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

Percentage Accurate: 88.8% → 97.3%

Time: 8.1s

Alternatives: 8

Speedup: 5.5×

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: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-33}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 7.3 \cdot 10^{-44}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -3.5e-33)
     (* t_0 (* eps (* x 5.0)))
     (if (<= x 7.3e-44)
       (* (pow eps 5.0) (fma 5.0 (/ x eps) 1.0))
       (* 5.0 (* eps (* x t_0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.5e-33) {
		tmp = t_0 * (eps * (x * 5.0));
	} else if (x <= 7.3e-44) {
		tmp = pow(eps, 5.0) * fma(5.0, (x / eps), 1.0);
	} else {
		tmp = 5.0 * (eps * (x * t_0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -3.5e-33)
		tmp = Float64(t_0 * Float64(eps * Float64(x * 5.0)));
	elseif (x <= 7.3e-44)
		tmp = Float64((eps ^ 5.0) * fma(5.0, Float64(x / eps), 1.0));
	else
		tmp = Float64(5.0 * Float64(eps * Float64(x * t_0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5e-33], N[(t$95$0 * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.3e-44], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(5.0 * N[(eps * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4999999999999999e-33

   1. Initial program 17.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      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 99.6

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

   5. Simplified 99.6%

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

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. cube-mult N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 99.6

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

   7. Applied egg-rr 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 99.7

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

   9. Applied egg-rr 99.7%

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

   if -3.4999999999999999e-33 < x < 7.29999999999999987e-44

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 7.29999999999999987e-44 < x

   1. Initial program 55.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      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 99.7

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

   5. Simplified 99.7%

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

      1. lift-pow.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 99.8

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

      6. lift-pow.f64 N/A

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

      7. sqr-pow N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. cube-mult N/A

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

      14. lower-*.f64 N/A

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-33}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 7.3 \cdot 10^{-44}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-318}:\\ \;\;\;\;\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-318)
     (* (fma 5.0 (/ x eps) 1.0) (* 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 <= -2e-318) {
		tmp = fma(5.0, (x / eps), 1.0) * (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 <= -2e-318)
		tmp = Float64(fma(5.0, Float64(x / eps), 1.0) * Float64(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, -2e-318], N[(N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $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))) < -2.0000024e-318

   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

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(\varepsilon \cdot {\varepsilon}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \]

      6. pow-sqr N/A

         \[\leadsto \left(\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)}\right) \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)}\right) \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right)\right) \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right)\right) \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \]

      10. unpow2 N/A

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

      11. lower-*.f64 99.5

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

   8. Simplified 99.5%

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

   if -2.0000024e-318 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 92.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      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 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]
   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 100.0

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. cube-mult N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

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

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 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 99.3

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

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
   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. lower-*.f64 N/A

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

      5. lower-*.f64 99.6

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

   10. Applied egg-rr 99.6%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-318}:\\ \;\;\;\;\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-318}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-318)
     (* (* (* eps eps) (* eps eps)) (fma 5.0 x 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 <= -2e-318) {
		tmp = ((eps * eps) * (eps * eps)) * fma(5.0, x, 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 <= -2e-318)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * 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(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, -2e-318], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(5.0 * x + eps), $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))) < -2.0000024e-318

   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 99.4

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

   8. Simplified 99.4%

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

   if -2.0000024e-318 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 92.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      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 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]
   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 100.0

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. cube-mult N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

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

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 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 99.3

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

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
   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. lower-*.f64 N/A

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

      5. lower-*.f64 99.6

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

   10. Applied egg-rr 99.6%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-318}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-318}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-318)
     (* 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 <= -2e-318) {
		tmp = 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;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-2d-318)) then
        tmp = eps * ((eps * eps) * (eps * eps))
    else if (t_0 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x * (x * (x * x))))
    else
        tmp = (eps * eps) * (eps * (eps * eps))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-318) {
		tmp = 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;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -2e-318:
		tmp = eps * ((eps * eps) * (eps * eps))
	elif 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 <= -2e-318)
		tmp = Float64(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

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -2e-318)
		tmp = eps * ((eps * eps) * (eps * eps));
	elseif (t_0 <= 0.0)
		tmp = eps * (5.0 * (x * (x * (x * x))));
	else
		tmp = (eps * eps) * (eps * (eps * eps));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-318], N[(eps * 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))) < -2.0000024e-318

   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 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 98.9

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

   8. Simplified 98.9%

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

   if -2.0000024e-318 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 92.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      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 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]
   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 100.0

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. cube-mult N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

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

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 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 99.3

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

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
   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. lower-*.f64 N/A

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

      5. lower-*.f64 99.6

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

   10. Applied egg-rr 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-318}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-33}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{-44}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -3.5e-33)
     (* t_0 (* eps (* x 5.0)))
     (if (<= x 6.7e-44) (pow eps 5.0) (* 5.0 (* eps (* x t_0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.5e-33) {
		tmp = t_0 * (eps * (x * 5.0));
	} else if (x <= 6.7e-44) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * (x * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= (-3.5d-33)) then
        tmp = t_0 * (eps * (x * 5.0d0))
    else if (x <= 6.7d-44) then
        tmp = eps ** 5.0d0
    else
        tmp = 5.0d0 * (eps * (x * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.5e-33) {
		tmp = t_0 * (eps * (x * 5.0));
	} else if (x <= 6.7e-44) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * (x * t_0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	tmp = 0
	if x <= -3.5e-33:
		tmp = t_0 * (eps * (x * 5.0))
	elif x <= 6.7e-44:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = 5.0 * (eps * (x * t_0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -3.5e-33)
		tmp = Float64(t_0 * Float64(eps * Float64(x * 5.0)));
	elseif (x <= 6.7e-44)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(5.0 * Float64(eps * Float64(x * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= -3.5e-33)
		tmp = t_0 * (eps * (x * 5.0));
	elseif (x <= 6.7e-44)
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5e-33], N[(t$95$0 * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.7e-44], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * N[(eps * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4999999999999999e-33

   1. Initial program 17.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      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 99.6

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

   5. Simplified 99.6%

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

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. cube-mult N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 99.6

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

   7. Applied egg-rr 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 99.7

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

   9. Applied egg-rr 99.7%

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

   if -3.4999999999999999e-33 < x < 6.7000000000000002e-44

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.9

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

   5. Simplified 99.9%

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

   if 6.7000000000000002e-44 < x

   1. Initial program 55.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      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 99.7

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

   5. Simplified 99.7%

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

      1. lift-pow.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 99.8

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

      6. lift-pow.f64 N/A

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

      7. sqr-pow N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. cube-mult N/A

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

      14. lower-*.f64 N/A

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-33}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{-44}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-33}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{-44}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -3.5e-33)
     (* t_0 (* eps (* x 5.0)))
     (if (<= x 6.7e-44)
       (* eps (* eps (* eps (* eps eps))))
       (* 5.0 (* eps (* x t_0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.5e-33) {
		tmp = t_0 * (eps * (x * 5.0));
	} else if (x <= 6.7e-44) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = 5.0 * (eps * (x * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= (-3.5d-33)) then
        tmp = t_0 * (eps * (x * 5.0d0))
    else if (x <= 6.7d-44) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = 5.0d0 * (eps * (x * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -3.5e-33) {
		tmp = t_0 * (eps * (x * 5.0));
	} else if (x <= 6.7e-44) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = 5.0 * (eps * (x * t_0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	tmp = 0
	if x <= -3.5e-33:
		tmp = t_0 * (eps * (x * 5.0))
	elif x <= 6.7e-44:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = 5.0 * (eps * (x * t_0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -3.5e-33)
		tmp = Float64(t_0 * Float64(eps * Float64(x * 5.0)));
	elseif (x <= 6.7e-44)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(5.0 * Float64(eps * Float64(x * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= -3.5e-33)
		tmp = t_0 * (eps * (x * 5.0));
	elseif (x <= 6.7e-44)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = 5.0 * (eps * (x * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5e-33], N[(t$95$0 * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.7e-44], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(5.0 * N[(eps * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4999999999999999e-33

   1. Initial program 17.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      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 99.6

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

   5. Simplified 99.6%

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

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. cube-mult N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 99.6

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

   7. Applied egg-rr 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 99.7

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

   9. Applied egg-rr 99.7%

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

   if -3.4999999999999999e-33 < x < 6.7000000000000002e-44

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 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 99.8

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

   8. Simplified 99.8%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
   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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.8

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 99.8

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

   10. Applied egg-rr 99.8%

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

   if 6.7000000000000002e-44 < x

   1. Initial program 55.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      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 99.7

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

   5. Simplified 99.7%

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

      1. lift-pow.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 99.8

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

      6. lift-pow.f64 N/A

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

      7. sqr-pow N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. cube-mult N/A

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

      14. lower-*.f64 N/A

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-33}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{-44}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.7% accurate, 10.0× speedup

Math

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

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(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[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
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 93.9

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

5. Simplified 93.9%

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

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 93.7

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

8. Simplified 93.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
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. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 93.7

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 93.8

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

10. Applied egg-rr 93.8%

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

11. Final simplification 93.8%

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

Alternative 8: 87.7% 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 93.9%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
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 93.9

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

5. Simplified 93.9%

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

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 93.7

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

8. Simplified 93.7%

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

Reproduce

herbie shell --seed 2024207 
(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)))