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

Percentage Accurate: 88.2% → 99.3%

Time: 4.5s

Alternatives: 10

Speedup: 0.4×

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: 88.2% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_0 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -2e-305)
     (- t_0 (* (* (* x x) (* x x)) x))
     (if (<= t_1 0.0) (* (* 5.0 eps) (pow x 4.0)) 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 <= -2e-305) {
		tmp = t_0 - (((x * x) * (x * x)) * x);
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} 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 <= (-2d-305)) then
        tmp = t_0 - (((x * x) * (x * x)) * x)
    else if (t_1 <= 0.0d0) then
        tmp = (5.0d0 * eps) * (x ** 4.0d0)
    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 <= -2e-305) {
		tmp = t_0 - (((x * x) * (x * x)) * x);
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * eps) * Math.pow(x, 4.0);
	} 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 <= -2e-305:
		tmp = t_0 - (((x * x) * (x * x)) * x)
	elif t_1 <= 0.0:
		tmp = (5.0 * eps) * math.pow(x, 4.0)
	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 <= -2e-305)
		tmp = Float64(t_0 - Float64(Float64(Float64(x * x) * Float64(x * x)) * x));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	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 <= -2e-305)
		tmp = t_0 - (((x * x) * (x * x)) * x);
	elseif (t_1 <= 0.0)
		tmp = (5.0 * eps) * (x ^ 4.0);
	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, -2e-305], N[(t$95$0 - N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $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))) < -1.99999999999999999e-305

   1. Initial program 97.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus-rev N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 97.4

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

   3. Applied rewrites 97.4%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{\color{blue}{2}}\right)}\right) \]

      8. sqr-pow N/A

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

      9. lift-pow.f64 99.6

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

   6. Applied rewrites 99.6%

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

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

   1. Initial program 98.3%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (- t_0 (* (* (* x x) (* x x)) x))))
   (if (<= t_1 -2e-305)
     t_2
     (if (<= t_1 0.0) (* (* 5.0 eps) (pow x 4.0)) t_2))))

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 t_2 = t_0 - (((x * x) * (x * x)) * x);
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    t_2 = t_0 - (((x * x) * (x * x)) * x)
    if (t_1 <= (-2d-305)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = (5.0d0 * eps) * (x ** 4.0d0)
    else
        tmp = t_2
    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 t_2 = t_0 - (((x * x) * (x * x)) * x);
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * eps) * Math.pow(x, 4.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	t_2 = t_0 - (((x * x) * (x * x)) * x)
	tmp = 0
	if t_1 <= -2e-305:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = (5.0 * eps) * math.pow(x, 4.0)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = Float64(t_0 - Float64(Float64(Float64(x * x) * Float64(x * x)) * x))
	tmp = 0.0
	if (t_1 <= -2e-305)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	t_2 = t_0 - (((x * x) * (x * x)) * x);
	tmp = 0.0;
	if (t_1 <= -2e-305)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = (5.0 * eps) * (x ^ 4.0);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(t$95$0 - N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-305], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus-rev N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 97.9

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

   3. Applied rewrites 97.9%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{\color{blue}{2}}\right)}\right) \]

      8. sqr-pow N/A

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

      9. lift-pow.f64 99.6

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

   6. Applied rewrites 99.6%

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

Alternative 3: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot \left(\frac{\varepsilon}{x} + 5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* eps eps) (* eps eps))))
   (if (<= t_0 -2e-305)
     (* t_1 (* x (+ (/ eps x) 5.0)))
     (if (<= t_0 0.0) (* (* 5.0 eps) (pow x 4.0)) (* t_1 (fma 5.0 x eps))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * eps);
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1 * (x * ((eps / x) + 5.0));
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = t_1 * fma(5.0, x, eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * Float64(eps * eps))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = Float64(t_1 * Float64(x * Float64(Float64(eps / x) + 5.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = Float64(t_1 * fma(5.0, 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]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], N[(t$95$1 * N[(x * N[(N[(eps / x), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(5.0 * x + 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))) < -1.99999999999999999e-305

   1. Initial program 97.4%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   3. 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. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 92.8

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot \color{blue}{x}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot \color{blue}{x}\right) \]

      12. metadata-eval N/A

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

      13. lower-*.f64 92.8

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

   10. Applied rewrites 92.8%

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

   11. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-/.f64 92.7

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

   13. Applied rewrites 92.7%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{\color{blue}{2}}\right)}\right) \]

      8. sqr-pow N/A

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

      9. lift-pow.f64 99.6

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

   6. Applied rewrites 99.6%

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

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

   1. Initial program 98.3%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   3. 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. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 94.1

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 94.0

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

   7. Applied rewrites 94.0%

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

   8. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot \color{blue}{x}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot \color{blue}{x}\right) \]

      12. metadata-eval N/A

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

      13. lower-*.f64 94.1

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

   10. Applied rewrites 94.1%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. metadata-eval N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot x\right) \]

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

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

       5. metadata-eval N/A

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

       6. distribute-rgt1-in N/A

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

       7. +-commutative N/A

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

       8. distribute-rgt1-in N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 94.1

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

   12. Applied rewrites 94.1%

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

Alternative 4: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot \left(\frac{\varepsilon}{x} + 5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot \left(t\_1 \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* eps eps) (* eps eps))))
   (if (<= t_0 -2e-305)
     (* t_1 (* x (+ (/ eps x) 5.0)))
     (if (<= t_0 0.0)
       (* (* 5.0 eps) (pow x 4.0))
       (* (fma 5.0 (/ x eps) 1.0) (* t_1 eps))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * eps);
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1 * (x * ((eps / x) + 5.0));
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = fma(5.0, (x / eps), 1.0) * (t_1 * eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * Float64(eps * eps))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = Float64(t_1 * Float64(x * Float64(Float64(eps / x) + 5.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = Float64(fma(5.0, Float64(x / eps), 1.0) * Float64(t_1 * 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]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], N[(t$95$1 * N[(x * N[(N[(eps / x), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$1 * 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))) < -1.99999999999999999e-305

   1. Initial program 97.4%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   3. 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. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 92.8

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot \color{blue}{x}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot \color{blue}{x}\right) \]

      12. metadata-eval N/A

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

      13. lower-*.f64 92.8

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

   10. Applied rewrites 92.8%

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

   11. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-/.f64 92.7

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

   13. Applied rewrites 92.7%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{\color{blue}{2}}\right)}\right) \]

      8. sqr-pow N/A

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

      9. lift-pow.f64 99.6

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

   6. Applied rewrites 99.6%

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

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

   1. Initial program 98.3%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   3. 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. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 94.1

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

   4. Applied rewrites 94.1%

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

Alternative 5: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot \left(\frac{\varepsilon}{x} + 5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* eps eps) (* eps eps))))
   (if (<= t_0 -2e-305)
     (* t_1 (* x (+ (/ eps x) 5.0)))
     (if (<= t_0 0.0)
       (* (* 5.0 eps) (* (* (* x x) x) x))
       (* t_1 (fma 5.0 x eps))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * eps);
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1 * (x * ((eps / x) + 5.0));
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * (((x * x) * x) * x);
	} else {
		tmp = t_1 * fma(5.0, x, eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * Float64(eps * eps))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = Float64(t_1 * Float64(x * Float64(Float64(eps / x) + 5.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * Float64(Float64(Float64(x * x) * x) * x));
	else
		tmp = Float64(t_1 * fma(5.0, 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]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], N[(t$95$1 * N[(x * N[(N[(eps / x), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(5.0 * x + 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))) < -1.99999999999999999e-305

   1. Initial program 97.4%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   3. 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. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 92.8

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot \color{blue}{x}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot \color{blue}{x}\right) \]

      12. metadata-eval N/A

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

      13. lower-*.f64 92.8

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

   10. Applied rewrites 92.8%

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

   11. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-/.f64 92.7

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

   13. Applied rewrites 92.7%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      8. pow-pow N/A

         \[\leadsto \frac{\color{blue}{{\left(x + \varepsilon\right)}^{\left(5 \cdot 3\right)}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(x + \varepsilon\right)}^{\left(5 \cdot 3\right)}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\varepsilon + x\right)}}^{\left(5 \cdot 3\right)} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\varepsilon + x\right)}}^{\left(5 \cdot 3\right)} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{\color{blue}{15}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      13. pow-pow N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - \color{blue}{{x}^{\left(5 \cdot 3\right)}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      14. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - \color{blue}{{x}^{\left(5 \cdot 3\right)}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - {x}^{\color{blue}{15}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - {x}^{15}}{\color{blue}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

   3. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\frac{{\left(\varepsilon + x\right)}^{15} - {x}^{15}}{{\left(\varepsilon + x\right)}^{10} + \left({x}^{10} + {\left(\left(\varepsilon + x\right) \cdot x\right)}^{5}\right)}} \]

   4. Taylor expanded in x around 0

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

      1. lower-pow.f64 85.9

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left({x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\left({x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \left({x}^{4} \cdot \color{blue}{\left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)}\right) \]

      3. sqr-pow N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\color{blue}{\varepsilon} + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      7. pow2 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\varepsilon} + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      11. associate-+r+ N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\varepsilon + 2 \cdot \varepsilon\right) + \color{blue}{\left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)}\right)\right) \]

      12. distribute-rgt1-in N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(2 + 1\right) \cdot \varepsilon + \left(\color{blue}{4 \cdot \varepsilon} + 8 \cdot \varepsilon\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(3 \cdot \varepsilon + \left(\color{blue}{4} \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \color{blue}{\varepsilon}, 4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right) \]

      15. distribute-rgt-out N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot \left(4 + 8\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot \left(4 + 8\right)\right)\right) \]

      17. metadata-eval 99.6

          \[\leadsto 0.3333333333333333 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot 12\right)\right) \]

   9. Applied rewrites 99.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot 12\right)\right)} \]

   10. Taylor expanded in eps around 0

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

       1. associate-*r* N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 99.6

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

       5. lift-pow.f64 N/A

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

       6. sqr-pow N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\color{blue}{\left(\frac{4}{2}\right)}}\right) \]

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. pow2 N/A

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

       10. associate-*r* N/A

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

       11. pow2 N/A

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

       12. unpow3 N/A

           \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{3} \cdot x\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{3} \cdot x\right) \]

       14. unpow3 N/A

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

       15. pow2 N/A

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

       16. lower-*.f64 N/A

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

       17. pow2 N/A

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

       18. lift-*.f64 99.6

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

   12. Applied rewrites 99.6%

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

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

   1. Initial program 98.3%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   3. 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. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 94.1

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 94.0

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

   7. Applied rewrites 94.0%

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

   8. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot \color{blue}{x}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot \color{blue}{x}\right) \]

      12. metadata-eval N/A

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

      13. lower-*.f64 94.1

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

   10. Applied rewrites 94.1%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. metadata-eval N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot x\right) \]

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

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

       5. metadata-eval N/A

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

       6. distribute-rgt1-in N/A

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

       7. +-commutative N/A

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

       8. distribute-rgt1-in N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 94.1

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

   12. Applied rewrites 94.1%

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

Alternative 6: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* (* eps eps) (* eps eps)) (fma 5.0 x eps))))
   (if (<= t_0 -2e-305)
     t_1
     (if (<= t_0 0.0) (* (* 5.0 eps) (* (* (* x x) x) x)) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = ((eps * eps) * (eps * eps)) * fma(5.0, x, eps);
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * (((x * x) * x) * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * fma(5.0, x, eps))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * Float64(Float64(Float64(x * x) * x) * x));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.9%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   3. 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. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 93.3

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot \color{blue}{x}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot \color{blue}{x}\right) \]

      12. metadata-eval N/A

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

      13. lower-*.f64 93.5

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

   10. Applied rewrites 93.5%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. metadata-eval N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon - \left(\mathsf{neg}\left(5\right)\right) \cdot x\right) \]

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

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

       5. metadata-eval N/A

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

       6. distribute-rgt1-in N/A

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

       7. +-commutative N/A

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

       8. distribute-rgt1-in N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 93.5

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

   12. Applied rewrites 93.5%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      8. pow-pow N/A

         \[\leadsto \frac{\color{blue}{{\left(x + \varepsilon\right)}^{\left(5 \cdot 3\right)}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(x + \varepsilon\right)}^{\left(5 \cdot 3\right)}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\varepsilon + x\right)}}^{\left(5 \cdot 3\right)} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\varepsilon + x\right)}}^{\left(5 \cdot 3\right)} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{\color{blue}{15}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      13. pow-pow N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - \color{blue}{{x}^{\left(5 \cdot 3\right)}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      14. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - \color{blue}{{x}^{\left(5 \cdot 3\right)}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - {x}^{\color{blue}{15}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - {x}^{15}}{\color{blue}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

   3. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\frac{{\left(\varepsilon + x\right)}^{15} - {x}^{15}}{{\left(\varepsilon + x\right)}^{10} + \left({x}^{10} + {\left(\left(\varepsilon + x\right) \cdot x\right)}^{5}\right)}} \]

   4. Taylor expanded in x around 0

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

      1. lower-pow.f64 85.9

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left({x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\left({x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \left({x}^{4} \cdot \color{blue}{\left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)}\right) \]

      3. sqr-pow N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\color{blue}{\varepsilon} + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      7. pow2 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\varepsilon} + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      11. associate-+r+ N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\varepsilon + 2 \cdot \varepsilon\right) + \color{blue}{\left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)}\right)\right) \]

      12. distribute-rgt1-in N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(2 + 1\right) \cdot \varepsilon + \left(\color{blue}{4 \cdot \varepsilon} + 8 \cdot \varepsilon\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(3 \cdot \varepsilon + \left(\color{blue}{4} \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \color{blue}{\varepsilon}, 4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right) \]

      15. distribute-rgt-out N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot \left(4 + 8\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot \left(4 + 8\right)\right)\right) \]

      17. metadata-eval 99.6

          \[\leadsto 0.3333333333333333 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot 12\right)\right) \]

   9. Applied rewrites 99.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot 12\right)\right)} \]

   10. Taylor expanded in eps around 0

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

       1. associate-*r* N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 99.6

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

       5. lift-pow.f64 N/A

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

       6. sqr-pow N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\color{blue}{\left(\frac{4}{2}\right)}}\right) \]

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. pow2 N/A

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

       10. associate-*r* N/A

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

       11. pow2 N/A

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

       12. unpow3 N/A

           \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{3} \cdot x\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{3} \cdot x\right) \]

       14. unpow3 N/A

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

       15. pow2 N/A

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

       16. lower-*.f64 N/A

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

       17. pow2 N/A

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

       18. lift-*.f64 99.6

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

   12. Applied rewrites 99.6%

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

Alternative 7: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-305)
     (pow eps 5.0)
     (if (<= t_0 0.0) (* (* 5.0 eps) (* (* (* x 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 <= -2e-305) {
		tmp = pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * (((x * x) * x) * x);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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 <= (-2d-305)) then
        tmp = eps ** 5.0d0
    else if (t_0 <= 0.0d0) then
        tmp = (5.0d0 * eps) * (((x * x) * x) * x)
    else
        tmp = eps ** 5.0d0
    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 <= -2e-305) {
		tmp = Math.pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * (((x * x) * x) * x);
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	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 <= -2e-305:
		tmp = math.pow(eps, 5.0)
	elif t_0 <= 0.0:
		tmp = (5.0 * eps) * (((x * x) * x) * x)
	else:
		tmp = math.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 <= -2e-305)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * Float64(Float64(Float64(x * x) * x) * x));
	else
		tmp = eps ^ 5.0;
	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 <= -2e-305)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = (5.0 * eps) * (((x * x) * x) * x);
	else
		tmp = eps ^ 5.0;
	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[LessEqual[t$95$0, -2e-305], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      8. pow-pow N/A

         \[\leadsto \frac{\color{blue}{{\left(x + \varepsilon\right)}^{\left(5 \cdot 3\right)}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(x + \varepsilon\right)}^{\left(5 \cdot 3\right)}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\varepsilon + x\right)}}^{\left(5 \cdot 3\right)} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\varepsilon + x\right)}}^{\left(5 \cdot 3\right)} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{\color{blue}{15}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      13. pow-pow N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - \color{blue}{{x}^{\left(5 \cdot 3\right)}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      14. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - \color{blue}{{x}^{\left(5 \cdot 3\right)}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - {x}^{\color{blue}{15}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - {x}^{15}}{\color{blue}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

   3. Applied rewrites 36.1%

      \[\leadsto \color{blue}{\frac{{\left(\varepsilon + x\right)}^{15} - {x}^{15}}{{\left(\varepsilon + x\right)}^{10} + \left({x}^{10} + {\left(\left(\varepsilon + x\right) \cdot x\right)}^{5}\right)}} \]

   4. Taylor expanded in x around 0

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

      1. lower-pow.f64 93.5

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

   6. Applied rewrites 93.5%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      8. pow-pow N/A

         \[\leadsto \frac{\color{blue}{{\left(x + \varepsilon\right)}^{\left(5 \cdot 3\right)}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(x + \varepsilon\right)}^{\left(5 \cdot 3\right)}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\varepsilon + x\right)}}^{\left(5 \cdot 3\right)} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\varepsilon + x\right)}}^{\left(5 \cdot 3\right)} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{\color{blue}{15}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      13. pow-pow N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - \color{blue}{{x}^{\left(5 \cdot 3\right)}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      14. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - \color{blue}{{x}^{\left(5 \cdot 3\right)}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - {x}^{\color{blue}{15}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - {x}^{15}}{\color{blue}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

   3. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\frac{{\left(\varepsilon + x\right)}^{15} - {x}^{15}}{{\left(\varepsilon + x\right)}^{10} + \left({x}^{10} + {\left(\left(\varepsilon + x\right) \cdot x\right)}^{5}\right)}} \]

   4. Taylor expanded in x around 0

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

      1. lower-pow.f64 85.9

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left({x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\left({x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \left({x}^{4} \cdot \color{blue}{\left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)}\right) \]

      3. sqr-pow N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\color{blue}{\varepsilon} + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      7. pow2 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\varepsilon} + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      11. associate-+r+ N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\varepsilon + 2 \cdot \varepsilon\right) + \color{blue}{\left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)}\right)\right) \]

      12. distribute-rgt1-in N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(2 + 1\right) \cdot \varepsilon + \left(\color{blue}{4 \cdot \varepsilon} + 8 \cdot \varepsilon\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(3 \cdot \varepsilon + \left(\color{blue}{4} \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \color{blue}{\varepsilon}, 4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right) \]

      15. distribute-rgt-out N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot \left(4 + 8\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot \left(4 + 8\right)\right)\right) \]

      17. metadata-eval 99.6

          \[\leadsto 0.3333333333333333 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot 12\right)\right) \]

   9. Applied rewrites 99.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot 12\right)\right)} \]

   10. Taylor expanded in eps around 0

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

       1. associate-*r* N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 99.6

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

       5. lift-pow.f64 N/A

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

       6. sqr-pow N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\color{blue}{\left(\frac{4}{2}\right)}}\right) \]

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. pow2 N/A

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

       10. associate-*r* N/A

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

       11. pow2 N/A

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

       12. unpow3 N/A

           \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{3} \cdot x\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{3} \cdot x\right) \]

       14. unpow3 N/A

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

       15. pow2 N/A

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

       16. lower-*.f64 N/A

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

       17. pow2 N/A

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

       18. lift-*.f64 99.6

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

   12. Applied rewrites 99.6%

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

Alternative 8: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* (* eps eps) (* eps eps)) eps)))
   (if (<= t_0 -2e-305)
     t_1
     (if (<= t_0 0.0) (* (* 5.0 eps) (* (* (* x x) x) x)) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = ((eps * eps) * (eps * eps)) * eps;
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * (((x * x) * x) * x);
	} 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) - (x ** 5.0d0)
    t_1 = ((eps * eps) * (eps * eps)) * eps
    if (t_0 <= (-2d-305)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (5.0d0 * eps) * (((x * x) * x) * x)
    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) - Math.pow(x, 5.0);
	double t_1 = ((eps * eps) * (eps * eps)) * eps;
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * (((x * x) * x) * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = ((eps * eps) * (eps * eps)) * eps
	tmp = 0
	if t_0 <= -2e-305:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (5.0 * eps) * (((x * x) * x) * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps)
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * Float64(Float64(Float64(x * x) * x) * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = ((eps * eps) * (eps * eps)) * eps;
	tmp = 0.0;
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = (5.0 * eps) * (((x * x) * x) * x);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      3. lower-*.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 92.7

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

   4. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3} - {\left({x}^{5}\right)}^{3}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      8. pow-pow N/A

         \[\leadsto \frac{\color{blue}{{\left(x + \varepsilon\right)}^{\left(5 \cdot 3\right)}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(x + \varepsilon\right)}^{\left(5 \cdot 3\right)}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\varepsilon + x\right)}}^{\left(5 \cdot 3\right)} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\varepsilon + x\right)}}^{\left(5 \cdot 3\right)} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{\color{blue}{15}} - {\left({x}^{5}\right)}^{3}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      13. pow-pow N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - \color{blue}{{x}^{\left(5 \cdot 3\right)}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      14. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - \color{blue}{{x}^{\left(5 \cdot 3\right)}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - {x}^{\color{blue}{15}}}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\varepsilon + x\right)}^{15} - {x}^{15}}{\color{blue}{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5} + \left({x}^{5} \cdot {x}^{5} + {\left(x + \varepsilon\right)}^{5} \cdot {x}^{5}\right)}} \]

   3. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\frac{{\left(\varepsilon + x\right)}^{15} - {x}^{15}}{{\left(\varepsilon + x\right)}^{10} + \left({x}^{10} + {\left(\left(\varepsilon + x\right) \cdot x\right)}^{5}\right)}} \]

   4. Taylor expanded in x around 0

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

      1. lower-pow.f64 85.9

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left({x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\left({x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \left({x}^{4} \cdot \color{blue}{\left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)}\right) \]

      3. sqr-pow N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\color{blue}{\varepsilon} + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      7. pow2 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\varepsilon} + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon + \left(2 \cdot \varepsilon + \left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right)\right) \]

      11. associate-+r+ N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\varepsilon + 2 \cdot \varepsilon\right) + \color{blue}{\left(4 \cdot \varepsilon + 8 \cdot \varepsilon\right)}\right)\right) \]

      12. distribute-rgt1-in N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(2 + 1\right) \cdot \varepsilon + \left(\color{blue}{4 \cdot \varepsilon} + 8 \cdot \varepsilon\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(3 \cdot \varepsilon + \left(\color{blue}{4} \cdot \varepsilon + 8 \cdot \varepsilon\right)\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \color{blue}{\varepsilon}, 4 \cdot \varepsilon + 8 \cdot \varepsilon\right)\right) \]

      15. distribute-rgt-out N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot \left(4 + 8\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot \left(4 + 8\right)\right)\right) \]

      17. metadata-eval 99.6

          \[\leadsto 0.3333333333333333 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot 12\right)\right) \]

   9. Applied rewrites 99.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(3, \varepsilon, \varepsilon \cdot 12\right)\right)} \]

   10. Taylor expanded in eps around 0

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

       1. associate-*r* N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 99.6

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

       5. lift-pow.f64 N/A

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

       6. sqr-pow N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\color{blue}{\left(\frac{4}{2}\right)}}\right) \]

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. pow2 N/A

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

       10. associate-*r* N/A

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

       11. pow2 N/A

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

       12. unpow3 N/A

           \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{3} \cdot x\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{3} \cdot x\right) \]

       14. unpow3 N/A

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

       15. pow2 N/A

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

       16. lower-*.f64 N/A

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

       17. pow2 N/A

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

       18. lift-*.f64 99.6

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

   12. Applied rewrites 99.6%

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

Alternative 9: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* (* eps eps) (* eps eps)) eps)))
   (if (<= t_0 -2e-305)
     t_1
     (if (<= t_0 0.0) (* 5.0 (* eps (* (* x x) (* x x)))) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = ((eps * eps) * (eps * eps)) * eps;
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 5.0 * (eps * ((x * x) * (x * x)));
	} 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) - (x ** 5.0d0)
    t_1 = ((eps * eps) * (eps * eps)) * eps
    if (t_0 <= (-2d-305)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = 5.0d0 * (eps * ((x * x) * (x * x)))
    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) - Math.pow(x, 5.0);
	double t_1 = ((eps * eps) * (eps * eps)) * eps;
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 5.0 * (eps * ((x * x) * (x * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = ((eps * eps) * (eps * eps)) * eps
	tmp = 0
	if t_0 <= -2e-305:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = 5.0 * (eps * ((x * x) * (x * x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps)
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(5.0 * Float64(eps * Float64(Float64(x * x) * Float64(x * x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = ((eps * eps) * (eps * eps)) * eps;
	tmp = 0.0;
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = 5.0 * (eps * ((x * x) * (x * x)));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$1, If[LessEqual[t$95$0, 0.0], N[(5.0 * N[(eps * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      3. lower-*.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 92.7

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

   4. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 85.9%

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

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   3. 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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sqr-pow N/A

         \[\leadsto 5 \cdot \left(\varepsilon \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\color{blue}{\left(\frac{4}{2}\right)}}\right)\right) \]

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 99.6

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

   10. Applied rewrites 99.6%

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

Alternative 10: 87.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \end{array} \]

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(Float64(eps * eps) * Float64(eps * eps)) * eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = ((eps * eps) * (eps * eps)) * eps;
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. pow-plus-rev N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   3. lower-*.f64 N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   4. metadata-eval N/A

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

   5. pow-prod-up N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 87.2

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

4. Applied rewrites 87.2%

   \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(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)))