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

Percentage Accurate: 88.9% → 99.5%

Time: 3.3s

Alternatives: 11

Speedup: 1.3×

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: 88.9% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-322}:\\ \;\;\;\;t\_0 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -5e-322)
     (- 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 <= -5e-322) {
		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 <= (-5d-322)) 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 <= -5e-322) {
		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 <= -5e-322:
		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 <= -5e-322)
		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 <= -5e-322)
		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, -5e-322], 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))) < -4.99006e-322

   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 -4.99006e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

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

   2. 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 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]
   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. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 99.9

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

   6. Applied rewrites 99.9%

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

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

   1. Initial program 97.6%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-322}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -5e-322)
     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 <= -5e-322) {
		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 <= (-5d-322)) 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 <= -5e-322) {
		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 <= -5e-322:
		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 <= -5e-322)
		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 <= -5e-322)
		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, -5e-322], 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))) < -4.99006e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 97.5

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

   3. Applied rewrites 97.5%

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

   if -4.99006e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

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

   2. 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 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]
   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. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 99.9

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

   6. Applied rewrites 99.9%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-55}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-59}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.55e-55)
   (* (* (* (* x x) x) x) (* 5.0 eps))
   (if (<= x 1.9e-59)
     (pow eps 5.0)
     (*
      (* (* (fma (fma 10.0 eps (* 5.0 x)) x (* (* eps eps) 10.0)) x) x)
      eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.55e-55) {
		tmp = (((x * x) * x) * x) * (5.0 * eps);
	} else if (x <= 1.9e-59) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = ((fma(fma(10.0, eps, (5.0 * x)), x, ((eps * eps) * 10.0)) * x) * x) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.55e-55)
		tmp = Float64(Float64(Float64(Float64(x * x) * x) * x) * Float64(5.0 * eps));
	elseif (x <= 1.9e-59)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(Float64(fma(fma(10.0, eps, Float64(5.0 * x)), x, Float64(Float64(eps * eps) * 10.0)) * x) * x) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.55e-55], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-59], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.54999999999999998e-55

   1. Initial program 43.7%

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

   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 89.5

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

   4. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-*l* N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

          \[\leadsto {x}^{\left(3 + 1\right)} \cdot \left(5 \cdot \varepsilon\right) \]

      17. pow-plus N/A

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

      18. lower-*.f64 N/A

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

      19. pow3 N/A

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

      20. pow2 N/A

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

      21. lower-*.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lower-*.f64 89.5

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

   8. Applied rewrites 89.5%

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

   if -1.54999999999999998e-55 < x < 1.89999999999999992e-59

   1. Initial program 100.0%

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

   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 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. unpow3 N/A

         \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

      10. pow-prod-up N/A

          \[\leadsto {\varepsilon}^{\color{blue}{\left(2 + 3\right)}} \]

      11. metadata-eval N/A

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

      12. lower-pow.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if 1.89999999999999992e-59 < x

   1. Initial program 46.1%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), 4, \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 4, \mathsf{fma}\left(\left(x \cdot x\right) \cdot 10, \varepsilon, \left(\left(x \cdot x\right) \cdot 6\right) \cdot x\right)\right), \varepsilon, \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

   9. Applied rewrites 91.5%

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

Alternative 4: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-55}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-59}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, 5, \left(\left(\varepsilon + x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.55e-55)
   (* (* (* (* x x) x) x) (* 5.0 eps))
   (if (<= x 1.9e-59)
     (pow eps 5.0)
     (* (* (* (fma (* x x) 5.0 (* (* (+ eps x) 10.0) eps)) x) x) eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.55e-55) {
		tmp = (((x * x) * x) * x) * (5.0 * eps);
	} else if (x <= 1.9e-59) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = ((fma((x * x), 5.0, (((eps + x) * 10.0) * eps)) * x) * x) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.55e-55)
		tmp = Float64(Float64(Float64(Float64(x * x) * x) * x) * Float64(5.0 * eps));
	elseif (x <= 1.9e-59)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 5.0, Float64(Float64(Float64(eps + x) * 10.0) * eps)) * x) * x) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.55e-55], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-59], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0 + N[(N[(N[(eps + x), $MachinePrecision] * 10.0), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.54999999999999998e-55

   1. Initial program 43.7%

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

   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 89.5

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

   4. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-*l* N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

          \[\leadsto {x}^{\left(3 + 1\right)} \cdot \left(5 \cdot \varepsilon\right) \]

      17. pow-plus N/A

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

      18. lower-*.f64 N/A

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

      19. pow3 N/A

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

      20. pow2 N/A

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

      21. lower-*.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lower-*.f64 89.5

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

   8. Applied rewrites 89.5%

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

   if -1.54999999999999998e-55 < x < 1.89999999999999992e-59

   1. Initial program 100.0%

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

   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 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. unpow3 N/A

         \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

      10. pow-prod-up N/A

          \[\leadsto {\varepsilon}^{\color{blue}{\left(2 + 3\right)}} \]

      11. metadata-eval N/A

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

      12. lower-pow.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if 1.89999999999999992e-59 < x

   1. Initial program 46.1%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), 4, \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 4, \mathsf{fma}\left(\left(x \cdot x\right) \cdot 10, \varepsilon, \left(\left(x \cdot x\right) \cdot 6\right) \cdot x\right)\right), \varepsilon, \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

   8. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 91.6

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

   10. Applied rewrites 91.6%

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

   12. Applied rewrites 91.5%

       \[\leadsto \left(\left(\mathsf{fma}\left(x \cdot x, 5, \left(\left(\varepsilon + x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \color{blue}{\varepsilon} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-55}:\\ \;\;\;\;t\_0 \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-59}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot t\_0\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* x x) x) x)))
   (if (<= x -1.55e-55)
     (* t_0 (* 5.0 eps))
     (if (<= x 3.3e-59)
       (pow eps 5.0)
       (* (* (fma (/ eps x) 10.0 5.0) t_0) eps)))))

C

double code(double x, double eps) {
	double t_0 = ((x * x) * x) * x;
	double tmp;
	if (x <= -1.55e-55) {
		tmp = t_0 * (5.0 * eps);
	} else if (x <= 3.3e-59) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (fma((eps / x), 10.0, 5.0) * t_0) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(x * x) * x) * x)
	tmp = 0.0
	if (x <= -1.55e-55)
		tmp = Float64(t_0 * Float64(5.0 * eps));
	elseif (x <= 3.3e-59)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * t_0) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.55e-55], N[(t$95$0 * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e-59], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * t$95$0), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.54999999999999998e-55

   1. Initial program 43.7%

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

   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 89.5

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

   4. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-*l* N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

          \[\leadsto {x}^{\left(3 + 1\right)} \cdot \left(5 \cdot \varepsilon\right) \]

      17. pow-plus N/A

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

      18. lower-*.f64 N/A

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

      19. pow3 N/A

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

      20. pow2 N/A

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

      21. lower-*.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lower-*.f64 89.5

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

   8. Applied rewrites 89.5%

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

   if -1.54999999999999998e-55 < x < 3.29999999999999982e-59

   1. Initial program 100.0%

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

   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 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. unpow3 N/A

         \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

      10. pow-prod-up N/A

          \[\leadsto {\varepsilon}^{\color{blue}{\left(2 + 3\right)}} \]

      11. metadata-eval N/A

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

      12. lower-pow.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if 3.29999999999999982e-59 < x

   1. Initial program 46.1%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), 4, \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 4, \mathsf{fma}\left(\left(x \cdot x\right) \cdot 10, \varepsilon, \left(\left(x \cdot x\right) \cdot 6\right) \cdot x\right)\right), \varepsilon, \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right) \cdot \varepsilon \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(5 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \frac{\varepsilon}{x} + 5\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{\varepsilon}{x} \cdot 10 + 5\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      7. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot {x}^{\left(3 + 1\right)}\right) \cdot \varepsilon \]

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 90.9

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

   7. Applied rewrites 90.9%

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

Alternative 6: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-55}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-59}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.55e-55)
   (* (* (* (* x x) x) x) (* 5.0 eps))
   (if (<= x 3.3e-59)
     (pow eps 5.0)
     (* (* (* (fma 10.0 eps (* 5.0 x)) x) (* x x)) eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.55e-55) {
		tmp = (((x * x) * x) * x) * (5.0 * eps);
	} else if (x <= 3.3e-59) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = ((fma(10.0, eps, (5.0 * x)) * x) * (x * x)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.55e-55)
		tmp = Float64(Float64(Float64(Float64(x * x) * x) * x) * Float64(5.0 * eps));
	elseif (x <= 3.3e-59)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * x) * Float64(x * x)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.55e-55], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e-59], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.54999999999999998e-55

   1. Initial program 43.7%

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

   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 89.5

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

   4. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-*l* N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

          \[\leadsto {x}^{\left(3 + 1\right)} \cdot \left(5 \cdot \varepsilon\right) \]

      17. pow-plus N/A

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

      18. lower-*.f64 N/A

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

      19. pow3 N/A

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

      20. pow2 N/A

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

      21. lower-*.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lower-*.f64 89.5

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

   8. Applied rewrites 89.5%

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

   if -1.54999999999999998e-55 < x < 3.29999999999999982e-59

   1. Initial program 100.0%

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

   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 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. unpow3 N/A

         \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

      10. pow-prod-up N/A

          \[\leadsto {\varepsilon}^{\color{blue}{\left(2 + 3\right)}} \]

      11. metadata-eval N/A

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

      12. lower-pow.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if 3.29999999999999982e-59 < x

   1. Initial program 46.1%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), 4, \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 4, \mathsf{fma}\left(\left(x \cdot x\right) \cdot 10, \varepsilon, \left(\left(x \cdot x\right) \cdot 6\right) \cdot x\right)\right), \varepsilon, \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 91.5

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

   7. Applied rewrites 91.5%

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

   8. Taylor expanded in eps around 0

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

      1. pow2 N/A

         \[\leadsto \left(\left(5 \cdot \left(x \cdot x\right) + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\left(5 \cdot x\right) \cdot x + 10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\left(5 \cdot x\right) \cdot x + \left(10 \cdot \varepsilon\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\left(x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]

      5. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \left(10 \cdot \varepsilon + 5 \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(10 \cdot \varepsilon + 5 \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(10 \cdot \varepsilon + 5 \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 91.0

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

   10. Applied rewrites 91.0%

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

Alternative 7: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-322}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-322)
     (* (fma 5.0 x eps) (* (* eps eps) (* eps eps)))
     (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 <= -5e-322) {
		tmp = fma(5.0, x, eps) * ((eps * eps) * (eps * eps));
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * ((x * x) * (x * x));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-322)
		tmp = Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * Float64(eps * eps)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * Float64(Float64(x * x) * Float64(x * x)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-322], N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.99006e-322

   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 93.2

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 93.2

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

   7. Applied rewrites 93.2%

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

   if -4.99006e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

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

   2. 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 99.9

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

   4. Applied rewrites 99.9%

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

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

   1. Initial program 97.6%

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

   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 91.1

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

   4. Applied rewrites 91.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. unpow3 N/A

         \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

      10. pow-prod-up N/A

          \[\leadsto {\varepsilon}^{\color{blue}{\left(2 + 3\right)}} \]

      11. metadata-eval N/A

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

      12. lower-pow.f64 91.8

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

   6. Applied rewrites 91.8%

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

Alternative 8: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-322}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-322)
     (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 <= -5e-322) {
		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 <= (-5d-322)) 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 <= -5e-322) {
		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 <= -5e-322:
		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 <= -5e-322)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * Float64(Float64(x * x) * Float64(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 <= -5e-322)
		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, -5e-322], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $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))) < -4.99006e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.5%

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

   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 91.5

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

   4. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. unpow3 N/A

         \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

      10. pow-prod-up N/A

          \[\leadsto {\varepsilon}^{\color{blue}{\left(2 + 3\right)}} \]

      11. metadata-eval N/A

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

      12. lower-pow.f64 92.2

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

   6. Applied rewrites 92.2%

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

   if -4.99006e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

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

   2. 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 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 9: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-322}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -5e-322)
     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 <= -5e-322) {
		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 <= (-5d-322)) 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 <= -5e-322) {
		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 <= -5e-322:
		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(Float64(eps * eps) * eps) * eps) * eps)
	tmp = 0.0
	if (t_0 <= -5e-322)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * 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 <= -5e-322)
		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[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-322], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $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))) < -4.99006e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.5%

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

   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 91.5

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

   4. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. unpow3 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow3 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 91.7

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

   6. Applied rewrites 91.7%

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

   if -4.99006e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.7%

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

   2. 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 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 10: 87.7% accurate, 3.0× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.9%

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

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 87.7

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

4. Applied rewrites 87.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. pow2 N/A

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

   7. unpow3 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow3 N/A

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

   10. pow2 N/A

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

   11. lower-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 87.7

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

6. Applied rewrites 87.7%

   \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]
7. Add Preprocessing

Alternative 11: 87.7% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* (* eps eps) (* (* eps eps) eps)))

C

double code(double x, double eps) {
	return (eps * eps) * ((eps * eps) * eps);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double eps) {
	return (eps * eps) * ((eps * eps) * eps);
}

Python

def code(x, eps):
	return (eps * eps) * ((eps * eps) * eps)

Julia

function code(x, eps)
	return Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.9%

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

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 87.7

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

4. Applied rewrites 87.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. pow2 N/A

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

   7. associate-*l* N/A

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

   8. pow2 N/A

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

   9. unpow3 N/A

      \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

   10. lower-*.f64 N/A

       \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

   11. pow2 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3} \]

   13. unpow3 N/A

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

   14. pow2 N/A

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

   15. lower-*.f64 N/A

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

   16. pow2 N/A

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

   17. lift-*.f64 87.7

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

6. Applied rewrites 87.7%

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

Reproduce

herbie shell --seed 2025115 
(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)))