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

Percentage Accurate: 88.2% → 99.3%

Time: 4.2s

Alternatives: 11

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-2d-305)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (5.0d0 * eps) * (x ** 4.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * Math.pow(x, 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -2e-305)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (5.0 * eps) * (x ^ 4.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.9%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 2: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(-4, x \cdot x, \varepsilon \cdot \left(-1 \cdot \varepsilon - \left(x + 4 \cdot x\right)\right)\right) - 6 \cdot \left(x \cdot x\right)}{\varepsilon \cdot \varepsilon} \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(-\frac{x \cdot \left(5 + 10 \cdot \frac{x}{\varepsilon}\right)}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-305)
     (-
      (*
       (/
        (-
         (fma -4.0 (* x x) (* eps (- (* -1.0 eps) (+ x (* 4.0 x)))))
         (* 6.0 (* x x)))
        (* eps eps))
       (pow eps 5.0)))
     (if (<= t_0 0.0)
       (* (* 5.0 eps) (pow x 4.0))
       (-
        (*
         (- (- (/ (* x (+ 5.0 (* 10.0 (/ x eps)))) eps)) 1.0)
         (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 <= -2e-305) {
		tmp = -(((fma(-4.0, (x * x), (eps * ((-1.0 * eps) - (x + (4.0 * x))))) - (6.0 * (x * x))) / (eps * eps)) * pow(eps, 5.0));
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = -((-((x * (5.0 + (10.0 * (x / eps)))) / eps) - 1.0) * pow(eps, 5.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = Float64(-Float64(Float64(Float64(fma(-4.0, Float64(x * x), Float64(eps * Float64(Float64(-1.0 * eps) - Float64(x + Float64(4.0 * x))))) - Float64(6.0 * Float64(x * x))) / Float64(eps * eps)) * (eps ^ 5.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = Float64(-Float64(Float64(Float64(-Float64(Float64(x * Float64(5.0 + Float64(10.0 * Float64(x / eps)))) / eps)) - 1.0) * (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, -2e-305], (-N[(N[(N[(N[(-4.0 * N[(x * x), $MachinePrecision] + N[(eps * N[(N[(-1.0 * eps), $MachinePrecision] - N[(x + N[(4.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(6.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], (-N[(N[((-N[(N[(x * N[(5.0 + N[(10.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]) - 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 97.4%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. 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

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 94.1%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 98.3%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. 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

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto -\left(\left(-\frac{x \cdot \left(5 + 10 \cdot \frac{x}{\varepsilon}\right)}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5} \]
   6. 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 95.5

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

   7. Applied rewrites 95.5%

      \[\leadsto -\left(\left(-\frac{x \cdot \left(5 + 10 \cdot \frac{x}{\varepsilon}\right)}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1
         (-
          (*
           (- (- (/ (* x (+ 5.0 (* 10.0 (/ x eps)))) eps)) 1.0)
           (pow eps 5.0)))))
   (if (<= t_0 -2e-305)
     t_1
     (if (<= t_0 0.0) (* (* 5.0 eps) (pow x 4.0)) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = -((-((x * (5.0 + (10.0 * (x / eps)))) / eps) - 1.0) * pow(eps, 5.0));
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = -((-((x * (5.0d0 + (10.0d0 * (x / eps)))) / eps) - 1.0d0) * (eps ** 5.0d0))
    if (t_0 <= (-2d-305)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (5.0d0 * eps) * (x ** 4.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double t_1 = -((-((x * (5.0 + (10.0 * (x / eps)))) / eps) - 1.0) * Math.pow(eps, 5.0));
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * Math.pow(x, 4.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = -((-((x * (5.0 + (10.0 * (x / eps)))) / eps) - 1.0) * math.pow(eps, 5.0))
	tmp = 0
	if t_0 <= -2e-305:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (5.0 * eps) * math.pow(x, 4.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(-Float64(Float64(Float64(-Float64(Float64(x * Float64(5.0 + Float64(10.0 * Float64(x / eps)))) / eps)) - 1.0) * (eps ^ 5.0)))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = -((-((x * (5.0 + (10.0 * (x / eps)))) / eps) - 1.0) * (eps ^ 5.0));
	tmp = 0.0;
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = (5.0 * eps) * (x ^ 4.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(N[((-N[(N[(x * N[(5.0 + N[(10.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]) - 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t$95$0, -2e-305], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.9%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. 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

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto -\left(\left(-\frac{x \cdot \left(5 + 10 \cdot \frac{x}{\varepsilon}\right)}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5} \]
   6. 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 94.9

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

   7. Applied rewrites 94.9%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 4: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1 \cdot \left(\mathsf{fma}\left(6, x \cdot x, \left(\left(\frac{\varepsilon}{x} + 5\right) \cdot \varepsilon\right) \cdot x\right) - -4 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(10, x, 5 \cdot \varepsilon\right), x, \varepsilon \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))) (t_1 (* (* eps eps) eps)))
   (if (<= t_0 -2e-305)
     (*
      t_1
      (- (fma 6.0 (* x x) (* (* (+ (/ eps x) 5.0) eps) x)) (* -4.0 (* x x))))
     (if (<= t_0 0.0)
       (* (* 5.0 eps) (pow x 4.0))
       (* t_1 (fma (fma 10.0 x (* 5.0 eps)) x (* eps eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * eps;
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1 * (fma(6.0, (x * x), ((((eps / x) + 5.0) * eps) * x)) - (-4.0 * (x * x)));
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = t_1 * fma(fma(10.0, x, (5.0 * eps)), x, (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * eps)
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = Float64(t_1 * Float64(fma(6.0, Float64(x * x), Float64(Float64(Float64(Float64(eps / x) + 5.0) * eps) * x)) - Float64(-4.0 * Float64(x * x))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = Float64(t_1 * fma(fma(10.0, x, Float64(5.0 * eps)), x, Float64(eps * eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], N[(t$95$1 * N[(N[(6.0 * N[(x * x), $MachinePrecision] + N[(N[(N[(N[(eps / x), $MachinePrecision] + 5.0), $MachinePrecision] * eps), $MachinePrecision] * x), $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[(t$95$1 * N[(N[(10.0 * x + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * x + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 97.4%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. 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

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in eps around 0

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

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

   7. Applied rewrites 93.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 93.3

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

   9. Applied rewrites 93.3%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. distribute-lft-in N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 N/A

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

       11. lower-/.f64 93.4

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

   12. Applied rewrites 93.4%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 98.3%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. 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

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in eps around 0

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

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

   7. Applied rewrites 95.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.7

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

   9. Applied rewrites 94.7%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 94.7

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

   12. Applied rewrites 94.7%

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

Alternative 5: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1 \cdot \left(\mathsf{fma}\left(6, x \cdot x, \left(\left(\frac{\varepsilon}{x} + 5\right) \cdot \varepsilon\right) \cdot x\right) - -4 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(10, x, 5 \cdot \varepsilon\right), x, \varepsilon \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))) (t_1 (* (* eps eps) eps)))
   (if (<= t_0 -2e-305)
     (*
      t_1
      (- (fma 6.0 (* x x) (* (* (+ (/ eps x) 5.0) eps) x)) (* -4.0 (* x x))))
     (if (<= t_0 0.0)
       (* 5.0 (* eps (pow x 4.0)))
       (* t_1 (fma (fma 10.0 x (* 5.0 eps)) x (* eps eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * eps;
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1 * (fma(6.0, (x * x), ((((eps / x) + 5.0) * eps) * x)) - (-4.0 * (x * x)));
	} else if (t_0 <= 0.0) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = t_1 * fma(fma(10.0, x, (5.0 * eps)), x, (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * eps)
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = Float64(t_1 * Float64(fma(6.0, Float64(x * x), Float64(Float64(Float64(Float64(eps / x) + 5.0) * eps) * x)) - Float64(-4.0 * Float64(x * x))));
	elseif (t_0 <= 0.0)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = Float64(t_1 * fma(fma(10.0, x, Float64(5.0 * eps)), x, Float64(eps * eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], N[(t$95$1 * N[(N[(6.0 * N[(x * x), $MachinePrecision] + N[(N[(N[(N[(eps / x), $MachinePrecision] + 5.0), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(10.0 * x + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * x + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 97.4%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. 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

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in eps around 0

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

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

   7. Applied rewrites 93.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 93.3

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

   9. Applied rewrites 93.3%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. distribute-lft-in N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 N/A

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

       11. lower-/.f64 93.4

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

   12. Applied rewrites 93.4%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   6. 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 99.6

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

   7. Applied rewrites 99.6%

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

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

   1. Initial program 98.3%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. 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

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in eps around 0

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

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

   7. Applied rewrites 95.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.7

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

   9. Applied rewrites 94.7%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 94.7

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

   12. Applied rewrites 94.7%

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

Alternative 6: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1 \cdot \left(\mathsf{fma}\left(6, x \cdot x, \left(\left(\frac{\varepsilon}{x} + 5\right) \cdot \varepsilon\right) \cdot x\right) - -4 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(10, x, 5 \cdot \varepsilon\right), x, \varepsilon \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))) (t_1 (* (* eps eps) eps)))
   (if (<= t_0 -2e-305)
     (*
      t_1
      (- (fma 6.0 (* x x) (* (* (+ (/ eps x) 5.0) eps) x)) (* -4.0 (* x x))))
     (if (<= t_0 0.0)
       (* 5.0 (* eps (* (* x x) (* x x))))
       (* t_1 (fma (fma 10.0 x (* 5.0 eps)) x (* eps eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * eps;
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1 * (fma(6.0, (x * x), ((((eps / x) + 5.0) * eps) * x)) - (-4.0 * (x * x)));
	} else if (t_0 <= 0.0) {
		tmp = 5.0 * (eps * ((x * x) * (x * x)));
	} else {
		tmp = t_1 * fma(fma(10.0, x, (5.0 * eps)), x, (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * eps)
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = Float64(t_1 * Float64(fma(6.0, Float64(x * x), Float64(Float64(Float64(Float64(eps / x) + 5.0) * eps) * x)) - Float64(-4.0 * Float64(x * x))));
	elseif (t_0 <= 0.0)
		tmp = Float64(5.0 * Float64(eps * Float64(Float64(x * x) * Float64(x * x))));
	else
		tmp = Float64(t_1 * fma(fma(10.0, x, Float64(5.0 * eps)), x, Float64(eps * eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], N[(t$95$1 * N[(N[(6.0 * N[(x * x), $MachinePrecision] + N[(N[(N[(N[(eps / x), $MachinePrecision] + 5.0), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(5.0 * N[(eps * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(10.0 * x + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * x + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 97.4%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. 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

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in eps around 0

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

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

   7. Applied rewrites 93.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 93.3

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

   9. Applied rewrites 93.3%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. distribute-lft-in N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 N/A

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

       11. lower-/.f64 93.4

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

   12. Applied rewrites 93.4%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   6. 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 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

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

   1. Initial program 98.3%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. 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

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in eps around 0

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

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

   7. Applied rewrites 95.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.7

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

   9. Applied rewrites 94.7%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 94.7

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

   12. Applied rewrites 94.7%

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

Alternative 7: 98.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1
         (* (* (* eps eps) eps) (fma (fma 10.0 x (* 5.0 eps)) x (* eps eps)))))
   (if (<= t_0 -2e-305)
     t_1
     (if (<= t_0 0.0) (* 5.0 (* eps (* (* x x) (* x x)))) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = ((eps * eps) * eps) * fma(fma(10.0, x, (5.0 * eps)), x, (eps * eps));
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 5.0 * (eps * ((x * x) * (x * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(eps * eps) * eps) * fma(fma(10.0, x, Float64(5.0 * eps)), x, Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(5.0 * Float64(eps * Float64(Float64(x * x) * Float64(x * x))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(10.0 * x + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * x + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$1, If[LessEqual[t$95$0, 0.0], N[(5.0 * N[(eps * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.9%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. 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

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in eps around 0

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

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

   7. Applied rewrites 94.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.1

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

   9. Applied rewrites 94.1%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 94.1

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

   12. Applied rewrites 94.1%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   6. 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 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

Alternative 8: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(5.0 * N[(eps * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 97.4%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. 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

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in eps around 0

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

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

   7. Applied rewrites 93.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 93.3

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

   9. Applied rewrites 93.3%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 92.8

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

   12. Applied rewrites 92.8%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   6. 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 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

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

   1. Initial program 98.3%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 94.5

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

   7. Applied rewrites 94.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 94.1

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

   9. Applied rewrites 94.1%

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

Alternative 9: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = ((eps * eps) * (eps * eps)) * (eps + (5.0d0 * x))
    if (t_0 <= (-2d-305)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = 5.0d0 * (eps * ((x * x) * (x * x)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = ((eps * eps) * (eps * eps)) * (eps + (5.0 * x));
	tmp = 0.0;
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = 5.0 * (eps * ((x * x) * (x * x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$1, If[LessEqual[t$95$0, 0.0], N[(5.0 * N[(eps * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.9%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 94.3

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 93.9

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

   7. Applied rewrites 93.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 93.5

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

   9. Applied rewrites 93.5%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   6. 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 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

Alternative 10: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = ((eps * eps) * eps) * (eps * eps)
    if (t_0 <= (-2d-305)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = 5.0d0 * (eps * ((x * x) * (x * x)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = ((eps * eps) * eps) * (eps * eps);
	tmp = 0.0;
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = 5.0 * (eps * ((x * x) * (x * x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$1, If[LessEqual[t$95$0, 0.0], N[(5.0 * N[(eps * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.9%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. 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

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in eps around 0

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

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

   7. Applied rewrites 94.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.1

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

   9. Applied rewrites 94.1%

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

   10. Taylor expanded in x around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 92.7

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

   12. Applied rewrites 92.7%

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

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

   1. Initial program 85.9%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   6. 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 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

Alternative 11: 87.2% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in eps around -inf

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
3. 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

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

4. Applied rewrites 77.7%

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

5. Taylor expanded in eps around 0

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

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

7. Applied rewrites 87.5%

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

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. pow2 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 87.5

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

9. Applied rewrites 87.5%

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

10. Taylor expanded in x around 0

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

    1. pow2 N/A

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

    2. lift-*.f64 87.2

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

12. Applied rewrites 87.2%

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

Reproduce

herbie shell --seed 2025106 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=5"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 5.0) (pow x 5.0)))