ENA, Section 1.4, Exercise 4b, n=5

Percentage Accurate: 88.5% → 97.6%

Time: 4.3s

Alternatives: 17

Speedup: 0.5×

Specification

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

\[\begin{array}{l} \\ {\left(x + \varepsilon\right)}^{5} - {x}^{5} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 5.0) (pow x 5.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 5.0) - math.pow(x, 5.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + \varepsilon\right)}^{5} - {x}^{5} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 5.0) (pow x 5.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 5.0) - math.pow(x, 5.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-249}:\\ \;\;\;\;t\_0 - {\left(x \cdot x\right)}^{2.5}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -5e-249)
     (- t_0 (pow (* x x) 2.5))
     (if (<= t_1 0.0) (* (* 5.0 (pow x 4.0)) eps) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -5e-249) {
		tmp = t_0 - pow((x * x), 2.5);
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	} else {
		tmp = t_1;
	}
	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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    if (t_1 <= (-5d-249)) then
        tmp = t_0 - ((x * x) ** 2.5d0)
    else if (t_1 <= 0.0d0) then
        tmp = (5.0d0 * (x ** 4.0d0)) * eps
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double tmp;
	if (t_1 <= -5e-249) {
		tmp = t_0 - Math.pow((x * x), 2.5);
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * Math.pow(x, 4.0)) * eps;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	tmp = 0
	if t_1 <= -5e-249:
		tmp = t_0 - math.pow((x * x), 2.5)
	elif t_1 <= 0.0:
		tmp = (5.0 * math.pow(x, 4.0)) * eps
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -5e-249)
		tmp = Float64(t_0 - (Float64(x * x) ^ 2.5));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	tmp = 0.0;
	if (t_1 <= -5e-249)
		tmp = t_0 - ((x * x) ^ 2.5);
	elseif (t_1 <= 0.0)
		tmp = (5.0 * (x ^ 4.0)) * eps;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-249], N[(t$95$0 - N[Power[N[(x * x), $MachinePrecision], 2.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{5}} \]

      2. sqr-pow N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{\left(\frac{5}{2}\right)} \cdot {x}^{\left(\frac{5}{2}\right)}} \]

      3. pow-prod-down N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{\left(x \cdot x\right)}^{\left(\frac{5}{2}\right)}} \]

      4. unpow2 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {\color{blue}{\left({x}^{2}\right)}}^{\left(\frac{5}{2}\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{\left({x}^{2}\right)}^{\left(\frac{5}{2}\right)}} \]

      6. unpow2 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {\color{blue}{\left(x \cdot x\right)}}^{\left(\frac{5}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {\color{blue}{\left(x \cdot x\right)}}^{\left(\frac{5}{2}\right)} \]

      8. metadata-eval 100.0

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {\left(x \cdot x\right)}^{\color{blue}{2.5}} \]

   4. Applied rewrites 100.0%

      \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{\left(x \cdot x\right)}^{2.5}} \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. lower-pow.f64 99.9

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(5 \cdot {x}^{4}\right) \cdot \varepsilon} \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-249) (not (<= t_0 0.0)))
     t_0
     (* (* 5.0 (pow x 4.0)) eps))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	}
	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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-5d-249)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (5.0d0 * (x ** 4.0d0)) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (5.0 * Math.pow(x, 4.0)) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -5e-249) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (5.0 * math.pow(x, 4.0)) * eps
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -5e-249) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (5.0 * (x ^ 4.0)) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-249], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. lower-pow.f64 99.9

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(5 \cdot {x}^{4}\right) \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-249} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon} - -1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-249) (not (<= t_0 0.0)))
     (* (- (/ (* (fma 10.0 (/ x eps) 5.0) x) eps) -1.0) (pow eps 5.0))
     (* (* 5.0 (pow x 4.0)) eps))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0)) {
		tmp = (((fma(10.0, (x / eps), 5.0) * x) / eps) - -1.0) * pow(eps, 5.0);
	} else {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(Float64(fma(10.0, Float64(x / eps), 5.0) * x) / eps) - -1.0) * (eps ^ 5.0));
	else
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-249], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(N[(N[(10.0 * N[(x / eps), $MachinePrecision] + 5.0), $MachinePrecision] * x), $MachinePrecision] / eps), $MachinePrecision] - -1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto -\left(\left(-\frac{x \cdot \left(5 + 10 \cdot \frac{x}{\varepsilon}\right)}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -\left(\left(-\frac{\left(5 + 10 \cdot \frac{x}{\varepsilon}\right) \cdot x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5} \]

      2. lower-*.f64 N/A

         \[\leadsto -\left(\left(-\frac{\left(5 + 10 \cdot \frac{x}{\varepsilon}\right) \cdot x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5} \]

      3. +-commutative N/A

         \[\leadsto -\left(\left(-\frac{\left(10 \cdot \frac{x}{\varepsilon} + 5\right) \cdot x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-fma.f64 N/A

         \[\leadsto -\left(\left(-\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5} \]

      5. lift-/.f64 96.3

         \[\leadsto -\left(\left(-\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5} \]

   8. Applied rewrites 96.3%

      \[\leadsto -\left(\left(-\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5} \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. lower-pow.f64 99.9

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(5 \cdot {x}^{4}\right) \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-249} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon} - -1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon} \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-249) (not (<= t_0 0.0)))
     (* (/ (fma 5.0 x eps) eps) (pow eps 5.0))
     (* (* 5.0 (pow x 4.0)) eps))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0)) {
		tmp = (fma(5.0, x, eps) / eps) * pow(eps, 5.0);
	} else {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0))
		tmp = Float64(Float64(fma(5.0, x, eps) / eps) * (eps ^ 5.0));
	else
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-249], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(5.0 * x + eps), $MachinePrecision] / eps), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right) \cdot {\color{blue}{\varepsilon}}^{5} \]

      4. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot \frac{x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

      5. metadata-eval N/A

         \[\leadsto \left(5 \cdot \frac{x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\color{blue}{\varepsilon}}^{5} \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{5} \]

      8. lower-pow.f64 95.9

         \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{\color{blue}{5}} \]

   5. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \frac{\varepsilon + 5 \cdot x}{\varepsilon} \cdot {\color{blue}{\varepsilon}}^{5} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\varepsilon + 5 \cdot x}{\varepsilon} \cdot {\varepsilon}^{5} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\varepsilon + \left(4 + 1\right) \cdot x}{\varepsilon} \cdot {\varepsilon}^{5} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{\varepsilon + \left(x + 4 \cdot x\right)}{\varepsilon} \cdot {\varepsilon}^{5} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(x + 4 \cdot x\right) + \varepsilon}{\varepsilon} \cdot {\varepsilon}^{5} \]

      5. distribute-rgt1-in N/A

         \[\leadsto \frac{\left(4 + 1\right) \cdot x + \varepsilon}{\varepsilon} \cdot {\varepsilon}^{5} \]

      6. metadata-eval N/A

         \[\leadsto \frac{5 \cdot x + \varepsilon}{\varepsilon} \cdot {\varepsilon}^{5} \]

      7. lower-fma.f64 96.0

         \[\leadsto \frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon} \cdot {\varepsilon}^{5} \]

   8. Applied rewrites 96.0%

      \[\leadsto \frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon} \cdot {\color{blue}{\varepsilon}}^{5} \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. lower-pow.f64 99.9

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(5 \cdot {x}^{4}\right) \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-249} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon} \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon + 5 \cdot x, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon} \cdot {\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-249)
     (*
      (- (fma (+ eps (* 5.0 x)) eps (* (* x x) 6.0)) (* (* x x) -4.0))
      (pow eps 3.0))
     (if (<= t_0 0.0)
       (* (* 5.0 (pow x 4.0)) eps)
       (* (/ (fma 5.0 x eps) eps) (pow eps 5.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-249) {
		tmp = (fma((eps + (5.0 * x)), eps, ((x * x) * 6.0)) - ((x * x) * -4.0)) * pow(eps, 3.0);
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	} else {
		tmp = (fma(5.0, x, eps) / eps) * pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-249)
		tmp = Float64(Float64(fma(Float64(eps + Float64(5.0 * x)), eps, Float64(Float64(x * x) * 6.0)) - Float64(Float64(x * x) * -4.0)) * (eps ^ 3.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	else
		tmp = Float64(Float64(fma(5.0, x, eps) / eps) * (eps ^ 5.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-249], N[(N[(N[(N[(eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(5.0 * x + eps), $MachinePrecision] / eps), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 97.8%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. lower-pow.f64 99.9

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(5 \cdot {x}^{4}\right) \cdot \varepsilon} \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right) \cdot {\color{blue}{\varepsilon}}^{5} \]

      4. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot \frac{x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

      5. metadata-eval N/A

         \[\leadsto \left(5 \cdot \frac{x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\color{blue}{\varepsilon}}^{5} \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{5} \]

      8. lower-pow.f64 95.3

         \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{\color{blue}{5}} \]

   5. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \frac{\varepsilon + 5 \cdot x}{\varepsilon} \cdot {\color{blue}{\varepsilon}}^{5} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\varepsilon + 5 \cdot x}{\varepsilon} \cdot {\varepsilon}^{5} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\varepsilon + \left(4 + 1\right) \cdot x}{\varepsilon} \cdot {\varepsilon}^{5} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{\varepsilon + \left(x + 4 \cdot x\right)}{\varepsilon} \cdot {\varepsilon}^{5} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(x + 4 \cdot x\right) + \varepsilon}{\varepsilon} \cdot {\varepsilon}^{5} \]

      5. distribute-rgt1-in N/A

         \[\leadsto \frac{\left(4 + 1\right) \cdot x + \varepsilon}{\varepsilon} \cdot {\varepsilon}^{5} \]

      6. metadata-eval N/A

         \[\leadsto \frac{5 \cdot x + \varepsilon}{\varepsilon} \cdot {\varepsilon}^{5} \]

      7. lower-fma.f64 95.4

         \[\leadsto \frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon} \cdot {\varepsilon}^{5} \]

   8. Applied rewrites 95.4%

      \[\leadsto \frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon} \cdot {\color{blue}{\varepsilon}}^{5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon + 5 \cdot x, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon} \cdot {\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-249)
     (*
      (-
       (*
        (+ (fma (/ x eps) 5.0 1.0) (/ (* (* x x) 6.0) (* eps eps)))
        (* eps eps))
       (* (* x x) -4.0))
      (* (* eps eps) eps))
     (if (<= t_0 0.0)
       (* (* 5.0 (pow x 4.0)) eps)
       (* (fma 5.0 x eps) (pow eps 4.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-249) {
		tmp = (((fma((x / eps), 5.0, 1.0) + (((x * x) * 6.0) / (eps * eps))) * (eps * eps)) - ((x * x) * -4.0)) * ((eps * eps) * eps);
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	} else {
		tmp = fma(5.0, x, eps) * pow(eps, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-249)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(x / eps), 5.0, 1.0) + Float64(Float64(Float64(x * x) * 6.0) / Float64(eps * eps))) * Float64(eps * eps)) - Float64(Float64(x * x) * -4.0)) * Float64(Float64(eps * eps) * eps));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	else
		tmp = Float64(fma(5.0, x, eps) * (eps ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-249], N[(N[(N[(N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * 6.0), $MachinePrecision] / N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[(N[(5.0 * x + eps), $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 97.8%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 97.0

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 97.0%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   11. Taylor expanded in eps around inf

       \[\leadsto \left({\varepsilon}^{2} \cdot \left(1 + \left(5 \cdot \frac{x}{\varepsilon} + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right)\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(1 + \left(5 \cdot \frac{x}{\varepsilon} + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right)\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(1 + \left(5 \cdot \frac{x}{\varepsilon} + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right)\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       3. associate-+r+ N/A

          \[\leadsto \left(\left(\left(1 + 5 \cdot \frac{x}{\varepsilon}\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       4. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + 5 \cdot \frac{x}{\varepsilon}\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\left(5 \cdot \frac{x}{\varepsilon} + 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{x}{\varepsilon} \cdot 5 + 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       9. associate-*r/ N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{6 \cdot {x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       10. lower-/.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{6 \cdot {x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       11. *-commutative N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{{x}^{2} \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       12. pow2 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       14. lift-*.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       15. pow2 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       17. pow2 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       18. lift-*.f64 97.0

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   13. Applied rewrites 97.0%

       \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. lower-pow.f64 99.9

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(5 \cdot {x}^{4}\right) \cdot \varepsilon} \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right) \cdot {\color{blue}{\varepsilon}}^{5} \]

      4. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot \frac{x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

      5. metadata-eval N/A

         \[\leadsto \left(5 \cdot \frac{x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\color{blue}{\varepsilon}}^{5} \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{5} \]

      8. lower-pow.f64 95.3

         \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{\color{blue}{5}} \]

   5. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\left(\varepsilon + 5 \cdot x\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\varepsilon + 5 \cdot x\right) \cdot {\varepsilon}^{\color{blue}{4}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\varepsilon + 5 \cdot x\right) \cdot {\varepsilon}^{\color{blue}{4}} \]

      3. metadata-eval N/A

         \[\leadsto \left(\varepsilon + \left(4 + 1\right) \cdot x\right) \cdot {\varepsilon}^{4} \]

      4. distribute-rgt1-in N/A

         \[\leadsto \left(\varepsilon + \left(x + 4 \cdot x\right)\right) \cdot {\varepsilon}^{4} \]

      5. +-commutative N/A

         \[\leadsto \left(\left(x + 4 \cdot x\right) + \varepsilon\right) \cdot {\varepsilon}^{4} \]

      6. distribute-rgt1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot x + \varepsilon\right) \cdot {\varepsilon}^{4} \]

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{4} \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{4} \]

      9. lower-pow.f64 94.9

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{4} \]

   8. Applied rewrites 94.9%

      \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{4}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-249)
     (*
      (-
       (*
        (+ (fma (/ x eps) 5.0 1.0) (/ (* (* x x) 6.0) (* eps eps)))
        (* eps eps))
       (* (* x x) -4.0))
      (* (* eps eps) eps))
     (if (<= t_0 0.0) (* (* 5.0 (pow x 4.0)) eps) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-249) {
		tmp = (((fma((x / eps), 5.0, 1.0) + (((x * x) * 6.0) / (eps * eps))) * (eps * eps)) - ((x * x) * -4.0)) * ((eps * eps) * eps);
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-249)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(x / eps), 5.0, 1.0) + Float64(Float64(Float64(x * x) * 6.0) / Float64(eps * eps))) * Float64(eps * eps)) - Float64(Float64(x * x) * -4.0)) * Float64(Float64(eps * eps) * eps));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-249], N[(N[(N[(N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * 6.0), $MachinePrecision] / N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 97.8%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 97.0

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 97.0%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   11. Taylor expanded in eps around inf

       \[\leadsto \left({\varepsilon}^{2} \cdot \left(1 + \left(5 \cdot \frac{x}{\varepsilon} + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right)\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(1 + \left(5 \cdot \frac{x}{\varepsilon} + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right)\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(1 + \left(5 \cdot \frac{x}{\varepsilon} + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right)\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       3. associate-+r+ N/A

          \[\leadsto \left(\left(\left(1 + 5 \cdot \frac{x}{\varepsilon}\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       4. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + 5 \cdot \frac{x}{\varepsilon}\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\left(5 \cdot \frac{x}{\varepsilon} + 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{x}{\varepsilon} \cdot 5 + 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       9. associate-*r/ N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{6 \cdot {x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       10. lower-/.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{6 \cdot {x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       11. *-commutative N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{{x}^{2} \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       12. pow2 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       14. lift-*.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       15. pow2 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       17. pow2 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       18. lift-*.f64 97.0

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   13. Applied rewrites 97.0%

       \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. lower-pow.f64 99.9

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(5 \cdot {x}^{4}\right) \cdot \varepsilon} \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 94.9

         \[\leadsto {\varepsilon}^{\color{blue}{5}} \]

   5. Applied rewrites 94.9%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-249)
     (*
      (-
       (*
        (+ (fma (/ x eps) 5.0 1.0) (/ (* (* x x) 6.0) (* eps eps)))
        (* eps eps))
       (* (* x x) -4.0))
      (* (* eps eps) eps))
     (if (<= t_0 0.0)
       (* (* (fma (* eps x) 5.0 (* (* eps eps) 10.0)) x) (* x x))
       (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-249) {
		tmp = (((fma((x / eps), 5.0, 1.0) + (((x * x) * 6.0) / (eps * eps))) * (eps * eps)) - ((x * x) * -4.0)) * ((eps * eps) * eps);
	} else if (t_0 <= 0.0) {
		tmp = (fma((eps * x), 5.0, ((eps * eps) * 10.0)) * x) * (x * x);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-249)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(x / eps), 5.0, 1.0) + Float64(Float64(Float64(x * x) * 6.0) / Float64(eps * eps))) * Float64(eps * eps)) - Float64(Float64(x * x) * -4.0)) * Float64(Float64(eps * eps) * eps));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)) * x) * Float64(x * x));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-249], N[(N[(N[(N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * 6.0), $MachinePrecision] / N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 97.8%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 97.0

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 97.0%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   11. Taylor expanded in eps around inf

       \[\leadsto \left({\varepsilon}^{2} \cdot \left(1 + \left(5 \cdot \frac{x}{\varepsilon} + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right)\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(1 + \left(5 \cdot \frac{x}{\varepsilon} + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right)\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(1 + \left(5 \cdot \frac{x}{\varepsilon} + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right)\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       3. associate-+r+ N/A

          \[\leadsto \left(\left(\left(1 + 5 \cdot \frac{x}{\varepsilon}\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       4. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + 5 \cdot \frac{x}{\varepsilon}\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\left(5 \cdot \frac{x}{\varepsilon} + 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{x}{\varepsilon} \cdot 5 + 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       9. associate-*r/ N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{6 \cdot {x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       10. lower-/.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{6 \cdot {x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       11. *-commutative N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{{x}^{2} \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       12. pow2 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       14. lift-*.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       15. pow2 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       17. pow2 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       18. lift-*.f64 97.0

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   13. Applied rewrites 97.0%

       \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(10, {\varepsilon}^{3}, \mathsf{fma}\left(5 \cdot \varepsilon, x, --10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(\varepsilon \cdot \left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\left({x}^{2} \cdot 5 + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({x}^{2}, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       5. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       9. lower-*.f64 99.9

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right) \cdot \left(x \cdot x\right) \]

   14. Applied rewrites 99.9%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right) \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 94.9

         \[\leadsto {\varepsilon}^{\color{blue}{5}} \]

   5. Applied rewrites 94.9%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-249) (not (<= t_0 0.0)))
     (*
      (-
       (fma (* (fma (/ x eps) 5.0 1.0) eps) eps (* (* x x) 6.0))
       (* (* x x) -4.0))
      (* (* eps eps) eps))
     (* (* (fma (* eps x) 5.0 (* (* eps eps) 10.0)) x) (* x x)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0)) {
		tmp = (fma((fma((x / eps), 5.0, 1.0) * eps), eps, ((x * x) * 6.0)) - ((x * x) * -4.0)) * ((eps * eps) * eps);
	} else {
		tmp = (fma((eps * x), 5.0, ((eps * eps) * 10.0)) * x) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0))
		tmp = Float64(Float64(fma(Float64(fma(Float64(x / eps), 5.0, 1.0) * eps), eps, Float64(Float64(x * x) * 6.0)) - Float64(Float64(x * x) * -4.0)) * Float64(Float64(eps * eps) * eps));
	else
		tmp = Float64(Float64(fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)) * x) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-249], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] * eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.9%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 95.3

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 95.3%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   11. Taylor expanded in eps around inf

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(1 + 5 \cdot \frac{x}{\varepsilon}\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(1 + 5 \cdot \frac{x}{\varepsilon}\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(5 \cdot \frac{x}{\varepsilon} + 1\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{x}{\varepsilon} \cdot 5 + 1\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       6. lower-/.f64 95.4

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   13. Applied rewrites 95.4%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(10, {\varepsilon}^{3}, \mathsf{fma}\left(5 \cdot \varepsilon, x, --10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(\varepsilon \cdot \left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\left({x}^{2} \cdot 5 + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({x}^{2}, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       5. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       9. lower-*.f64 99.9

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right) \cdot \left(x \cdot x\right) \]

   14. Applied rewrites 99.9%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-249} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right)\\ t_1 := \left(x \cdot x\right) \cdot -4\\ t_2 := \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\\ t_3 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_4 := \left(x \cdot x\right) \cdot 6\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(t\_0 + \frac{t\_4}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_0 \cdot \varepsilon, \varepsilon, t\_4\right) - t\_1\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (/ x eps) 5.0 1.0))
        (t_1 (* (* x x) -4.0))
        (t_2 (* (* eps eps) eps))
        (t_3 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_4 (* (* x x) 6.0)))
   (if (<= t_3 -5e-249)
     (* (- (* (+ t_0 (/ t_4 (* eps eps))) (* eps eps)) t_1) t_2)
     (if (<= t_3 0.0)
       (* (* (fma (* eps x) 5.0 (* (* eps eps) 10.0)) x) (* x x))
       (* (- (fma (* t_0 eps) eps t_4) t_1) t_2)))))

C

double code(double x, double eps) {
	double t_0 = fma((x / eps), 5.0, 1.0);
	double t_1 = (x * x) * -4.0;
	double t_2 = (eps * eps) * eps;
	double t_3 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_4 = (x * x) * 6.0;
	double tmp;
	if (t_3 <= -5e-249) {
		tmp = (((t_0 + (t_4 / (eps * eps))) * (eps * eps)) - t_1) * t_2;
	} else if (t_3 <= 0.0) {
		tmp = (fma((eps * x), 5.0, ((eps * eps) * 10.0)) * x) * (x * x);
	} else {
		tmp = (fma((t_0 * eps), eps, t_4) - t_1) * t_2;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(Float64(x / eps), 5.0, 1.0)
	t_1 = Float64(Float64(x * x) * -4.0)
	t_2 = Float64(Float64(eps * eps) * eps)
	t_3 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_4 = Float64(Float64(x * x) * 6.0)
	tmp = 0.0
	if (t_3 <= -5e-249)
		tmp = Float64(Float64(Float64(Float64(t_0 + Float64(t_4 / Float64(eps * eps))) * Float64(eps * eps)) - t_1) * t_2);
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)) * x) * Float64(x * x));
	else
		tmp = Float64(Float64(fma(Float64(t_0 * eps), eps, t_4) - t_1) * t_2);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * -4.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x * x), $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-249], N[(N[(N[(N[(t$95$0 + N[(t$95$4 / N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * eps), $MachinePrecision] * eps + t$95$4), $MachinePrecision] - t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 97.8%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 97.0

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 97.0%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   11. Taylor expanded in eps around inf

       \[\leadsto \left({\varepsilon}^{2} \cdot \left(1 + \left(5 \cdot \frac{x}{\varepsilon} + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right)\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(1 + \left(5 \cdot \frac{x}{\varepsilon} + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right)\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(1 + \left(5 \cdot \frac{x}{\varepsilon} + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right)\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       3. associate-+r+ N/A

          \[\leadsto \left(\left(\left(1 + 5 \cdot \frac{x}{\varepsilon}\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       4. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + 5 \cdot \frac{x}{\varepsilon}\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\left(5 \cdot \frac{x}{\varepsilon} + 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{x}{\varepsilon} \cdot 5 + 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + 6 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       9. associate-*r/ N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{6 \cdot {x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       10. lower-/.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{6 \cdot {x}^{2}}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       11. *-commutative N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{{x}^{2} \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       12. pow2 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       14. lift-*.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{{\varepsilon}^{2}}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       15. pow2 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot {\varepsilon}^{2} - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       17. pow2 N/A

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       18. lift-*.f64 97.0

           \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   13. Applied rewrites 97.0%

       \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) + \frac{\left(x \cdot x\right) \cdot 6}{\varepsilon \cdot \varepsilon}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(10, {\varepsilon}^{3}, \mathsf{fma}\left(5 \cdot \varepsilon, x, --10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(\varepsilon \cdot \left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\left({x}^{2} \cdot 5 + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({x}^{2}, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       5. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       9. lower-*.f64 99.9

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right) \cdot \left(x \cdot x\right) \]

   14. Applied rewrites 99.9%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right) \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.0%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 94.5

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 94.5%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   11. Taylor expanded in eps around inf

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(1 + 5 \cdot \frac{x}{\varepsilon}\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(1 + 5 \cdot \frac{x}{\varepsilon}\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(5 \cdot \frac{x}{\varepsilon} + 1\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{x}{\varepsilon} \cdot 5 + 1\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       6. lower-/.f64 94.5

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   13. Applied rewrites 94.5%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \varepsilon, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon + 5 \cdot x, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-249)
     (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) (* eps eps)) eps)
     (if (<= t_0 0.0)
       (* (* (fma (* eps x) 5.0 (* (* eps eps) 10.0)) x) (* x x))
       (*
        (- (fma (+ eps (* 5.0 x)) eps (* (* x x) 6.0)) (* (* x x) -4.0))
        (* (* eps eps) eps))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-249) {
		tmp = (fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * (eps * eps)) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (fma((eps * x), 5.0, ((eps * eps) * 10.0)) * x) * (x * x);
	} else {
		tmp = (fma((eps + (5.0 * x)), eps, ((x * x) * 6.0)) - ((x * x) * -4.0)) * ((eps * eps) * eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-249)
		tmp = Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * Float64(eps * eps)) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)) * x) * Float64(x * x));
	else
		tmp = Float64(Float64(fma(Float64(eps + Float64(5.0 * x)), eps, Float64(Float64(x * x) * 6.0)) - Float64(Float64(x * x) * -4.0)) * Float64(Float64(eps * eps) * eps));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-249], N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 97.8%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 97.0

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 97.0%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   12. Applied rewrites 97.0%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(10, {\varepsilon}^{3}, \mathsf{fma}\left(5 \cdot \varepsilon, x, --10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(\varepsilon \cdot \left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\left({x}^{2} \cdot 5 + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({x}^{2}, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       5. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       9. lower-*.f64 99.9

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right) \cdot \left(x \cdot x\right) \]

   14. Applied rewrites 99.9%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right) \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.0%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 94.5

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 94.5%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon + 5 \cdot x, \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-249) (not (<= t_0 0.0)))
     (* (fma (* eps x) 5.0 (* eps eps)) (* (* eps eps) eps))
     (* (* (fma (* eps x) 5.0 (* (* eps eps) 10.0)) x) (* x x)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0)) {
		tmp = fma((eps * x), 5.0, (eps * eps)) * ((eps * eps) * eps);
	} else {
		tmp = (fma((eps * x), 5.0, ((eps * eps) * 10.0)) * x) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0))
		tmp = Float64(fma(Float64(eps * x), 5.0, Float64(eps * eps)) * Float64(Float64(eps * eps) * eps));
	else
		tmp = Float64(Float64(fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)) * x) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-249], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.9%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 95.3

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 95.3%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(5 \cdot \left(\varepsilon \cdot x\right) + {\varepsilon}^{2}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\varepsilon \cdot x\right) \cdot 5 + {\varepsilon}^{2}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, {\varepsilon}^{2}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, {\varepsilon}^{2}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       5. lift-*.f64 95.0

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   13. Applied rewrites 95.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(10, {\varepsilon}^{3}, \mathsf{fma}\left(5 \cdot \varepsilon, x, --10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(\varepsilon \cdot \left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\left({x}^{2} \cdot 5 + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({x}^{2}, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       5. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       9. lower-*.f64 99.9

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right) \cdot \left(x \cdot x\right) \]

   14. Applied rewrites 99.9%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-249} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-249)
     (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) (* eps eps)) eps)
     (if (<= t_0 0.0)
       (* (* (fma (* eps x) 5.0 (* (* eps eps) 10.0)) x) (* x x))
       (* (fma (* eps x) 5.0 (* eps eps)) (* (* eps eps) eps))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-249) {
		tmp = (fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * (eps * eps)) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (fma((eps * x), 5.0, ((eps * eps) * 10.0)) * x) * (x * x);
	} else {
		tmp = fma((eps * x), 5.0, (eps * eps)) * ((eps * eps) * eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-249)
		tmp = Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * Float64(eps * eps)) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)) * x) * Float64(x * x));
	else
		tmp = Float64(fma(Float64(eps * x), 5.0, Float64(eps * eps)) * Float64(Float64(eps * eps) * eps));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-249], N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 97.8%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 97.0

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 97.0%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   12. Applied rewrites 97.0%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(10, {\varepsilon}^{3}, \mathsf{fma}\left(5 \cdot \varepsilon, x, --10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(\varepsilon \cdot \left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\left({x}^{2} \cdot 5 + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({x}^{2}, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       5. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

       9. lower-*.f64 99.9

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right) \cdot \left(x \cdot x\right) \]

   14. Applied rewrites 99.9%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \left(x \cdot x\right) \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.0%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 94.5

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 94.5%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(5 \cdot \left(\varepsilon \cdot x\right) + {\varepsilon}^{2}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\varepsilon \cdot x\right) \cdot 5 + {\varepsilon}^{2}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, {\varepsilon}^{2}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, {\varepsilon}^{2}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       5. lift-*.f64 94.4

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   13. Applied rewrites 94.4%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-249) (not (<= t_0 0.0)))
     (* (fma (* eps x) 5.0 (* eps eps)) (* (* eps eps) eps))
     (* (* (* (* x x) 5.0) (* x x)) eps))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0)) {
		tmp = fma((eps * x), 5.0, (eps * eps)) * ((eps * eps) * eps);
	} else {
		tmp = (((x * x) * 5.0) * (x * x)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0))
		tmp = Float64(fma(Float64(eps * x), 5.0, Float64(eps * eps)) * Float64(Float64(eps * eps) * eps));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * Float64(x * x)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-249], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.9%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 95.3

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 95.3%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(5 \cdot \left(\varepsilon \cdot x\right) + {\varepsilon}^{2}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\varepsilon \cdot x\right) \cdot 5 + {\varepsilon}^{2}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, {\varepsilon}^{2}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, {\varepsilon}^{2}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       5. lift-*.f64 95.0

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   13. Applied rewrites 95.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. lower-pow.f64 99.9

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(5 \cdot {x}^{4}\right) \cdot \varepsilon} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      2. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{\left(2 + 2\right)}\right) \cdot \varepsilon \]

      3. pow-prod-up N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      4. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      5. pow2 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      6. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      7. pow2 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      8. lift-*.f64 99.9

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   7. Applied rewrites 99.9%

      \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      2. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      3. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      4. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      5. pow2 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      6. pow2 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(5 \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

      9. *-commutative N/A

         \[\leadsto \left(\left({x}^{2} \cdot 5\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left({x}^{2} \cdot 5\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

      11. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

      13. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]

      14. lift-*.f64 99.9

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]

   9. Applied rewrites 99.9%

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-249} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot x, 5, \varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-249) (not (<= t_0 0.0)))
     (* (* eps eps) (* (* eps eps) eps))
     (* (* (* (* x x) 5.0) (* x x)) eps))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0)) {
		tmp = (eps * eps) * ((eps * eps) * eps);
	} else {
		tmp = (((x * x) * 5.0) * (x * x)) * eps;
	}
	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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-5d-249)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = (eps * eps) * ((eps * eps) * eps)
    else
        tmp = (((x * x) * 5.0d0) * (x * x)) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0)) {
		tmp = (eps * eps) * ((eps * eps) * eps);
	} else {
		tmp = (((x * x) * 5.0) * (x * x)) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -5e-249) or not (t_0 <= 0.0):
		tmp = (eps * eps) * ((eps * eps) * eps)
	else:
		tmp = (((x * x) * 5.0) * (x * x)) * eps
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0))
		tmp = Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * Float64(x * x)) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -5e-249) || ~((t_0 <= 0.0)))
		tmp = (eps * eps) * ((eps * eps) * eps);
	else
		tmp = (((x * x) * 5.0) * (x * x)) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-249], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.9%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 95.3

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 95.3%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto {\varepsilon}^{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   12. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       2. lift-*.f64 94.5

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   13. Applied rewrites 94.5%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. lower-pow.f64 99.9

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(5 \cdot {x}^{4}\right) \cdot \varepsilon} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      2. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{\left(2 + 2\right)}\right) \cdot \varepsilon \]

      3. pow-prod-up N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      4. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      5. pow2 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      6. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      7. pow2 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      8. lift-*.f64 99.9

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   7. Applied rewrites 99.9%

      \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      2. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      3. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      4. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      5. pow2 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      6. pow2 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(5 \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

      9. *-commutative N/A

         \[\leadsto \left(\left({x}^{2} \cdot 5\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left({x}^{2} \cdot 5\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

      11. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

      13. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]

      14. lift-*.f64 99.9

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]

   9. Applied rewrites 99.9%

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-249} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-249} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(\varepsilon \cdot x\right)\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-249) (not (<= t_0 0.0)))
     (* (* eps eps) (* (* eps eps) eps))
     (* (* (* x (* eps x)) 5.0) (* x x)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0)) {
		tmp = (eps * eps) * ((eps * eps) * eps);
	} else {
		tmp = ((x * (eps * x)) * 5.0) * (x * x);
	}
	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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-5d-249)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = (eps * eps) * ((eps * eps) * eps)
    else
        tmp = ((x * (eps * x)) * 5.0d0) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0)) {
		tmp = (eps * eps) * ((eps * eps) * eps);
	} else {
		tmp = ((x * (eps * x)) * 5.0) * (x * x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -5e-249) or not (t_0 <= 0.0):
		tmp = (eps * eps) * ((eps * eps) * eps)
	else:
		tmp = ((x * (eps * x)) * 5.0) * (x * x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-249) || !(t_0 <= 0.0))
		tmp = Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps));
	else
		tmp = Float64(Float64(Float64(x * Float64(eps * x)) * 5.0) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -5e-249) || ~((t_0 <= 0.0)))
		tmp = (eps * eps) * ((eps * eps) * eps);
	else
		tmp = ((x * (eps * x)) * 5.0) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-249], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(eps * x), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999999e-249 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.9%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 95.3

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   10. Applied rewrites 95.3%

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto {\varepsilon}^{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   12. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

       2. lift-*.f64 94.5

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   13. Applied rewrites 94.5%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   if -4.9999999999999999e-249 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(10, {\varepsilon}^{3}, \mathsf{fma}\left(5 \cdot \varepsilon, x, --10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\varepsilon \cdot {x}^{2}\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\varepsilon \cdot {x}^{2}\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\left({x}^{2} \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left({x}^{2} \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]

       5. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]

       6. lift-*.f64 99.8

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]

   11. Applied rewrites 99.8%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\left(x \cdot \left(x \cdot \varepsilon\right)\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \left(\varepsilon \cdot x\right)\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \left(\varepsilon \cdot x\right)\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]

       6. lift-*.f64 99.9

          \[\leadsto \left(\left(x \cdot \left(\varepsilon \cdot x\right)\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]

   13. Applied rewrites 99.9%

       \[\leadsto \left(\left(x \cdot \left(\varepsilon \cdot x\right)\right) \cdot 5\right) \cdot \left(x \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-249} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(\varepsilon \cdot x\right)\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 87.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* eps eps) (* (* eps eps) eps)))

C

double code(double x, double eps) {
	return (eps * eps) * ((eps * eps) * eps);
}

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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * eps) * ((eps * eps) * eps)
end function

Java

public static double code(double x, double eps) {
	return (eps * eps) * ((eps * eps) * eps);
}

Python

def code(x, eps):
	return (eps * eps) * ((eps * eps) * eps)

Julia

function code(x, eps)
	return Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * eps) * ((eps * eps) * eps);
end

Wolfram

code[x_, eps_] := N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.5%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in eps around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right) \]

   2. lower-neg.f64 N/A

      \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

   3. *-commutative N/A

      \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   4. lower-*.f64 N/A

      \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

5. Applied rewrites 78.2%

   \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

6. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 88.8%

   \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

   2. unpow3 N/A

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   3. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

   5. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   6. lift-*.f64 88.7

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

10. Applied rewrites 88.7%

    \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

11. Taylor expanded in x around 0

    \[\leadsto {\varepsilon}^{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
12. Step-by-step derivation

    1. pow2 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

    2. lift-*.f64 88.4

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

13. Applied rewrites 88.4%

    \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025061 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=5"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 5.0) (pow x 5.0)))