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

Percentage Accurate: 88.7% → 99.2%

Time: 4.2s

Alternatives: 12

Speedup: 0.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-303}:\\ \;\;\;\;t\_0 - {\left(x \cdot x\right)}^{2.5}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -4e-303)
     (- t_0 (pow (* x x) 2.5))
     (if (<= t_1 0.0) (* (* 5.0 (pow x 4.0)) eps) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -4e-303) {
		tmp = t_0 - pow((x * x), 2.5);
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    if (t_1 <= (-4d-303)) then
        tmp = t_0 - ((x * x) ** 2.5d0)
    else if (t_1 <= 0.0d0) then
        tmp = (5.0d0 * (x ** 4.0d0)) * eps
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double tmp;
	if (t_1 <= -4e-303) {
		tmp = t_0 - Math.pow((x * x), 2.5);
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * Math.pow(x, 4.0)) * eps;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	tmp = 0
	if t_1 <= -4e-303:
		tmp = t_0 - math.pow((x * x), 2.5)
	elif t_1 <= 0.0:
		tmp = (5.0 * math.pow(x, 4.0)) * eps
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -4e-303)
		tmp = Float64(t_0 - (Float64(x * x) ^ 2.5));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	tmp = 0.0;
	if (t_1 <= -4e-303)
		tmp = t_0 - ((x * x) ^ 2.5);
	elseif (t_1 <= 0.0)
		tmp = (5.0 * (x ^ 4.0)) * eps;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-303], N[(t$95$0 - N[Power[N[(x * x), $MachinePrecision], 2.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.99999999999999972e-303

   1. Initial program 96.4%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. unpow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 96.4

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

   4. Applied rewrites 96.4%

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

   if -3.99999999999999972e-303 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 96.1%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-303} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -4e-303) (not (<= t_0 0.0)))
     t_0
     (* (* 5.0 (pow x 4.0)) eps))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-303) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-4d-303)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (5.0d0 * (x ** 4.0d0)) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-303) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (5.0 * Math.pow(x, 4.0)) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -4e-303) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (5.0 * math.pow(x, 4.0)) * eps
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -4e-303) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -4e-303) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (5.0 * (x ^ 4.0)) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-303], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.99999999999999972e-303 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.2%

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

   if -3.99999999999999972e-303 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-303} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-303} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\frac{x}{\varepsilon}, 10, 5\right) \cdot x}{\varepsilon} - -1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -4e-303) (not (<= t_0 0.0)))
     (* (- (/ (* (fma (/ x eps) 10.0 5.0) x) eps) -1.0) (pow eps 5.0))
     (* (* 5.0 (pow x 4.0)) eps))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-303) || !(t_0 <= 0.0)) {
		tmp = (((fma((x / eps), 10.0, 5.0) * x) / eps) - -1.0) * pow(eps, 5.0);
	} else {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -4e-303) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(Float64(fma(Float64(x / eps), 10.0, 5.0) * x) / eps) - -1.0) * (eps ^ 5.0));
	else
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-303], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(N[(N[(N[(x / eps), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * x), $MachinePrecision] / eps), $MachinePrecision] - -1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.99999999999999972e-303 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.2%

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

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 91.5

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

   8. Applied rewrites 91.5%

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

   10. Applied rewrites 91.5%

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

   if -3.99999999999999972e-303 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-303} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\frac{x}{\varepsilon}, 10, 5\right) \cdot x}{\varepsilon} - -1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-303)
     (* (fma 5.0 (/ x eps) 1.0) (pow eps 5.0))
     (if (<= t_0 0.0) (* (* 5.0 (pow x 4.0)) eps) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-303) {
		tmp = fma(5.0, (x / eps), 1.0) * pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-303)
		tmp = Float64(fma(5.0, Float64(x / eps), 1.0) * (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-303], N[(N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.99999999999999972e-303

   1. Initial program 96.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 94.9

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

   5. Applied rewrites 94.9%

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

   if -3.99999999999999972e-303 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.0

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

   5. Applied rewrites 87.0%

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

Alternative 5: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(6, x \cdot x, \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) - -4 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-303)
     (*
      (* (* eps eps) eps)
      (- (fma 6.0 (* x x) (* eps (+ eps (* 5.0 x)))) (* -4.0 (* x x))))
     (if (<= t_0 0.0) (* (* 5.0 (pow x 4.0)) eps) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-303) {
		tmp = ((eps * eps) * eps) * (fma(6.0, (x * x), (eps * (eps + (5.0 * x)))) - (-4.0 * (x * x)));
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-303)
		tmp = Float64(Float64(Float64(eps * eps) * eps) * Float64(fma(6.0, Float64(x * x), Float64(eps * Float64(eps + Float64(5.0 * x)))) - Float64(-4.0 * Float64(x * x))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-303], N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(6.0 * N[(x * x), $MachinePrecision] + N[(eps * N[(eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.99999999999999972e-303

   1. Initial program 96.4%

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

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

   8. Applied rewrites 94.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.8

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

   10. Applied rewrites 94.8%

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

   if -3.99999999999999972e-303 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(6, x \cdot x, \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) - -4 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(6, x \cdot x, \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) - -4 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-303)
     (*
      (* (* eps eps) eps)
      (- (fma 6.0 (* x x) (* eps (+ eps (* 5.0 x)))) (* -4.0 (* x x))))
     (if (<= t_0 0.0) (* (* 5.0 eps) (pow x 4.0)) (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 <= -4e-303) {
		tmp = ((eps * eps) * eps) * (fma(6.0, (x * x), (eps * (eps + (5.0 * x)))) - (-4.0 * (x * x)));
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} 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 <= -4e-303)
		tmp = Float64(Float64(Float64(eps * eps) * eps) * Float64(fma(6.0, Float64(x * x), Float64(eps * Float64(eps + Float64(5.0 * x)))) - Float64(-4.0 * Float64(x * x))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	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, -4e-303], N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(6.0 * N[(x * x), $MachinePrecision] + N[(eps * N[(eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $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))) < -3.99999999999999972e-303

   1. Initial program 96.4%

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

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

   8. Applied rewrites 94.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.8

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

   10. Applied rewrites 94.8%

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

   if -3.99999999999999972e-303 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   4. 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. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(6, x \cdot x, \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) - -4 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(6, x \cdot x, \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) - -4 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot x\right) \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-303)
     (*
      (* (* eps eps) eps)
      (- (fma 6.0 (* x x) (* eps (+ eps (* 5.0 x)))) (* -4.0 (* x x))))
     (if (<= t_0 0.0) (* (* (* x x) x) (* (* eps x) 5.0)) (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 <= -4e-303) {
		tmp = ((eps * eps) * eps) * (fma(6.0, (x * x), (eps * (eps + (5.0 * x)))) - (-4.0 * (x * x)));
	} else if (t_0 <= 0.0) {
		tmp = ((x * x) * x) * ((eps * x) * 5.0);
	} 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 <= -4e-303)
		tmp = Float64(Float64(Float64(eps * eps) * eps) * Float64(fma(6.0, Float64(x * x), Float64(eps * Float64(eps + Float64(5.0 * x)))) - Float64(-4.0 * Float64(x * x))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(x * x) * x) * Float64(Float64(eps * x) * 5.0));
	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, -4e-303], N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(6.0 * N[(x * x), $MachinePrecision] + N[(eps * N[(eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(eps * x), $MachinePrecision] * 5.0), $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))) < -3.99999999999999972e-303

   1. Initial program 96.4%

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

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

   8. Applied rewrites 94.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.8

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

   10. Applied rewrites 94.8%

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

   if -3.99999999999999972e-303 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10 \cdot {\varepsilon}^{2}}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot \color{blue}{x}, 10 \cdot {\varepsilon}^{2}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {\varepsilon}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {\varepsilon}^{2}\right) \]

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

      \[\leadsto {x}^{3} \cdot \color{blue}{\mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 99.9

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 99.9

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

   13. Applied rewrites 99.9%

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

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

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(6, x \cdot x, \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) - -4 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot x\right) \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-303} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(6, x \cdot x, \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) - -4 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot x\right) \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -4e-303) (not (<= t_0 0.0)))
     (*
      (* (* eps eps) eps)
      (- (fma 6.0 (* x x) (* eps (+ eps (* 5.0 x)))) (* -4.0 (* x x))))
     (* (* (* x x) x) (* (* eps x) 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 <= -4e-303) || !(t_0 <= 0.0)) {
		tmp = ((eps * eps) * eps) * (fma(6.0, (x * x), (eps * (eps + (5.0 * x)))) - (-4.0 * (x * x)));
	} else {
		tmp = ((x * x) * x) * ((eps * x) * 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 <= -4e-303) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(eps * eps) * eps) * Float64(fma(6.0, Float64(x * x), Float64(eps * Float64(eps + Float64(5.0 * x)))) - Float64(-4.0 * Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(x * x) * x) * Float64(Float64(eps * x) * 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[Or[LessEqual[t$95$0, -4e-303], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(6.0 * N[(x * x), $MachinePrecision] + N[(eps * N[(eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(eps * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -3.99999999999999972e-303 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.2%

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

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

   8. Applied rewrites 90.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 90.7

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

   10. Applied rewrites 90.7%

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

   if -3.99999999999999972e-303 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10 \cdot {\varepsilon}^{2}}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot \color{blue}{x}, 10 \cdot {\varepsilon}^{2}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {\varepsilon}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {\varepsilon}^{2}\right) \]

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

      \[\leadsto {x}^{3} \cdot \color{blue}{\mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 99.9

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 99.9

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

   13. Applied rewrites 99.9%

       \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot x\right) \cdot 5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-303} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\mathsf{fma}\left(6, x \cdot x, \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) - -4 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot x\right) \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (* x x) x) (fma 5.0 (* eps x) (* 10.0 (* eps eps)))))

C

double code(double x, double eps) {
	return ((x * x) * x) * fma(5.0, (eps * x), (10.0 * (eps * eps)));
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(x * x) * x) * fma(5.0, Float64(eps * x), Float64(10.0 * Float64(eps * eps))))
end

Wolfram

code[x_, eps_] := N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(5.0 * N[(eps * x), $MachinePrecision] + N[(10.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.7%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 85.7%

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

6. Taylor expanded in x around 0

   \[\leadsto {x}^{3} \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10 \cdot {\varepsilon}^{2}}\right) \]

   2. lift-pow.f64 N/A

      \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot \color{blue}{x}, 10 \cdot {\varepsilon}^{2}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {\varepsilon}^{2}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {\varepsilon}^{2}\right) \]

   6. unpow2 N/A

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

   7. lower-*.f64 85.7

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

8. Applied rewrites 85.7%

   \[\leadsto {x}^{3} \cdot \color{blue}{\mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. pow2 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lower-*.f64 85.7

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

10. Applied rewrites 85.7%

    \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(5, \color{blue}{\varepsilon} \cdot x, 10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]
11. Add Preprocessing

Alternative 10: 82.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (* x x) x) (* (fma 10.0 eps (* 5.0 x)) eps)))

C

double code(double x, double eps) {
	return ((x * x) * x) * (fma(10.0, eps, (5.0 * x)) * eps);
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(x * x) * x) * Float64(fma(10.0, eps, Float64(5.0 * x)) * eps))
end

Wolfram

code[x_, eps_] := N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.7%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 85.7%

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

6. Taylor expanded in x around 0

   \[\leadsto {x}^{3} \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10 \cdot {\varepsilon}^{2}}\right) \]

   2. lift-pow.f64 N/A

      \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot \color{blue}{x}, 10 \cdot {\varepsilon}^{2}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {\varepsilon}^{2}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {\varepsilon}^{2}\right) \]

   6. unpow2 N/A

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

   7. lower-*.f64 85.7

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

8. Applied rewrites 85.7%

   \[\leadsto {x}^{3} \cdot \color{blue}{\mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. pow2 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lower-*.f64 85.7

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

10. Applied rewrites 85.7%

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

11. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. lift-*.f64 85.6

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

13. Applied rewrites 85.6%

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

Alternative 11: 82.3% accurate, 8.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* (* (* x x) x) (* (* eps x) 5.0)))

C

double code(double x, double eps) {
	return ((x * x) * x) * ((eps * 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 * x) * x) * ((eps * x) * 5.0d0)
end function

Java

public static double code(double x, double eps) {
	return ((x * x) * x) * ((eps * x) * 5.0);
}

Python

def code(x, eps):
	return ((x * x) * x) * ((eps * x) * 5.0)

Julia

function code(x, eps)
	return Float64(Float64(Float64(x * x) * x) * Float64(Float64(eps * x) * 5.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x * x) * x) * ((eps * x) * 5.0);
end

Wolfram

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

Derivation

1. Initial program 85.7%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 85.7%

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

6. Taylor expanded in x around 0

   \[\leadsto {x}^{3} \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10 \cdot {\varepsilon}^{2}}\right) \]

   2. lift-pow.f64 N/A

      \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot \color{blue}{x}, 10 \cdot {\varepsilon}^{2}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {\varepsilon}^{2}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto {x}^{3} \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {\varepsilon}^{2}\right) \]

   6. unpow2 N/A

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

   7. lower-*.f64 85.7

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

8. Applied rewrites 85.7%

   \[\leadsto {x}^{3} \cdot \color{blue}{\mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. pow2 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lower-*.f64 85.7

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

10. Applied rewrites 85.7%

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

11. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lift-*.f64 85.5

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

13. Applied rewrites 85.5%

    \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot x\right) \cdot 5\right) \]
14. Add Preprocessing

Alternative 12: 82.3% accurate, 8.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* 5.0 (* eps (* (* x x) (* x x)))))

C

double code(double x, double eps) {
	return 5.0 * (eps * ((x * x) * (x * x)));
}

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 = 5.0d0 * (eps * ((x * x) * (x * x)))
end function

Java

public static double code(double x, double eps) {
	return 5.0 * (eps * ((x * x) * (x * x)));
}

Python

def code(x, eps):
	return 5.0 * (eps * ((x * x) * (x * x)))

Julia

function code(x, eps)
	return Float64(5.0 * Float64(eps * Float64(Float64(x * x) * Float64(x * x))))
end

MATLAB

function tmp = code(x, eps)
	tmp = 5.0 * (eps * ((x * x) * (x * x)));
end

Wolfram

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

Derivation

1. Initial program 85.7%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 85.7%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-pow.f64 85.4

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

8. Applied rewrites 85.4%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lower-*.f64 85.4

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

10. Applied rewrites 85.4%

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

Reproduce

herbie shell --seed 2025085 
(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)))