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

Percentage Accurate: 88.4% → 99.5%

Time: 8.2s

Alternatives: 15

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

real(8) function code(x, eps)
    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: 88.4% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -4e-322) (not (<= t_0 0.0)))
     t_0
     (* (* 5.0 eps) (pow x 4.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-322) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (5.0 * eps) * pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-4d-322)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (5.0d0 * eps) * (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-322) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (5.0 * eps) * Math.pow(x, 4.0);
	}
	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 <= -4e-322) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (5.0 * eps) * math.pow(x, 4.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -4e-322) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	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 <= -4e-322) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (5.0 * eps) * (x ^ 4.0);
	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[Or[LessEqual[t$95$0, -4e-322], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.2%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-322} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-322}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right)\right)}^{-1}\right)}^{-1} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-322)
     (* (fma (/ x eps) 5.0 1.0) (pow eps 5.0))
     (if (<= t_0 0.0)
       (* (* 5.0 eps) (pow x 4.0))
       (*
        (*
         (pow (pow (fma (fma x 5.0 eps) eps (* (* 10.0 x) x)) -1.0) -1.0)
         (* eps eps))
        eps)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-322) {
		tmp = fma((x / eps), 5.0, 1.0) * pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = (pow(pow(fma(fma(x, 5.0, eps), eps, ((10.0 * x) * x)), -1.0), -1.0) * (eps * eps)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-322)
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = Float64(Float64(((fma(fma(x, 5.0, eps), eps, Float64(Float64(10.0 * x) * x)) ^ -1.0) ^ -1.0) * Float64(eps * eps)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-322], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $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[Power[N[Power[N[(N[(x * 5.0 + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], -1.0], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.9%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.5%

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

   10. Applied rewrites 93.2%

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

   12. Applied rewrites 93.2%

       \[\leadsto \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right)}} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-322}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right)\right)}^{-1}\right)}^{-1} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-322}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{3} \cdot \varepsilon\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right)\right)}^{-1}\right)}^{-1} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-322)
     (* (fma 5.0 x eps) (* (pow eps 3.0) eps))
     (if (<= t_0 0.0)
       (* (* 5.0 eps) (pow x 4.0))
       (*
        (*
         (pow (pow (fma (fma x 5.0 eps) eps (* (* 10.0 x) x)) -1.0) -1.0)
         (* eps eps))
        eps)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-322) {
		tmp = fma(5.0, x, eps) * (pow(eps, 3.0) * eps);
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = (pow(pow(fma(fma(x, 5.0, eps), eps, ((10.0 * x) * x)), -1.0), -1.0) * (eps * eps)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-322)
		tmp = Float64(fma(5.0, x, eps) * Float64((eps ^ 3.0) * eps));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = Float64(Float64(((fma(fma(x, 5.0, eps), eps, Float64(Float64(10.0 * x) * x)) ^ -1.0) ^ -1.0) * Float64(eps * eps)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-322], N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[Power[eps, 3.0], $MachinePrecision] * eps), $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[Power[N[Power[N[(N[(x * 5.0 + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], -1.0], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 95.9

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

   8. Applied rewrites 95.9%

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

   10. Applied rewrites 96.2%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.9%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.5%

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

   10. Applied rewrites 93.2%

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

   12. Applied rewrites 93.2%

       \[\leadsto \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right)}} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-322}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{3} \cdot \varepsilon\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right)\right)}^{-1}\right)}^{-1} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right)\right)}^{-1}\right)}^{-1} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-322)
     (* (* (* (* (fma x 5.0 eps) eps) eps) eps) eps)
     (if (<= t_0 0.0)
       (* (* 5.0 eps) (pow x 4.0))
       (*
        (*
         (pow (pow (fma (fma x 5.0 eps) eps (* (* 10.0 x) x)) -1.0) -1.0)
         (* eps eps))
        eps)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-322) {
		tmp = (((fma(x, 5.0, eps) * eps) * eps) * eps) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = (pow(pow(fma(fma(x, 5.0, eps), eps, ((10.0 * x) * x)), -1.0), -1.0) * (eps * eps)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-322)
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, eps) * eps) * eps) * eps) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = Float64(Float64(((fma(fma(x, 5.0, eps), eps, Float64(Float64(10.0 * x) * x)) ^ -1.0) ^ -1.0) * Float64(eps * eps)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-322], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[Power[N[(N[(x * 5.0 + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], -1.0], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 95.9

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

   8. Applied rewrites 95.9%

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

   10. Applied rewrites 95.8%

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

   12. Applied rewrites 96.0%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.9%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.5%

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

   10. Applied rewrites 93.2%

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

   12. Applied rewrites 93.2%

       \[\leadsto \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right)}} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right)\right)}^{-1}\right)}^{-1} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, 5, \left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left({\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right)\right)}^{-1}\right)}^{-1} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-322)
     (* (* (* (* (fma x 5.0 eps) eps) eps) eps) eps)
     (if (<= t_0 0.0)
       (* (* (* (fma (* x x) 5.0 (* (* 10.0 eps) (+ eps x))) x) x) eps)
       (*
        (*
         (pow (pow (fma (fma x 5.0 eps) eps (* (* 10.0 x) x)) -1.0) -1.0)
         (* eps eps))
        eps)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-322) {
		tmp = (((fma(x, 5.0, eps) * eps) * eps) * eps) * eps;
	} else if (t_0 <= 0.0) {
		tmp = ((fma((x * x), 5.0, ((10.0 * eps) * (eps + x))) * x) * x) * eps;
	} else {
		tmp = (pow(pow(fma(fma(x, 5.0, eps), eps, ((10.0 * x) * x)), -1.0), -1.0) * (eps * eps)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-322)
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, eps) * eps) * eps) * eps) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 5.0, Float64(Float64(10.0 * eps) * Float64(eps + x))) * x) * x) * eps);
	else
		tmp = Float64(Float64(((fma(fma(x, 5.0, eps), eps, Float64(Float64(10.0 * x) * x)) ^ -1.0) ^ -1.0) * Float64(eps * eps)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-322], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0 + N[(N[(10.0 * eps), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[Power[N[Power[N[(N[(x * 5.0 + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], -1.0], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 95.9

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

   8. Applied rewrites 95.9%

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

   10. Applied rewrites 95.8%

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

   12. Applied rewrites 96.0%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.9%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.5%

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

   10. Applied rewrites 93.2%

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

   12. Applied rewrites 93.2%

       \[\leadsto \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right)}} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, 5, \left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left({\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right)\right)}^{-1}\right)}^{-1} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-322} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -4e-322) (not (<= t_0 0.0)))
     (* (+ (/ (* (fma 10.0 (/ x eps) 5.0) x) eps) 1.0) (pow eps 5.0))
     (* (* 5.0 eps) (pow x 4.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-322) || !(t_0 <= 0.0)) {
		tmp = (((fma(10.0, (x / eps), 5.0) * x) / eps) + 1.0) * pow(eps, 5.0);
	} else {
		tmp = (5.0 * eps) * pow(x, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -4e-322) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(Float64(fma(10.0, Float64(x / eps), 5.0) * x) / eps) + 1.0) * (eps ^ 5.0));
	else
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-322], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(N[(N[(10.0 * N[(x / eps), $MachinePrecision] + 5.0), $MachinePrecision] * x), $MachinePrecision] / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.2%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.1%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-322} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-322}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot {\varepsilon}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-322)
     (* (fma (/ x eps) 5.0 1.0) (pow eps 5.0))
     (if (<= t_0 0.0)
       (* (* 5.0 eps) (pow x 4.0))
       (* (fma (fma 5.0 x eps) eps (* (* 10.0 x) x)) (pow eps 3.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-322) {
		tmp = fma((x / eps), 5.0, 1.0) * pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = fma(fma(5.0, x, eps), eps, ((10.0 * x) * x)) * pow(eps, 3.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-322)
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(10.0 * x) * x)) * (eps ^ 3.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-322], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $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[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.9%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.5%

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

Alternative 8: 98.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-322)
     (* (* (* (* (fma x 5.0 eps) eps) eps) eps) eps)
     (if (<= t_0 0.0)
       (* (* (* (fma (* x x) 5.0 (* (* 10.0 eps) (+ eps x))) x) x) eps)
       (* (* (fma (fma 5.0 x eps) eps (* (* 10.0 x) x)) (* eps eps)) eps)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-322) {
		tmp = (((fma(x, 5.0, eps) * eps) * eps) * eps) * eps;
	} else if (t_0 <= 0.0) {
		tmp = ((fma((x * x), 5.0, ((10.0 * eps) * (eps + x))) * x) * x) * eps;
	} else {
		tmp = (fma(fma(5.0, x, eps), eps, ((10.0 * x) * x)) * (eps * eps)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-322)
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, eps) * eps) * eps) * eps) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 5.0, Float64(Float64(10.0 * eps) * Float64(eps + x))) * x) * x) * eps);
	else
		tmp = Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(10.0 * x) * x)) * Float64(eps * eps)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-322], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0 + N[(N[(10.0 * eps), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 95.9

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

   8. Applied rewrites 95.9%

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

   10. Applied rewrites 95.8%

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

   12. Applied rewrites 96.0%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.9%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.5%

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

   10. Applied rewrites 93.2%

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

Alternative 9: 98.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-322)
     (* (* (* (* (fma x 5.0 eps) eps) eps) eps) eps)
     (if (<= t_0 0.0)
       (* (* eps (fma (* x x) 5.0 (* (* 10.0 eps) (+ eps x)))) (* x x))
       (* (* (fma (fma 5.0 x eps) eps (* (* 10.0 x) x)) (* eps eps)) eps)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-322) {
		tmp = (((fma(x, 5.0, eps) * eps) * eps) * eps) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (eps * fma((x * x), 5.0, ((10.0 * eps) * (eps + x)))) * (x * x);
	} else {
		tmp = (fma(fma(5.0, x, eps), eps, ((10.0 * x) * x)) * (eps * eps)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-322)
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, eps) * eps) * eps) * eps) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(eps * fma(Float64(x * x), 5.0, Float64(Float64(10.0 * eps) * Float64(eps + x)))) * Float64(x * x));
	else
		tmp = Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(10.0 * x) * x)) * Float64(eps * eps)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-322], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(eps * N[(N[(x * x), $MachinePrecision] * 5.0 + N[(N[(10.0 * eps), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 95.9

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

   8. Applied rewrites 95.9%

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

   10. Applied rewrites 95.8%

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

   12. Applied rewrites 96.0%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.9%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.5%

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

   10. Applied rewrites 93.2%

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

Alternative 10: 98.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-322)
     (* (* (* (* (fma x 5.0 eps) eps) eps) eps) eps)
     (if (<= t_0 0.0)
       (* (* (* (* x x) 5.0) eps) (* x x))
       (* (* (fma (fma 5.0 x eps) eps (* (* 10.0 x) x)) (* eps eps)) eps)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-322) {
		tmp = (((fma(x, 5.0, eps) * eps) * eps) * eps) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	} else {
		tmp = (fma(fma(5.0, x, eps), eps, ((10.0 * x) * x)) * (eps * eps)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-322)
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, eps) * eps) * eps) * eps) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	else
		tmp = Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(10.0 * x) * x)) * Float64(eps * eps)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-322], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 95.9

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

   8. Applied rewrites 95.9%

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

   10. Applied rewrites 95.8%

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

   12. Applied rewrites 96.0%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.9%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.5%

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

   10. Applied rewrites 93.2%

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

Alternative 11: 98.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -4e-322) (not (<= t_0 0.0)))
     (* (* (* (* (fma x 5.0 eps) eps) eps) eps) eps)
     (* (* (* (* x x) 5.0) eps) (* x x)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-322) || !(t_0 <= 0.0)) {
		tmp = (((fma(x, 5.0, eps) * eps) * eps) * eps) * eps;
	} else {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -4e-322) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, eps) * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-322], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.2%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 94.8

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 94.3

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

   8. Applied rewrites 94.3%

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

   10. Applied rewrites 94.2%

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

   12. Applied rewrites 94.3%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-322} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-322)
     (* (* (* (* (fma x 5.0 eps) eps) eps) eps) eps)
     (if (<= t_0 0.0)
       (* (* (* (* x x) 5.0) eps) (* x x))
       (* (fma 5.0 x eps) (* (* eps eps) (* eps eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-322) {
		tmp = (((fma(x, 5.0, eps) * eps) * eps) * eps) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	} else {
		tmp = fma(5.0, x, eps) * ((eps * eps) * (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-322)
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, eps) * eps) * eps) * eps) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	else
		tmp = Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * 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]}, If[LessEqual[t$95$0, -4e-322], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * 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))) < -4.00193e-322

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 95.9

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

   8. Applied rewrites 95.9%

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

   10. Applied rewrites 95.8%

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

   12. Applied rewrites 96.0%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.9%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 92.5

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

   8. Applied rewrites 92.5%

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

   10. Applied rewrites 92.5%

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

Alternative 13: 98.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -4e-322)
     (* (* (* (* (fma x 5.0 eps) eps) eps) eps) eps)
     (if (<= t_0 0.0)
       (* (* (* (* x x) 5.0) eps) (* x x))
       (* (* (fma 5.0 x eps) (* eps eps)) (* eps eps))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -4e-322) {
		tmp = (((fma(x, 5.0, eps) * eps) * eps) * eps) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	} else {
		tmp = (fma(5.0, x, eps) * (eps * eps)) * (eps * eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -4e-322)
		tmp = Float64(Float64(Float64(Float64(fma(x, 5.0, eps) * eps) * eps) * eps) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	else
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(eps * eps)) * 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]}, If[LessEqual[t$95$0, -4e-322], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 95.9

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

   8. Applied rewrites 95.9%

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

   10. Applied rewrites 95.8%

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

   12. Applied rewrites 96.0%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.9%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 92.5

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

   8. Applied rewrites 92.5%

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

   10. Applied rewrites 92.4%

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

Alternative 14: 98.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -4e-322) (not (<= t_0 0.0)))
     (* (* (* eps eps) (* eps eps)) eps)
     (* (* (* (* x x) 5.0) eps) (* x x)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-322) || !(t_0 <= 0.0)) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-4d-322)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = ((eps * eps) * (eps * eps)) * eps
    else
        tmp = (((x * x) * 5.0d0) * eps) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-322) || !(t_0 <= 0.0)) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -4e-322) or not (t_0 <= 0.0):
		tmp = ((eps * eps) * (eps * eps)) * eps
	else:
		tmp = (((x * x) * 5.0) * eps) * (x * x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -4e-322) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps);
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -4e-322) || ~((t_0 <= 0.0)))
		tmp = ((eps * eps) * (eps * eps)) * eps;
	else
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-322], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.00193e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.2%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 94.8

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.8%

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

   10. Applied rewrites 94.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 93.5%

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

   if -4.00193e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-322} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 87.3% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.8%

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

3. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 88.0

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

5. Applied rewrites 88.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 88.1%

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

10. Applied rewrites 88.0%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 87.8%

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

Reproduce

herbie shell --seed 2024318 
(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)))