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

Percentage Accurate: 88.8% → 99.0%

Time: 8.2s

Alternatives: 15

Speedup: 1.5×

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: 88.8% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-318)
     (* (pow eps 5.0) (fma 5.0 (/ x eps) 1.0))
     (if (<= t_0 0.0) (* (pow x 4.0) (* eps 5.0)) t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = pow(eps, 5.0) * fma(5.0, (x / eps), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = Float64((eps ^ 5.0) * fma(5.0, Float64(x / eps), 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = t_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, -1e-318], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

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))) < -9.9999875e-319

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 99.9

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

   8. Simplified 99.9%

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

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

   1. Initial program 97.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-318)
     (* (pow eps 5.0) (fma 5.0 (/ x eps) 1.0))
     (if (<= t_0 0.0) (* (pow x 4.0) (* eps 5.0)) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = pow(eps, 5.0) * fma(5.0, (x / eps), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = Float64((eps ^ 5.0) * fma(5.0, Float64(x / eps), 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-318], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9999875e-319

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 99.9

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

   8. Simplified 99.9%

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

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

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 96.6

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

   5. Simplified 96.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\frac{\frac{\mathsf{fma}\left(x, -5, \varepsilon\right)}{\mathsf{fma}\left(5, x, \varepsilon\right)}}{\mathsf{fma}\left(x, -5, \varepsilon\right)}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-318)
     (/
      (* eps (* eps (* eps eps)))
      (/ (/ (fma x -5.0 eps) (fma 5.0 x eps)) (fma x -5.0 eps)))
     (if (<= t_0 0.0) (* (pow x 4.0) (* eps 5.0)) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = (eps * (eps * (eps * eps))) / ((fma(x, -5.0, eps) / fma(5.0, x, eps)) / fma(x, -5.0, eps));
	} else if (t_0 <= 0.0) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) / Float64(Float64(fma(x, -5.0, eps) / fma(5.0, x, eps)) / fma(x, -5.0, eps)));
	elseif (t_0 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-318], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * -5.0 + eps), $MachinePrecision] / N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision] / N[(x * -5.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9999875e-319

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 99.0

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

   8. Simplified 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. cube-unmult N/A

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

      14. lower-*.f64 N/A

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

      15. cube-unmult N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

   10. Applied egg-rr 99.4%

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

       1. lift-*.f64 N/A

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

       2. +-commutative N/A

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

       3. flip-+ N/A

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

       4. clear-num N/A

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

       5. difference-of-squares N/A

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

       6. +-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. sub-neg N/A

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

       13. +-commutative N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. distribute-rgt-neg-in N/A

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

       17. lower-fma.f64 N/A

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

       18. metadata-eval N/A

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

       19. sub-neg N/A

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

       20. +-commutative N/A

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

   12. Applied egg-rr 99.6%

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

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 99.9

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

   8. Simplified 99.9%

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

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

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 96.6

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

   5. Simplified 96.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\frac{\frac{\mathsf{fma}\left(x, -5, \varepsilon\right)}{\mathsf{fma}\left(5, x, \varepsilon\right)}}{\mathsf{fma}\left(x, -5, \varepsilon\right)}}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\frac{\frac{\mathsf{fma}\left(x, -5, \varepsilon\right)}{\mathsf{fma}\left(5, x, \varepsilon\right)}}{\mathsf{fma}\left(x, -5, \varepsilon\right)}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-318)
     (/
      (* eps (* eps (* eps eps)))
      (/ (/ (fma x -5.0 eps) (fma 5.0 x eps)) (fma x -5.0 eps)))
     (if (<= t_0 0.0) (* eps (* 5.0 (pow x 4.0))) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = (eps * (eps * (eps * eps))) / ((fma(x, -5.0, eps) / fma(5.0, x, eps)) / fma(x, -5.0, eps));
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) / Float64(Float64(fma(x, -5.0, eps) / fma(5.0, x, eps)) / fma(x, -5.0, eps)));
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-318], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * -5.0 + eps), $MachinePrecision] / N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision] / N[(x * -5.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9999875e-319

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 99.0

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

   8. Simplified 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. cube-unmult N/A

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

      14. lower-*.f64 N/A

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

      15. cube-unmult N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

   10. Applied egg-rr 99.4%

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

       1. lift-*.f64 N/A

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

       2. +-commutative N/A

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

       3. flip-+ N/A

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

       4. clear-num N/A

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

       5. difference-of-squares N/A

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

       6. +-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. sub-neg N/A

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

       13. +-commutative N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. distribute-rgt-neg-in N/A

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

       17. lower-fma.f64 N/A

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

       18. metadata-eval N/A

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

       19. sub-neg N/A

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

       20. +-commutative N/A

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

   12. Applied egg-rr 99.6%

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

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.9

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

   5. Simplified 99.9%

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

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

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 96.6

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

   5. Simplified 96.6%

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

Alternative 5: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\frac{\frac{\mathsf{fma}\left(x, -5, \varepsilon\right)}{\mathsf{fma}\left(5, x, \varepsilon\right)}}{\mathsf{fma}\left(x, -5, \varepsilon\right)}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\varepsilon \cdot t\_1\right), 5, \varepsilon \cdot \left(\varepsilon \cdot \left(t\_1 \cdot 10\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))) (t_1 (* x (* x x))))
   (if (<= t_0 -1e-318)
     (/
      (* eps (* eps (* eps eps)))
      (/ (/ (fma x -5.0 eps) (fma 5.0 x eps)) (fma x -5.0 eps)))
     (if (<= t_0 0.0)
       (fma (* x (* eps t_1)) 5.0 (* eps (* eps (* t_1 10.0))))
       (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = x * (x * x);
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = (eps * (eps * (eps * eps))) / ((fma(x, -5.0, eps) / fma(5.0, x, eps)) / fma(x, -5.0, eps));
	} else if (t_0 <= 0.0) {
		tmp = fma((x * (eps * t_1)), 5.0, (eps * (eps * (t_1 * 10.0))));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) / Float64(Float64(fma(x, -5.0, eps) / fma(5.0, x, eps)) / fma(x, -5.0, eps)));
	elseif (t_0 <= 0.0)
		tmp = fma(Float64(x * Float64(eps * t_1)), 5.0, Float64(eps * Float64(eps * Float64(t_1 * 10.0))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-318], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * -5.0 + eps), $MachinePrecision] / N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision] / N[(x * -5.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision] * 5.0 + N[(eps * N[(eps * N[(t$95$1 * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9999875e-319

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 99.0

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

   8. Simplified 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. cube-unmult N/A

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

      14. lower-*.f64 N/A

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

      15. cube-unmult N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

   10. Applied egg-rr 99.4%

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

       1. lift-*.f64 N/A

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

       2. +-commutative N/A

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

       3. flip-+ N/A

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

       4. clear-num N/A

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

       5. difference-of-squares N/A

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

       6. +-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. sub-neg N/A

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

       13. +-commutative N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. distribute-rgt-neg-in N/A

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

       17. lower-fma.f64 N/A

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

       18. metadata-eval N/A

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

       19. sub-neg N/A

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

       20. +-commutative N/A

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

   12. Applied egg-rr 99.6%

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

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 96.6

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

   5. Simplified 96.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\frac{\frac{\mathsf{fma}\left(x, -5, \varepsilon\right)}{\mathsf{fma}\left(5, x, \varepsilon\right)}}{\mathsf{fma}\left(x, -5, \varepsilon\right)}}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 5, \varepsilon \cdot \left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\frac{\frac{\mathsf{fma}\left(x, -5, \varepsilon\right)}{\mathsf{fma}\left(5, x, \varepsilon\right)}}{\mathsf{fma}\left(x, -5, \varepsilon\right)}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\varepsilon \cdot t\_1\right), 5, \varepsilon \cdot \left(\varepsilon \cdot \left(t\_1 \cdot 10\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))) (t_1 (* x (* x x))))
   (if (<= t_0 -1e-318)
     (/
      (* eps (* eps (* eps eps)))
      (/ (/ (fma x -5.0 eps) (fma 5.0 x eps)) (fma x -5.0 eps)))
     (if (<= t_0 0.0)
       (fma (* x (* eps t_1)) 5.0 (* eps (* eps (* t_1 10.0))))
       (/ eps (/ 1.0 (* (* eps eps) (* eps eps))))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = x * (x * x);
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = (eps * (eps * (eps * eps))) / ((fma(x, -5.0, eps) / fma(5.0, x, eps)) / fma(x, -5.0, eps));
	} else if (t_0 <= 0.0) {
		tmp = fma((x * (eps * t_1)), 5.0, (eps * (eps * (t_1 * 10.0))));
	} else {
		tmp = eps / (1.0 / ((eps * eps) * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) / Float64(Float64(fma(x, -5.0, eps) / fma(5.0, x, eps)) / fma(x, -5.0, eps)));
	elseif (t_0 <= 0.0)
		tmp = fma(Float64(x * Float64(eps * t_1)), 5.0, Float64(eps * Float64(eps * Float64(t_1 * 10.0))));
	else
		tmp = Float64(eps / Float64(1.0 / 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]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-318], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * -5.0 + eps), $MachinePrecision] / N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision] / N[(x * -5.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision] * 5.0 + N[(eps * N[(eps * N[(t$95$1 * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(1.0 / N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $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))) < -9.9999875e-319

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 99.0

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

   8. Simplified 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. cube-unmult N/A

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

      14. lower-*.f64 N/A

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

      15. cube-unmult N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

   10. Applied egg-rr 99.4%

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

       1. lift-*.f64 N/A

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

       2. +-commutative N/A

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

       3. flip-+ N/A

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

       4. clear-num N/A

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

       5. difference-of-squares N/A

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

       6. +-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. associate-/r* N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. sub-neg N/A

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

       13. +-commutative N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. distribute-rgt-neg-in N/A

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

       17. lower-fma.f64 N/A

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

       18. metadata-eval N/A

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

       19. sub-neg N/A

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

       20. +-commutative N/A

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

   12. Applied egg-rr 99.6%

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

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 97.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 96.4

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

   5. Simplified 96.4%

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

   6. Taylor expanded in eps around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 95.8

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

   8. Simplified 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. cube-unmult N/A

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

      6. lower-*.f64 N/A

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

      7. cube-unmult N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 95.9

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

   10. Applied egg-rr 95.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. remove-double-div N/A

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

       6. un-div-inv N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 95.9

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

   12. Applied egg-rr 95.9%

       \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\frac{\frac{\mathsf{fma}\left(x, -5, \varepsilon\right)}{\mathsf{fma}\left(5, x, \varepsilon\right)}}{\mathsf{fma}\left(x, -5, \varepsilon\right)}}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 5, \varepsilon \cdot \left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\frac{1}{\mathsf{fma}\left(5, x, \varepsilon\right)}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\varepsilon \cdot t\_1\right), 5, \varepsilon \cdot \left(\varepsilon \cdot \left(t\_1 \cdot 10\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))) (t_1 (* x (* x x))))
   (if (<= t_0 -1e-318)
     (/ (* eps (* eps (* eps eps))) (/ 1.0 (fma 5.0 x eps)))
     (if (<= t_0 0.0)
       (fma (* x (* eps t_1)) 5.0 (* eps (* eps (* t_1 10.0))))
       (/ eps (/ 1.0 (* (* eps eps) (* eps eps))))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = x * (x * x);
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = (eps * (eps * (eps * eps))) / (1.0 / fma(5.0, x, eps));
	} else if (t_0 <= 0.0) {
		tmp = fma((x * (eps * t_1)), 5.0, (eps * (eps * (t_1 * 10.0))));
	} else {
		tmp = eps / (1.0 / ((eps * eps) * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) / Float64(1.0 / fma(5.0, x, eps)));
	elseif (t_0 <= 0.0)
		tmp = fma(Float64(x * Float64(eps * t_1)), 5.0, Float64(eps * Float64(eps * Float64(t_1 * 10.0))));
	else
		tmp = Float64(eps / Float64(1.0 / 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]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-318], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision] * 5.0 + N[(eps * N[(eps * N[(t$95$1 * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(1.0 / N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $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))) < -9.9999875e-319

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 99.0

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

   8. Simplified 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. cube-unmult N/A

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

      14. lower-*.f64 N/A

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

      15. cube-unmult N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

   10. Applied egg-rr 99.4%

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

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 97.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 96.4

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

   5. Simplified 96.4%

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

   6. Taylor expanded in eps around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 95.8

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

   8. Simplified 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. cube-unmult N/A

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

      6. lower-*.f64 N/A

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

      7. cube-unmult N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 95.9

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

   10. Applied egg-rr 95.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. remove-double-div N/A

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

       6. un-div-inv N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 95.9

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

   12. Applied egg-rr 95.9%

       \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\frac{1}{\mathsf{fma}\left(5, x, \varepsilon\right)}}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 5, \varepsilon \cdot \left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\frac{1}{\mathsf{fma}\left(5, x, \varepsilon\right)}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-318)
     (/ (* eps (* eps (* eps eps))) (/ 1.0 (fma 5.0 x eps)))
     (if (<= t_0 0.0)
       (* eps (* 5.0 (* x (* x (* x x)))))
       (/ eps (/ 1.0 (* (* 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 <= -1e-318) {
		tmp = (eps * (eps * (eps * eps))) / (1.0 / fma(5.0, x, eps));
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * (x * (x * (x * x))));
	} else {
		tmp = eps / (1.0 / ((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 <= -1e-318)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) / Float64(1.0 / fma(5.0, x, eps)));
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))));
	else
		tmp = Float64(eps / Float64(1.0 / 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, -1e-318], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(1.0 / N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $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))) < -9.9999875e-319

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 99.0

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

   8. Simplified 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. cube-unmult N/A

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

      14. lower-*.f64 N/A

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

      15. cube-unmult N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

   10. Applied egg-rr 99.4%

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

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.9

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

   5. Simplified 99.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.9

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 97.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 96.4

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

   5. Simplified 96.4%

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

   6. Taylor expanded in eps around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 95.8

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

   8. Simplified 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. cube-unmult N/A

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

      6. lower-*.f64 N/A

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

      7. cube-unmult N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 95.9

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

   10. Applied egg-rr 95.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. remove-double-div N/A

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

       6. un-div-inv N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 95.9

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

   12. Applied egg-rr 95.9%

       \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\frac{1}{\mathsf{fma}\left(5, x, \varepsilon\right)}}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* eps eps) (* eps eps))))
   (if (<= t_0 -1e-318)
     (* (fma 5.0 x eps) t_1)
     (if (<= t_0 0.0)
       (* eps (* 5.0 (* x (* x (* x x)))))
       (/ eps (/ 1.0 t_1))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * eps);
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = fma(5.0, x, eps) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * (x * (x * (x * x))));
	} else {
		tmp = eps / (1.0 / t_1);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * Float64(eps * eps))
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = Float64(fma(5.0, x, eps) * t_1);
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))));
	else
		tmp = Float64(eps / Float64(1.0 / t_1));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-318], N[(N[(5.0 * x + eps), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(1.0 / t$95$1), $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))) < -9.9999875e-319

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 99.0

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

   8. Simplified 99.0%

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

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.9

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

   5. Simplified 99.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.9

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 97.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 96.4

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

   5. Simplified 96.4%

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

   6. Taylor expanded in eps around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 95.8

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

   8. Simplified 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. cube-unmult N/A

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

      6. lower-*.f64 N/A

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

      7. cube-unmult N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 95.9

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

   10. Applied egg-rr 95.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. remove-double-div N/A

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

       6. un-div-inv N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 95.9

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

   12. Applied egg-rr 95.9%

       \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{1}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-318)
     (* (fma 5.0 x eps) (* (* eps eps) (* eps eps)))
     (if (<= t_0 0.0)
       (* eps (* 5.0 (* x (* x (* x 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 <= -1e-318) {
		tmp = fma(5.0, x, eps) * ((eps * eps) * (eps * eps));
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * (x * (x * (x * x))));
	} else {
		tmp = (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 <= -1e-318)
		tmp = Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * Float64(eps * eps)));
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))));
	else
		tmp = Float64(Float64(eps * eps) * Float64(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, -1e-318], N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(eps * 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))) < -9.9999875e-319

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 99.0

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

   8. Simplified 99.0%

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

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.9

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

   5. Simplified 99.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.9

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 97.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 96.4

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

   5. Simplified 96.4%

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

   6. Taylor expanded in eps around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 95.8

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

   8. Simplified 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. cube-unmult N/A

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

      6. lower-*.f64 N/A

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

      7. cube-unmult N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 95.9

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

   10. Applied egg-rr 95.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* eps eps) (* eps (* eps eps)))))
   (if (<= t_0 -1e-318)
     t_1
     (if (<= t_0 0.0) (* eps (* 5.0 (* x (* x (* x x))))) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * (eps * eps));
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * (x * (x * (x * x))));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = (eps * eps) * (eps * (eps * eps))
    if (t_0 <= (-1d-318)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x * (x * (x * x))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * (eps * eps));
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * (x * (x * (x * x))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = (eps * eps) * (eps * (eps * eps))
	tmp = 0
	if t_0 <= -1e-318:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = eps * (5.0 * (x * (x * (x * x))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = (eps * eps) * (eps * (eps * eps));
	tmp = 0.0;
	if (t_0 <= -1e-318)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = eps * (5.0 * (x * (x * (x * x))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-318], t$95$1, If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9999875e-319 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 97.6

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

   5. Simplified 97.6%

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

   6. Taylor expanded in eps around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \]

      8. unpow2 N/A

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

      11. lower-*.f64 96.2

          \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   8. Simplified 96.2%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]

      5. cube-unmult N/A

         \[\leadsto \color{blue}{{\varepsilon}^{3}} \cdot \left(\varepsilon \cdot \varepsilon\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

      7. cube-unmult N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \cdot \left(\varepsilon \cdot \varepsilon\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]

      9. lower-*.f64 96.3

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \cdot \left(\varepsilon \cdot \varepsilon\right) \]

   10. Applied egg-rr 96.3%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {x}^{4} + \left(4 \cdot \varepsilon\right) \cdot {x}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \varepsilon \cdot {x}^{4} + \color{blue}{\left(\varepsilon \cdot 4\right)} \cdot {x}^{4} \]

      3. associate-*r* N/A

         \[\leadsto \varepsilon \cdot {x}^{4} + \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4}\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4}\right) + \varepsilon \cdot {x}^{4}} \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]

      7. distribute-lft1-in N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(4 + 1\right) \cdot {x}^{4}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \varepsilon \cdot \left(\color{blue}{5} \cdot {x}^{4}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(5 \cdot {x}^{4}\right)} \]

      10. lower-pow.f64 99.9

          \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(5 \cdot {x}^{4}\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 \cdot {x}^{4}\right) \cdot \varepsilon} \]

      4. lower-*.f64 99.9

         \[\leadsto \color{blue}{\left(5 \cdot {x}^{4}\right) \cdot \varepsilon} \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(5 \cdot \color{blue}{{x}^{4}}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{\color{blue}{\left(2 + 2\right)}}\right) \cdot \varepsilon \]

      7. pow-prod-up N/A

         \[\leadsto \left(5 \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right) \cdot \varepsilon \]

      8. pow2 N/A

         \[\leadsto \left(5 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      9. pow2 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \varepsilon \]

      10. lift-*.f64 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \varepsilon \]

      11. associate-*l* N/A

          \[\leadsto \left(5 \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right) \cdot \varepsilon \]

      12. lift-*.f64 N/A

          \[\leadsto \left(5 \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \varepsilon \]

      13. lower-*.f64 99.9

          \[\leadsto \left(5 \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right) \cdot \varepsilon \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(x \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* eps eps) (* eps (* eps eps)))))
   (if (<= t_0 -1e-318)
     t_1
     (if (<= t_0 0.0) (* eps (* x (* x (* 5.0 (* x x))))) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * (eps * eps));
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (x * (x * (5.0 * (x * x))));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = (eps * eps) * (eps * (eps * eps))
    if (t_0 <= (-1d-318)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = eps * (x * (x * (5.0d0 * (x * x))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * (eps * eps));
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (x * (x * (5.0 * (x * x))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = (eps * eps) * (eps * (eps * eps))
	tmp = 0
	if t_0 <= -1e-318:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = eps * (x * (x * (5.0 * (x * x))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(x * Float64(x * Float64(5.0 * Float64(x * x)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = (eps * eps) * (eps * (eps * eps));
	tmp = 0.0;
	if (t_0 <= -1e-318)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = eps * (x * (x * (5.0 * (x * x))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-318], t$95$1, If[LessEqual[t$95$0, 0.0], N[(eps * N[(x * N[(x * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.9999875e-319 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5}} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \]

      5. metadata-eval N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

      7. lower-/.f64 97.6

         \[\leadsto {\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \color{blue}{\frac{x}{\varepsilon}}, 1\right) \]

   5. Simplified 97.6%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\color{blue}{\left(4 + 1\right)}} \]

      2. pow-plus N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

      5. metadata-eval N/A

         \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(2 \cdot 2\right)}} \]

      6. pow-sqr N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \]

      8. unpow2 N/A

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

      11. lower-*.f64 96.2

          \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   8. Simplified 96.2%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]

      5. cube-unmult N/A

         \[\leadsto \color{blue}{{\varepsilon}^{3}} \cdot \left(\varepsilon \cdot \varepsilon\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

      7. cube-unmult N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \cdot \left(\varepsilon \cdot \varepsilon\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]

      9. lower-*.f64 96.3

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \cdot \left(\varepsilon \cdot \varepsilon\right) \]

   10. Applied egg-rr 96.3%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   if -9.9999875e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {x}^{4} + \left(4 \cdot \varepsilon\right) \cdot {x}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \varepsilon \cdot {x}^{4} + \color{blue}{\left(\varepsilon \cdot 4\right)} \cdot {x}^{4} \]

      3. associate-*r* N/A

         \[\leadsto \varepsilon \cdot {x}^{4} + \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4}\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4}\right) + \varepsilon \cdot {x}^{4}} \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]

      7. distribute-lft1-in N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(4 + 1\right) \cdot {x}^{4}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \varepsilon \cdot \left(\color{blue}{5} \cdot {x}^{4}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(5 \cdot {x}^{4}\right)} \]

      10. lower-pow.f64 99.9

          \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{x}^{4}}\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \]

      2. pow-pow N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{{\left({x}^{2}\right)}^{2}}\right) \]

      3. pow2 N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {\color{blue}{\left(x \cdot x\right)}}^{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {\color{blue}{\left(x \cdot x\right)}}^{2}\right) \]

      5. pow2 N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(5 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(\left(5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\left(5 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\left(5 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\left(5 \cdot \left(x \cdot x\right)\right) \cdot x\right)} \cdot x\right) \]

      11. lower-*.f64 99.9

          \[\leadsto \varepsilon \cdot \left(\left(\color{blue}{\left(5 \cdot \left(x \cdot x\right)\right)} \cdot x\right) \cdot x\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\left(5 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(x \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-45}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-48}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.4e-45)
   (* (pow x 4.0) (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)))
   (if (<= x 4.5e-48)
     (* (pow eps 5.0) (fma 5.0 (/ x eps) 1.0))
     (* (pow x 4.0) (* eps 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.4e-45) {
		tmp = pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} else if (x <= 4.5e-48) {
		tmp = pow(eps, 5.0) * fma(5.0, (x / eps), 1.0);
	} else {
		tmp = pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.4e-45)
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)));
	elseif (x <= 4.5e-48)
		tmp = Float64((eps ^ 5.0) * fma(5.0, Float64(x / eps), 1.0));
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.4e-45], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-48], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.3999999999999997e-45

   1. Initial program 21.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{x}^{4}} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. associate-+r+ N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      7. lower--.f64 N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      9. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\varepsilon \cdot 5 - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   if -5.3999999999999997e-45 < x < 4.49999999999999988e-48

   1. Initial program 99.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5}} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \]

      5. metadata-eval N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

      7. lower-/.f64 99.6

         \[\leadsto {\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \color{blue}{\frac{x}{\varepsilon}}, 1\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

   if 4.49999999999999988e-48 < x

   1. Initial program 41.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{x}^{4}} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. associate-+r+ N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      7. lower--.f64 N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      9. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\varepsilon \cdot 5 - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {x}^{4} \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 99.8

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]

   8. Simplified 99.8%

      \[\leadsto {x}^{4} \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-45}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-48}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 87.6% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* eps eps) (* eps (* eps eps))))

C

double code(double x, double eps) {
	return (eps * eps) * (eps * (eps * eps));
}

Fortran

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(eps * eps) * Float64(eps * Float64(eps * eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * eps) * (eps * (eps * eps));
end

Wolfram

code[x_, eps_] := N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.3%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
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. lower-*.f64 N/A

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]

   2. lower-pow.f64 N/A

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \]

   5. metadata-eval N/A

      \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

   7. lower-/.f64 86.1

      \[\leadsto {\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \color{blue}{\frac{x}{\varepsilon}}, 1\right) \]

5. Simplified 86.1%

   \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

6. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
7. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\color{blue}{\left(4 + 1\right)}} \]

   2. pow-plus N/A

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

   5. metadata-eval N/A

      \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(2 \cdot 2\right)}} \]

   6. pow-sqr N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \]

   8. unpow2 N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \]

   10. unpow2 N/A

       \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   11. lower-*.f64 85.9

       \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

8. Simplified 85.9%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]

   5. cube-unmult N/A

      \[\leadsto \color{blue}{{\varepsilon}^{3}} \cdot \left(\varepsilon \cdot \varepsilon\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   7. cube-unmult N/A

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \cdot \left(\varepsilon \cdot \varepsilon\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]

   9. lower-*.f64 85.9

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \cdot \left(\varepsilon \cdot \varepsilon\right) \]

10. Applied egg-rr 85.9%

    \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

11. Final simplification 85.9%

    \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]
12. Add Preprocessing

Alternative 15: 87.6% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (* (* eps eps) (* eps eps))))

C

double code(double x, double eps) {
	return eps * ((eps * eps) * (eps * eps));
}

Fortran

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(eps * Float64(Float64(eps * eps) * Float64(eps * eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * eps) * (eps * eps));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.3%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
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. lower-*.f64 N/A

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]

   2. lower-pow.f64 N/A

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \]

   5. metadata-eval N/A

      \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

   7. lower-/.f64 86.1

      \[\leadsto {\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \color{blue}{\frac{x}{\varepsilon}}, 1\right) \]

5. Simplified 86.1%

   \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

6. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
7. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\color{blue}{\left(4 + 1\right)}} \]

   2. pow-plus N/A

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

   5. metadata-eval N/A

      \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(2 \cdot 2\right)}} \]

   6. pow-sqr N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \]

   8. unpow2 N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \]

   10. unpow2 N/A

       \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   11. lower-*.f64 85.9

       \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

8. Simplified 85.9%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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)))