x2, \left(9 \cdot x1 - 1\right) \cdot x1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -780000000000:\\ \;\;\;\;{x1}^{4} \cdot 6\\ \mathbf{elif}\;x1 \leq 1700:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(12, x1, -12\right) \cdot x1\right), x2, \left(9 \cdot x1 - 1\right) \cdot x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(6, x1, -3\right) \cdot {x1}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -780000000000.0)
   (* (pow x1 4.0) 6.0)
   (if (<= x1 1700.0)
     (fma
      x2
      -6.0
      (fma
       (fma (* x1 x2) 8.0 (* (fma 12.0 x1 -12.0) x1))
       x2
       (* (- (* 9.0 x1) 1.0) x1)))
     (* (fma 6.0 x1 -3.0) (pow x1 3.0)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -780000000000.0) {
		tmp = pow(x1, 4.0) * 6.0;
	} else if (x1 <= 1700.0) {
		tmp = fma(x2, -6.0, fma(fma((x1 * x2), 8.0, (fma(12.0, x1, -12.0) * x1)), x2, (((9.0 * x1) - 1.0) * x1)));
	} else {
		tmp = fma(6.0, x1, -3.0) * pow(x1, 3.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -780000000000.0)
		tmp = Float64((x1 ^ 4.0) * 6.0);
	elseif (x1 <= 1700.0)
		tmp = fma(x2, -6.0, fma(fma(Float64(x1 * x2), 8.0, Float64(fma(12.0, x1, -12.0) * x1)), x2, Float64(Float64(Float64(9.0 * x1) - 1.0) * x1)));
	else
		tmp = Float64(fma(6.0, x1, -3.0) * (x1 ^ 3.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -780000000000.0], N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[x1, 1700.0], N[(x2 * -6.0 + N[(N[(N[(x1 * x2), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(N[(9.0 * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * N[Power[x1, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -7.8e11

   1. Initial program 36.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.8

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

   5. Applied rewrites 1.8%

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

   6. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 92.2

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

   8. Applied rewrites 92.2%

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

   if -7.8e11 < x1 < 1700

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 86.9%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(x2, -6, x1 \cdot \left(9 \cdot x1 - 1\right) + x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 98.3%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(12, x1, -12\right) \cdot x1\right), x2, \left(9 \cdot x1 - 1\right) \cdot x1\right)\right) \]

   if 1700 < x1

   1. Initial program 37.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 97.0%

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

Alternative 8: 94.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -780000000000:\\ \;\;\;\;{x1}^{4} \cdot 6\\ \mathbf{elif}\;x1 \leq 1700:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(12, x1, -12\right) \cdot x1\right), x2, \left(9 \cdot x1 - 1\right) \cdot x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(6 - \frac{3}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -780000000000.0)
   (* (pow x1 4.0) 6.0)
   (if (<= x1 1700.0)
     (fma
      x2
      -6.0
      (fma
       (fma (* x1 x2) 8.0 (* (fma 12.0 x1 -12.0) x1))
       x2
       (* (- (* 9.0 x1) 1.0) x1)))
     (* (* (- 6.0 (/ 3.0 x1)) (* x1 x1)) (* x1 x1)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -780000000000.0) {
		tmp = pow(x1, 4.0) * 6.0;
	} else if (x1 <= 1700.0) {
		tmp = fma(x2, -6.0, fma(fma((x1 * x2), 8.0, (fma(12.0, x1, -12.0) * x1)), x2, (((9.0 * x1) - 1.0) * x1)));
	} else {
		tmp = ((6.0 - (3.0 / x1)) * (x1 * x1)) * (x1 * x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -780000000000.0)
		tmp = Float64((x1 ^ 4.0) * 6.0);
	elseif (x1 <= 1700.0)
		tmp = fma(x2, -6.0, fma(fma(Float64(x1 * x2), 8.0, Float64(fma(12.0, x1, -12.0) * x1)), x2, Float64(Float64(Float64(9.0 * x1) - 1.0) * x1)));
	else
		tmp = Float64(Float64(Float64(6.0 - Float64(3.0 / x1)) * Float64(x1 * x1)) * Float64(x1 * x1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -780000000000.0], N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[x1, 1700.0], N[(x2 * -6.0 + N[(N[(N[(x1 * x2), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(N[(9.0 * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -7.8e11

   1. Initial program 36.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.8

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

   5. Applied rewrites 1.8%

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

   6. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 92.2

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

   8. Applied rewrites 92.2%

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

   if -7.8e11 < x1 < 1700

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 86.9%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(x2, -6, x1 \cdot \left(9 \cdot x1 - 1\right) + x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 98.3%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(12, x1, -12\right) \cdot x1\right), x2, \left(9 \cdot x1 - 1\right) \cdot x1\right)\right) \]

   if 1700 < x1

   1. Initial program 37.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 97.0

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

   5. Applied rewrites 97.0%

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

   7. Applied rewrites 97.0%

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

Alternative 9: 94.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -780000000000 \lor \neg \left(x1 \leq 1700\right):\\ \;\;\;\;\left(\left(6 - \frac{3}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(12, x1, -12\right) \cdot x1\right), x2, \left(9 \cdot x1 - 1\right) \cdot x1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -780000000000.0) (not (<= x1 1700.0)))
   (* (* (- 6.0 (/ 3.0 x1)) (* x1 x1)) (* x1 x1))
   (fma
    x2
    -6.0
    (fma
     (fma (* x1 x2) 8.0 (* (fma 12.0 x1 -12.0) x1))
     x2
     (* (- (* 9.0 x1) 1.0) x1)))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -780000000000.0) || !(x1 <= 1700.0)) {
		tmp = ((6.0 - (3.0 / x1)) * (x1 * x1)) * (x1 * x1);
	} else {
		tmp = fma(x2, -6.0, fma(fma((x1 * x2), 8.0, (fma(12.0, x1, -12.0) * x1)), x2, (((9.0 * x1) - 1.0) * x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -780000000000.0) || !(x1 <= 1700.0))
		tmp = Float64(Float64(Float64(6.0 - Float64(3.0 / x1)) * Float64(x1 * x1)) * Float64(x1 * x1));
	else
		tmp = fma(x2, -6.0, fma(fma(Float64(x1 * x2), 8.0, Float64(fma(12.0, x1, -12.0) * x1)), x2, Float64(Float64(Float64(9.0 * x1) - 1.0) * x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -780000000000.0], N[Not[LessEqual[x1, 1700.0]], $MachinePrecision]], N[(N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], N[(x2 * -6.0 + N[(N[(N[(x1 * x2), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(N[(9.0 * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -7.8e11 or 1700 < x1

   1. Initial program 36.9%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 94.6%

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

   if -7.8e11 < x1 < 1700

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 86.9%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(x2, -6, x1 \cdot \left(9 \cdot x1 - 1\right) + x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 98.3%

       \[\leadsto \mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(12, x1, -12\right) \cdot x1\right), x2, \left(9 \cdot x1 - 1\right) \cdot x1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -780000000000 \lor \neg \left(x1 \leq 1700\right):\\ \;\;\;\;\left(\left(6 - \frac{3}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(12, x1, -12\right) \cdot x1\right), x2, \left(9 \cdot x1 - 1\right) \cdot x1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -780000000000 \lor \neg \left(x1 \leq 1700\right):\\ \;\;\;\;\left(\left(6 - \frac{3}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \left(9 \cdot x1 - 1\right) \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -780000000000.0) (not (<= x1 1700.0)))
   (* (* (- 6.0 (/ 3.0 x1)) (* x1 x1)) (* x1 x1))
   (fma
    (fma (* x1 x2) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
    x2
    (* (- (* 9.0 x1) 1.0) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -780000000000.0) || !(x1 <= 1700.0)) {
		tmp = ((6.0 - (3.0 / x1)) * (x1 * x1)) * (x1 * x1);
	} else {
		tmp = fma(fma((x1 * x2), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (((9.0 * x1) - 1.0) * x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -780000000000.0) || !(x1 <= 1700.0))
		tmp = Float64(Float64(Float64(6.0 - Float64(3.0 / x1)) * Float64(x1 * x1)) * Float64(x1 * x1));
	else
		tmp = fma(fma(Float64(x1 * x2), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(Float64(Float64(9.0 * x1) - 1.0) * x1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -780000000000.0], N[Not[LessEqual[x1, 1700.0]], $MachinePrecision]], N[(N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x2), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(N[(9.0 * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -7.8e11 or 1700 < x1

   1. Initial program 36.9%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 94.6%

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

   if -7.8e11 < x1 < 1700

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), \color{blue}{x2}, \left(9 \cdot x1 - 1\right) \cdot x1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -780000000000 \lor \neg \left(x1 \leq 1700\right):\\ \;\;\;\;\left(\left(6 - \frac{3}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \left(9 \cdot x1 - 1\right) \cdot x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -780000000000 \lor \neg \left(x1 \leq 1450\right):\\ \;\;\;\;\left(\left(6 - \frac{3}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \left(\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -780000000000.0) (not (<= x1 1450.0)))
   (* (* (- 6.0 (/ 3.0 x1)) (* x1 x1)) (* x1 x1))
   (fma x2 -6.0 (* (- (* (* (fma 2.0 x2 -3.0) x2) 4.0) 1.0) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -780000000000.0) || !(x1 <= 1450.0)) {
		tmp = ((6.0 - (3.0 / x1)) * (x1 * x1)) * (x1 * x1);
	} else {
		tmp = fma(x2, -6.0, ((((fma(2.0, x2, -3.0) * x2) * 4.0) - 1.0) * x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -780000000000.0) || !(x1 <= 1450.0))
		tmp = Float64(Float64(Float64(6.0 - Float64(3.0 / x1)) * Float64(x1 * x1)) * Float64(x1 * x1));
	else
		tmp = fma(x2, -6.0, Float64(Float64(Float64(Float64(fma(2.0, x2, -3.0) * x2) * 4.0) - 1.0) * x1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -780000000000.0], N[Not[LessEqual[x1, 1450.0]], $MachinePrecision]], N[(N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], N[(x2 * -6.0 + N[(N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -7.8e11 or 1450 < x1

   1. Initial program 36.9%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 94.6%

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

   if -7.8e11 < x1 < 1450

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 86.9%

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

   8. Taylor expanded in x1 around 0

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

   10. Applied rewrites 86.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -780000000000 \lor \neg \left(x1 \leq 1450\right):\\ \;\;\;\;\left(\left(6 - \frac{3}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, \left(\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2\right) \cdot 4 - 1\right) \cdot x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.0% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \leq -8 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(8, x1, \frac{-6}{x2}\right) \cdot \left(x2 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x2 -8e+209)
   (* (fma 8.0 x1 (/ -6.0 x2)) (* x2 x2))
   (fma -6.0 x2 (* (fma 9.0 x1 -1.0) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x2 <= -8e+209) {
		tmp = fma(8.0, x1, (-6.0 / x2)) * (x2 * x2);
	} else {
		tmp = fma(-6.0, x2, (fma(9.0, x1, -1.0) * x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x2 <= -8e+209)
		tmp = Float64(fma(8.0, x1, Float64(-6.0 / x2)) * Float64(x2 * x2));
	else
		tmp = fma(-6.0, x2, Float64(fma(9.0, x1, -1.0) * x1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x2, -8e+209], N[(N[(8.0 * x1 + N[(-6.0 / x2), $MachinePrecision]), $MachinePrecision] * N[(x2 * x2), $MachinePrecision]), $MachinePrecision], N[(-6.0 * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x2 < -8.0000000000000006e209

   1. Initial program 82.6%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in x2 around -inf

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

   8. Applied rewrites 62.8%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 75.8%

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

   if -8.0000000000000006e209 < x2

   1. Initial program 65.6%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right), \color{blue}{x2}, \left(9 \cdot x1 - 1\right) \cdot x1\right) \]

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 71.9%

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

   13. Applied rewrites 71.9%

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

Alternative 13: 55.7% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.56 \cdot 10^{-114} \lor \neg \left(x1 \leq 1.56 \cdot 10^{-128}\right):\\ \;\;\;\;\mathsf{fma}\left(x1, 9, -1\right) \cdot x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -1.56e-114) (not (<= x1 1.56e-128)))
   (* (fma x1 9.0 -1.0) x1)
   (* -6.0 x2)))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -1.56e-114) || !(x1 <= 1.56e-128)) {
		tmp = fma(x1, 9.0, -1.0) * x1;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -1.56e-114) || !(x1 <= 1.56e-128))
		tmp = Float64(fma(x1, 9.0, -1.0) * x1);
	else
		tmp = Float64(-6.0 * x2);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -1.56e-114], N[Not[LessEqual[x1, 1.56e-128]], $MachinePrecision]], N[(N[(x1 * 9.0 + -1.0), $MachinePrecision] * x1), $MachinePrecision], N[(-6.0 * x2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.5599999999999999e-114 or 1.56000000000000004e-128 < x1

   1. Initial program 53.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 54.1%

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

   10. Applied rewrites 54.1%

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

   if -1.5599999999999999e-114 < x1 < 1.56000000000000004e-128

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.56 \cdot 10^{-114} \lor \neg \left(x1 \leq 1.56 \cdot 10^{-128}\right):\\ \;\;\;\;\mathsf{fma}\left(x1, 9, -1\right) \cdot x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.0% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \leq -8 \cdot 10^{+209}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x2 -8e+209)
   (* (* (* x2 x2) x1) 8.0)
   (fma -6.0 x2 (* (fma 9.0 x1 -1.0) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x2 <= -8e+209) {
		tmp = ((x2 * x2) * x1) * 8.0;
	} else {
		tmp = fma(-6.0, x2, (fma(9.0, x1, -1.0) * x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x2 <= -8e+209)
		tmp = Float64(Float64(Float64(x2 * x2) * x1) * 8.0);
	else
		tmp = fma(-6.0, x2, Float64(fma(9.0, x1, -1.0) * x1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x2, -8e+209], N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision], N[(-6.0 * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x2 < -8.0000000000000006e209

   1. Initial program 82.6%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 75.6%

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

   if -8.0000000000000006e209 < x2

   1. Initial program 65.6%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right), \color{blue}{x2}, \left(9 \cdot x1 - 1\right) \cdot x1\right) \]

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 71.9%

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

   13. Applied rewrites 71.9%

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

Alternative 15: 32.0% accurate, 16.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \leq -5.5 \cdot 10^{-125} \lor \neg \left(x2 \leq 4.4 \cdot 10^{-96}\right):\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;-x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x2 -5.5e-125) (not (<= x2 4.4e-96))) (* -6.0 x2) (- x1)))

C

double code(double x1, double x2) {
	double tmp;
	if ((x2 <= -5.5e-125) || !(x2 <= 4.4e-96)) {
		tmp = -6.0 * x2;
	} else {
		tmp = -x1;
	}
	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 ((x2 <= (-5.5d-125)) .or. (.not. (x2 <= 4.4d-96))) then
        tmp = (-6.0d0) * x2
    else
        tmp = -x1
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((x2 <= -5.5e-125) || !(x2 <= 4.4e-96)) {
		tmp = -6.0 * x2;
	} else {
		tmp = -x1;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x2 <= -5.5e-125) or not (x2 <= 4.4e-96):
		tmp = -6.0 * x2
	else:
		tmp = -x1
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x2 <= -5.5e-125) || !(x2 <= 4.4e-96))
		tmp = Float64(-6.0 * x2);
	else
		tmp = Float64(-x1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x2 <= -5.5e-125) || ~((x2 <= 4.4e-96)))
		tmp = -6.0 * x2;
	else
		tmp = -x1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x2, -5.5e-125], N[Not[LessEqual[x2, 4.4e-96]], $MachinePrecision]], N[(-6.0 * x2), $MachinePrecision], (-x1)]

Derivation

1. Split input into 2 regimes

2. if x2 < -5.4999999999999997e-125 or 4.39999999999999959e-96 < x2

   1. Initial program 67.6%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 33.9

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

   5. Applied rewrites 33.9%

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

   if -5.4999999999999997e-125 < x2 < 4.39999999999999959e-96

   1. Initial program 66.3%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 63.2%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto -1 \cdot x1 \]
   10. Step-by-step derivation

   11. Applied rewrites 35.0%

       \[\leadsto -x1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x2 \leq -5.5 \cdot 10^{-125} \lor \neg \left(x2 \leq 4.4 \cdot 10^{-96}\right):\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;-x1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 13.8% accurate, 99.3× speedup

Math

\[\begin{array}{l} \\ -x1 \end{array} \]

FPCore

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

C

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

Fortran

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 = -x1
end function

Java

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

Python

def code(x1, x2):
	return -x1

Julia

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

MATLAB

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

Wolfram

code[x1_, x2_] := (-x1)

Derivation

1. Initial program 67.2%

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

3. Taylor expanded in x1 around 0

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in x2 around 0

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

8. Applied rewrites 42.8%

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

9. Taylor expanded in x1 around 0

   \[\leadsto -1 \cdot x1 \]
10. Step-by-step derivation

11. Applied rewrites 14.7%

    \[\leadsto -x1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025007 
(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))))))