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

Percentage Accurate: 88.2% → 98.4%

Time: 19.8s

Alternatives: 17

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: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-51}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, x \cdot 5, 10 \cdot \left(x \cdot x\right)\right), 10 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.1e-48)
   (*
    eps
    (*
     x
     (fma
      eps
      (* eps (fma 5.0 eps (* x 10.0)))
      (* x (* x (fma 5.0 x (* eps 10.0)))))))
   (if (<= x 4.5e-51)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (*
      eps
      (fma
       eps
       (fma eps (fma eps (* x 5.0) (* 10.0 (* x x))) (* 10.0 (* x (* x x))))
       (* 5.0 (pow x 4.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.1e-48) {
		tmp = eps * (x * fma(eps, (eps * fma(5.0, eps, (x * 10.0))), (x * (x * fma(5.0, x, (eps * 10.0))))));
	} else if (x <= 4.5e-51) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = eps * fma(eps, fma(eps, fma(eps, (x * 5.0), (10.0 * (x * x))), (10.0 * (x * (x * x)))), (5.0 * pow(x, 4.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.1e-48)
		tmp = Float64(eps * Float64(x * fma(eps, Float64(eps * fma(5.0, eps, Float64(x * 10.0))), Float64(x * Float64(x * fma(5.0, x, Float64(eps * 10.0)))))));
	elseif (x <= 4.5e-51)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(eps * fma(eps, fma(eps, fma(eps, Float64(x * 5.0), Float64(10.0 * Float64(x * x))), Float64(10.0 * Float64(x * Float64(x * x)))), Float64(5.0 * (x ^ 4.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.1e-48], N[(eps * N[(x * N[(eps * N[(eps * N[(5.0 * eps + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(5.0 * x + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-51], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(eps * N[(eps * N[(eps * N[(x * 5.0), $MachinePrecision] + N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(10.0 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.10000000000000016e-48

   1. Initial program 27.0%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 96.9%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{5 \cdot x + 0}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{x \cdot 5} + 0\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{\mathsf{fma}\left(x, 5, 0\right)}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

   8. Simplified 96.9%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \mathsf{fma}\left(x, 5, 0\right)\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)} \]

   9. Taylor expanded in x around 0

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

       1. distribute-rgt-in N/A

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

       2. associate-*r* N/A

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

       3. associate-+r+ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. distribute-lft-in N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 96.9%

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

   if -3.10000000000000016e-48 < x < 4.49999999999999974e-51

   1. Initial program 99.6%

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

   if 4.49999999999999974e-51 < x

   1. Initial program 34.9%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-51}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, x \cdot 5, 10 \cdot \left(x \cdot x\right)\right), 10 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 5 \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{-49}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-51}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, x \cdot 5, 10 \cdot \left(x \cdot x\right)\right), 10 \cdot t\_0\right), 5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -4e-49)
     (*
      eps
      (*
       x
       (fma
        eps
        (* eps (fma 5.0 eps (* x 10.0)))
        (* x (* x (fma 5.0 x (* eps 10.0)))))))
     (if (<= x 1.85e-51)
       (- (pow (+ x eps) 5.0) (* (* x x) t_0))
       (*
        eps
        (fma
         eps
         (fma eps (fma eps (* x 5.0) (* 10.0 (* x x))) (* 10.0 t_0))
         (* 5.0 (pow x 4.0))))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -4e-49) {
		tmp = eps * (x * fma(eps, (eps * fma(5.0, eps, (x * 10.0))), (x * (x * fma(5.0, x, (eps * 10.0))))));
	} else if (x <= 1.85e-51) {
		tmp = pow((x + eps), 5.0) - ((x * x) * t_0);
	} else {
		tmp = eps * fma(eps, fma(eps, fma(eps, (x * 5.0), (10.0 * (x * x))), (10.0 * t_0)), (5.0 * pow(x, 4.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -4e-49)
		tmp = Float64(eps * Float64(x * fma(eps, Float64(eps * fma(5.0, eps, Float64(x * 10.0))), Float64(x * Float64(x * fma(5.0, x, Float64(eps * 10.0)))))));
	elseif (x <= 1.85e-51)
		tmp = Float64((Float64(x + eps) ^ 5.0) - Float64(Float64(x * x) * t_0));
	else
		tmp = Float64(eps * fma(eps, fma(eps, fma(eps, Float64(x * 5.0), Float64(10.0 * Float64(x * x))), Float64(10.0 * t_0)), Float64(5.0 * (x ^ 4.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e-49], N[(eps * N[(x * N[(eps * N[(eps * N[(5.0 * eps + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(5.0 * x + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.85e-51], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(eps * N[(eps * N[(eps * N[(eps * N[(x * 5.0), $MachinePrecision] + N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(10.0 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999975e-49

   1. Initial program 27.0%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 96.9%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{5 \cdot x + 0}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{x \cdot 5} + 0\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{\mathsf{fma}\left(x, 5, 0\right)}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

   8. Simplified 96.9%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \mathsf{fma}\left(x, 5, 0\right)\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)} \]

   9. Taylor expanded in x around 0

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

       1. distribute-rgt-in N/A

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

       2. associate-*r* N/A

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

       3. associate-+r+ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. distribute-lft-in N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 96.9%

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

   if -3.99999999999999975e-49 < x < 1.84999999999999987e-51

   1. Initial program 99.6%

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

      1. --lowering--.f64 N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. cube-mult N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   if 1.84999999999999987e-51 < x

   1. Initial program 34.9%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-49}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-51}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, x \cdot 5, 10 \cdot \left(x \cdot x\right)\right), 10 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 5 \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{if}\;x \leq -1.12 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-51}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          eps
          (*
           x
           (fma
            eps
            (* eps (fma 5.0 eps (* x 10.0)))
            (* x (* x (fma 5.0 x (* eps 10.0)))))))))
   (if (<= x -1.12e-48)
     t_0
     (if (<= x 1.85e-51)
       (- (pow (+ x eps) 5.0) (* (* x x) (* x (* x x))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (x * fma(eps, (eps * fma(5.0, eps, (x * 10.0))), (x * (x * fma(5.0, x, (eps * 10.0))))));
	double tmp;
	if (x <= -1.12e-48) {
		tmp = t_0;
	} else if (x <= 1.85e-51) {
		tmp = pow((x + eps), 5.0) - ((x * x) * (x * (x * x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(x * fma(eps, Float64(eps * fma(5.0, eps, Float64(x * 10.0))), Float64(x * Float64(x * fma(5.0, x, Float64(eps * 10.0)))))))
	tmp = 0.0
	if (x <= -1.12e-48)
		tmp = t_0;
	elseif (x <= 1.85e-51)
		tmp = Float64((Float64(x + eps) ^ 5.0) - Float64(Float64(x * x) * Float64(x * Float64(x * x))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(x * N[(eps * N[(eps * N[(5.0 * eps + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(5.0 * x + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.12e-48], t$95$0, If[LessEqual[x, 1.85e-51], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.11999999999999999e-48 or 1.84999999999999987e-51 < x

   1. Initial program 31.9%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{5 \cdot x + 0}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{x \cdot 5} + 0\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{\mathsf{fma}\left(x, 5, 0\right)}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

   8. Simplified 98.4%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \mathsf{fma}\left(x, 5, 0\right)\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)} \]

   9. Taylor expanded in x around 0

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

       1. distribute-rgt-in N/A

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

       2. associate-*r* N/A

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

       3. associate-+r+ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. distribute-lft-in N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 98.5%

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

   if -1.11999999999999999e-48 < x < 1.84999999999999987e-51

   1. Initial program 99.6%

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

      1. --lowering--.f64 N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. cube-mult N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 99.6

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

   4. Applied egg-rr 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-51}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(10, x \cdot t\_1, \frac{\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\varepsilon \cdot t\_1\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          eps
          (*
           x
           (fma
            eps
            (* eps (fma 5.0 eps (* x 10.0)))
            (* x (* x (fma 5.0 x (* eps 10.0))))))))
        (t_1 (* eps (* eps eps))))
   (if (<= x -2.5e-48)
     t_0
     (if (<= x 2e-51)
       (* x (fma 10.0 (* x t_1) (/ (* (fma x 5.0 eps) (* eps t_1)) x)))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (x * fma(eps, (eps * fma(5.0, eps, (x * 10.0))), (x * (x * fma(5.0, x, (eps * 10.0))))));
	double t_1 = eps * (eps * eps);
	double tmp;
	if (x <= -2.5e-48) {
		tmp = t_0;
	} else if (x <= 2e-51) {
		tmp = x * fma(10.0, (x * t_1), ((fma(x, 5.0, eps) * (eps * t_1)) / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(x * fma(eps, Float64(eps * fma(5.0, eps, Float64(x * 10.0))), Float64(x * Float64(x * fma(5.0, x, Float64(eps * 10.0)))))))
	t_1 = Float64(eps * Float64(eps * eps))
	tmp = 0.0
	if (x <= -2.5e-48)
		tmp = t_0;
	elseif (x <= 2e-51)
		tmp = Float64(x * fma(10.0, Float64(x * t_1), Float64(Float64(fma(x, 5.0, eps) * Float64(eps * t_1)) / x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(x * N[(eps * N[(eps * N[(5.0 * eps + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(5.0 * x + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-48], t$95$0, If[LessEqual[x, 2e-51], N[(x * N[(10.0 * N[(x * t$95$1), $MachinePrecision] + N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4999999999999999e-48 or 2e-51 < x

   1. Initial program 31.9%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{5 \cdot x + 0}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{x \cdot 5} + 0\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{\mathsf{fma}\left(x, 5, 0\right)}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

   8. Simplified 98.4%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \mathsf{fma}\left(x, 5, 0\right)\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)} \]

   9. Taylor expanded in x around 0

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

       1. distribute-rgt-in N/A

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

       2. associate-*r* N/A

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

       3. associate-+r+ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. distribute-lft-in N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 98.5%

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

   if -2.4999999999999999e-48 < x < 2e-51

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \frac{-5 \cdot {\varepsilon}^{4} + -1 \cdot \frac{{\varepsilon}^{5}}{x}}{x} + 10 \cdot {\varepsilon}^{3}\right)} \]

   8. Simplified 99.4%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(10, x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \frac{\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(10, x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \frac{\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-51}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right), \varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          eps
          (*
           x
           (fma
            eps
            (* eps (fma 5.0 eps (* x 10.0)))
            (* x (* x (fma 5.0 x (* eps 10.0)))))))))
   (if (<= x -4e-49)
     t_0
     (if (<= x 2.25e-51)
       (* (* eps eps) (* eps (fma x (fma 10.0 x (* eps 5.0)) (* eps eps))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (x * fma(eps, (eps * fma(5.0, eps, (x * 10.0))), (x * (x * fma(5.0, x, (eps * 10.0))))));
	double tmp;
	if (x <= -4e-49) {
		tmp = t_0;
	} else if (x <= 2.25e-51) {
		tmp = (eps * eps) * (eps * fma(x, fma(10.0, x, (eps * 5.0)), (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(x * fma(eps, Float64(eps * fma(5.0, eps, Float64(x * 10.0))), Float64(x * Float64(x * fma(5.0, x, Float64(eps * 10.0)))))))
	tmp = 0.0
	if (x <= -4e-49)
		tmp = t_0;
	elseif (x <= 2.25e-51)
		tmp = Float64(Float64(eps * eps) * Float64(eps * fma(x, fma(10.0, x, Float64(eps * 5.0)), Float64(eps * eps))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(x * N[(eps * N[(eps * N[(5.0 * eps + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(5.0 * x + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e-49], t$95$0, If[LessEqual[x, 2.25e-51], N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(x * N[(10.0 * x + N[(eps * 5.0), $MachinePrecision]), $MachinePrecision] + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.99999999999999975e-49 or 2.24999999999999987e-51 < x

   1. Initial program 31.9%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{5 \cdot x + 0}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{x \cdot 5} + 0\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{\mathsf{fma}\left(x, 5, 0\right)}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

   8. Simplified 98.4%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \mathsf{fma}\left(x, 5, 0\right)\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)} \]

   9. Taylor expanded in x around 0

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

       1. distribute-rgt-in N/A

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

       2. associate-*r* N/A

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

       3. associate-+r+ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. distribute-lft-in N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 98.5%

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

   if -3.99999999999999975e-49 < x < 2.24999999999999987e-51

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + {\varepsilon}^{5}} \]

   8. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right), \varepsilon \cdot \varepsilon\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-49}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-51}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right), \varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, x \cdot \left(\varepsilon \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{-47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-57}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          eps
          (*
           x
           (fma
            eps
            (* x (* eps 10.0))
            (* x (* x (fma 5.0 x (* eps 10.0)))))))))
   (if (<= x -2e-47)
     t_0
     (if (<= x 4.4e-57)
       (* (* eps (* eps eps)) (fma eps eps (* x (fma 5.0 eps (* x 10.0)))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (x * fma(eps, (x * (eps * 10.0)), (x * (x * fma(5.0, x, (eps * 10.0))))));
	double tmp;
	if (x <= -2e-47) {
		tmp = t_0;
	} else if (x <= 4.4e-57) {
		tmp = (eps * (eps * eps)) * fma(eps, eps, (x * fma(5.0, eps, (x * 10.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(x * fma(eps, Float64(x * Float64(eps * 10.0)), Float64(x * Float64(x * fma(5.0, x, Float64(eps * 10.0)))))))
	tmp = 0.0
	if (x <= -2e-47)
		tmp = t_0;
	elseif (x <= 4.4e-57)
		tmp = Float64(Float64(eps * Float64(eps * eps)) * fma(eps, eps, Float64(x * fma(5.0, eps, Float64(x * 10.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(x * N[(eps * N[(x * N[(eps * 10.0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(5.0 * x + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e-47], t$95$0, If[LessEqual[x, 4.4e-57], N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps + N[(x * N[(5.0 * eps + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999999e-47 or 4.39999999999999997e-57 < x

   1. Initial program 33.9%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 97.2%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{5 \cdot x + 0}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{x \cdot 5} + 0\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{\mathsf{fma}\left(x, 5, 0\right)}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

   8. Simplified 97.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \mathsf{fma}\left(x, 5, 0\right)\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)} \]

   9. Taylor expanded in x around 0

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

       1. distribute-rgt-in N/A

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

       2. associate-*r* N/A

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

       3. associate-+r+ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. distribute-lft-in N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 97.1%

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

   12. Taylor expanded in eps around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 96.7

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

   14. Simplified 96.7%

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

   if -1.9999999999999999e-47 < x < 4.39999999999999997e-57

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 96.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right) + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{{\varepsilon}^{5}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{{\varepsilon}^{\color{blue}{\left(4 + 1\right)}}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      7. pow-plus N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{{\varepsilon}^{4} \cdot \varepsilon}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{\varepsilon \cdot {\varepsilon}^{4}}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\varepsilon \cdot \frac{{\varepsilon}^{4}}{{x}^{2}}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{{\varepsilon}^{4}}{{x}^{2}}, 5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)} \]

   8. Simplified 44.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + {\varepsilon}^{5}} \]

   11. Simplified 99.9%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \mathsf{fma}\left(5, \varepsilon, 10 \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, x \cdot \left(\varepsilon \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-57}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, x \cdot \left(\varepsilon \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, \varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-58}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(10, \varepsilon \cdot \left(x + \varepsilon\right), 5 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.7e-48)
   (* eps (fma (* x (* x (* x x))) 5.0 (* eps (* x (* x (* x 10.0))))))
   (if (<= x 9e-58)
     (* (* eps (* eps eps)) (fma eps eps (* x (fma 5.0 eps (* x 10.0)))))
     (* eps (* (* x x) (fma 10.0 (* eps (+ x eps)) (* 5.0 (* x x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.7e-48) {
		tmp = eps * fma((x * (x * (x * x))), 5.0, (eps * (x * (x * (x * 10.0)))));
	} else if (x <= 9e-58) {
		tmp = (eps * (eps * eps)) * fma(eps, eps, (x * fma(5.0, eps, (x * 10.0))));
	} else {
		tmp = eps * ((x * x) * fma(10.0, (eps * (x + eps)), (5.0 * (x * x))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.7e-48)
		tmp = Float64(eps * fma(Float64(x * Float64(x * Float64(x * x))), 5.0, Float64(eps * Float64(x * Float64(x * Float64(x * 10.0))))));
	elseif (x <= 9e-58)
		tmp = Float64(Float64(eps * Float64(eps * eps)) * fma(eps, eps, Float64(x * fma(5.0, eps, Float64(x * 10.0)))));
	else
		tmp = Float64(eps * Float64(Float64(x * x) * fma(10.0, Float64(eps * Float64(x + eps)), Float64(5.0 * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.7e-48], N[(eps * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 5.0 + N[(eps * N[(x * N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9e-58], N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps + N[(x * N[(5.0 * eps + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(10.0 * N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.70000000000000014e-48

   1. Initial program 27.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 96.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 96.2

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

   7. Applied egg-rr 96.2%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 96.3

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

   9. Applied egg-rr 96.3%

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

   if -1.70000000000000014e-48 < x < 9.0000000000000006e-58

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 96.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right) + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{{\varepsilon}^{5}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{{\varepsilon}^{\color{blue}{\left(4 + 1\right)}}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      7. pow-plus N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{{\varepsilon}^{4} \cdot \varepsilon}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{\varepsilon \cdot {\varepsilon}^{4}}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\varepsilon \cdot \frac{{\varepsilon}^{4}}{{x}^{2}}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{{\varepsilon}^{4}}{{x}^{2}}, 5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)} \]

   8. Simplified 44.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + {\varepsilon}^{5}} \]

   11. Simplified 99.9%

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

   if 9.0000000000000006e-58 < x

   1. Initial program 37.8%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{5 \cdot x + 0}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{x \cdot 5} + 0\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{\mathsf{fma}\left(x, 5, 0\right)}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

   8. Simplified 97.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \mathsf{fma}\left(x, 5, 0\right)\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \varepsilon \cdot \color{blue}{\left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)\right)} \]

   11. Simplified 96.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 5, \varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-58}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(10, \varepsilon \cdot \left(x + \varepsilon\right), 5 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(10, \varepsilon \cdot \left(x + \varepsilon\right), 5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-57}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* (* x x) (fma 10.0 (* eps (+ x eps)) (* 5.0 (* x x)))))))
   (if (<= x -1.95e-47)
     t_0
     (if (<= x 6e-57)
       (* (* eps (* eps eps)) (fma eps eps (* x (fma 5.0 eps (* x 10.0)))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * ((x * x) * fma(10.0, (eps * (x + eps)), (5.0 * (x * x))));
	double tmp;
	if (x <= -1.95e-47) {
		tmp = t_0;
	} else if (x <= 6e-57) {
		tmp = (eps * (eps * eps)) * fma(eps, eps, (x * fma(5.0, eps, (x * 10.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(Float64(x * x) * fma(10.0, Float64(eps * Float64(x + eps)), Float64(5.0 * Float64(x * x)))))
	tmp = 0.0
	if (x <= -1.95e-47)
		tmp = t_0;
	elseif (x <= 6e-57)
		tmp = Float64(Float64(eps * Float64(eps * eps)) * fma(eps, eps, Float64(x * fma(5.0, eps, Float64(x * 10.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(N[(x * x), $MachinePrecision] * N[(10.0 * N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.95e-47], t$95$0, If[LessEqual[x, 6e-57], N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps + N[(x * N[(5.0 * eps + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.94999999999999989e-47 or 6.00000000000000001e-57 < x

   1. Initial program 33.9%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 97.2%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{5 \cdot x + 0}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{x \cdot 5} + 0\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{\mathsf{fma}\left(x, 5, 0\right)}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

   8. Simplified 97.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \mathsf{fma}\left(x, 5, 0\right)\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \varepsilon \cdot \color{blue}{\left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)\right)} \]

   11. Simplified 96.6%

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

   if -1.94999999999999989e-47 < x < 6.00000000000000001e-57

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 96.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right) + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{{\varepsilon}^{5}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{{\varepsilon}^{\color{blue}{\left(4 + 1\right)}}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      7. pow-plus N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{{\varepsilon}^{4} \cdot \varepsilon}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{\varepsilon \cdot {\varepsilon}^{4}}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\varepsilon \cdot \frac{{\varepsilon}^{4}}{{x}^{2}}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{{\varepsilon}^{4}}{{x}^{2}}, 5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)} \]

   8. Simplified 44.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + {\varepsilon}^{5}} \]

   11. Simplified 99.9%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \mathsf{fma}\left(5, \varepsilon, 10 \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(10, \varepsilon \cdot \left(x + \varepsilon\right), 5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-57}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \mathsf{fma}\left(5, \varepsilon, x \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(10, \varepsilon \cdot \left(x + \varepsilon\right), 5 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-51}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right), \varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x (* x x)) (* eps (fma 5.0 x (* eps 10.0))))))
   (if (<= x -1.1e-46)
     t_0
     (if (<= x 2.05e-51)
       (* (* eps eps) (* eps (fma x (fma 10.0 x (* eps 5.0)) (* eps eps))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = (x * (x * x)) * (eps * fma(5.0, x, (eps * 10.0)));
	double tmp;
	if (x <= -1.1e-46) {
		tmp = t_0;
	} else if (x <= 2.05e-51) {
		tmp = (eps * eps) * (eps * fma(x, fma(10.0, x, (eps * 5.0)), (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * Float64(x * x)) * Float64(eps * fma(5.0, x, Float64(eps * 10.0))))
	tmp = 0.0
	if (x <= -1.1e-46)
		tmp = t_0;
	elseif (x <= 2.05e-51)
		tmp = Float64(Float64(eps * eps) * Float64(eps * fma(x, fma(10.0, x, Float64(eps * 5.0)), Float64(eps * eps))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(5.0 * x + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.1e-46], t$95$0, If[LessEqual[x, 2.05e-51], N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(x * N[(10.0 * x + N[(eps * 5.0), $MachinePrecision]), $MachinePrecision] + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e-46 or 2.04999999999999987e-51 < x

   1. Initial program 31.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 97.7%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 97.7

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

   7. Applied egg-rr 97.7%

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

   8. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-out N/A

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

      15. *-lowering-*.f64 N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 97.6

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

   10. Simplified 97.6%

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

   if -1.1e-46 < x < 2.04999999999999987e-51

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + {\varepsilon}^{5}} \]

   8. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right), \varepsilon \cdot \varepsilon\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 98.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x (* x x)) (* eps (fma 5.0 x (* eps 10.0))))))
   (if (<= x -1.4e-46)
     t_0
     (if (<= x 2.5e-50) (* (fma x 5.0 eps) (* eps (* eps (* eps eps)))) t_0))))

C

double code(double x, double eps) {
	double t_0 = (x * (x * x)) * (eps * fma(5.0, x, (eps * 10.0)));
	double tmp;
	if (x <= -1.4e-46) {
		tmp = t_0;
	} else if (x <= 2.5e-50) {
		tmp = fma(x, 5.0, eps) * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * Float64(x * x)) * Float64(eps * fma(5.0, x, Float64(eps * 10.0))))
	tmp = 0.0
	if (x <= -1.4e-46)
		tmp = t_0;
	elseif (x <= 2.5e-50)
		tmp = Float64(fma(x, 5.0, eps) * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(5.0 * x + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e-46], t$95$0, If[LessEqual[x, 2.5e-50], N[(N[(x * 5.0 + eps), $MachinePrecision] * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999e-46 or 2.49999999999999984e-50 < x

   1. Initial program 31.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 97.7%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 97.7

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

   7. Applied egg-rr 97.7%

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

   8. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-out N/A

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

      15. *-lowering-*.f64 N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 97.6

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

   10. Simplified 97.6%

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

   if -1.3999999999999999e-46 < x < 2.49999999999999984e-50

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-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. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 99.4

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

   5. Simplified 99.4%

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

   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. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 99.3

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

   8. Simplified 99.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-46}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-48}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot t\_0\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x -1.25e-48)
     (* 5.0 (* eps t_0))
     (if (<= x 2.6e-50)
       (* (fma x 5.0 eps) (* eps (* eps (* eps eps))))
       (* eps (* 5.0 t_0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -1.25e-48) {
		tmp = 5.0 * (eps * t_0);
	} else if (x <= 2.6e-50) {
		tmp = fma(x, 5.0, eps) * (eps * (eps * (eps * eps)));
	} else {
		tmp = eps * (5.0 * t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= -1.25e-48)
		tmp = Float64(5.0 * Float64(eps * t_0));
	elseif (x <= 2.6e-50)
		tmp = Float64(fma(x, 5.0, eps) * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(eps * Float64(5.0 * t_0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e-48], N[(5.0 * N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e-50], N[(N[(x * 5.0 + eps), $MachinePrecision] * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(5.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25e-48

   1. Initial program 27.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 96.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 96.2

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

   7. Applied egg-rr 96.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 96.2

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

   10. Simplified 96.2%

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

   if -1.25e-48 < x < 2.6000000000000001e-50

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-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. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 99.4

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

   5. Simplified 99.4%

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

   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. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 99.3

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

   8. Simplified 99.3%

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

   if 2.6000000000000001e-50 < x

   1. Initial program 34.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 98.7%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 98.6

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 96.4

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

   10. Simplified 96.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-48}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-47}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot t\_0\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-51}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x -1e-47)
     (* 5.0 (* eps t_0))
     (if (<= x 6.8e-51)
       (* (* eps (* eps eps)) (* eps (fma 5.0 x eps)))
       (* eps (* 5.0 t_0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -1e-47) {
		tmp = 5.0 * (eps * t_0);
	} else if (x <= 6.8e-51) {
		tmp = (eps * (eps * eps)) * (eps * fma(5.0, x, eps));
	} else {
		tmp = eps * (5.0 * t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= -1e-47)
		tmp = Float64(5.0 * Float64(eps * t_0));
	elseif (x <= 6.8e-51)
		tmp = Float64(Float64(eps * Float64(eps * eps)) * Float64(eps * fma(5.0, x, eps)));
	else
		tmp = Float64(eps * Float64(5.0 * t_0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e-47], N[(5.0 * N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e-51], N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(5.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999997e-48

   1. Initial program 27.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 96.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 96.2

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

   7. Applied egg-rr 96.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 96.2

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

   10. Simplified 96.2%

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

   if -9.9999999999999997e-48 < x < 6.80000000000000005e-51

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + \left(10 \cdot {\varepsilon}^{3} + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right) + \frac{{\varepsilon}^{5}}{{x}^{2}}\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{{\varepsilon}^{5}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{{\varepsilon}^{\color{blue}{\left(4 + 1\right)}}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      7. pow-plus N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{{\varepsilon}^{4} \cdot \varepsilon}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{\varepsilon \cdot {\varepsilon}^{4}}}{{x}^{2}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\varepsilon \cdot \frac{{\varepsilon}^{4}}{{x}^{2}}} + \left(5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{{\varepsilon}^{4}}{{x}^{2}}, 5 \cdot \frac{{\varepsilon}^{4}}{x} + 10 \cdot {\varepsilon}^{3}\right)} \]

   8. Simplified 44.6%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. metadata-eval N/A

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

       5. pow-plus N/A

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

       6. distribute-lft-in N/A

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

       7. +-commutative N/A

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

       8. metadata-eval N/A

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

       9. pow-plus N/A

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

       10. associate-*r* N/A

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

       11. *-lowering-*.f64 N/A

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

       12. cube-mult N/A

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 N/A

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

       17. *-lowering-*.f64 N/A

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

       18. +-commutative N/A

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

       19. accelerator-lowering-fma.f64 99.3

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

   11. Simplified 99.3%

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

   if 6.80000000000000005e-51 < x

   1. Initial program 34.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 98.7%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 98.6

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 96.4

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

   10. Simplified 96.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-47}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-51}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-47}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot t\_0\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-57}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x -1.75e-47)
     (* 5.0 (* eps t_0))
     (if (<= x 4.7e-57)
       (* eps (* eps (* eps (* eps eps))))
       (* eps (* 5.0 t_0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -1.75e-47) {
		tmp = 5.0 * (eps * t_0);
	} else if (x <= 4.7e-57) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = eps * (5.0 * 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 * x))
    if (x <= (-1.75d-47)) then
        tmp = 5.0d0 * (eps * t_0)
    else if (x <= 4.7d-57) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = eps * (5.0d0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -1.75e-47) {
		tmp = 5.0 * (eps * t_0);
	} else if (x <= 4.7e-57) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = eps * (5.0 * t_0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * (x * x))
	tmp = 0
	if x <= -1.75e-47:
		tmp = 5.0 * (eps * t_0)
	elif x <= 4.7e-57:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = eps * (5.0 * t_0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= -1.75e-47)
		tmp = Float64(5.0 * Float64(eps * t_0));
	elseif (x <= 4.7e-57)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(eps * Float64(5.0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * (x * x));
	tmp = 0.0;
	if (x <= -1.75e-47)
		tmp = 5.0 * (eps * t_0);
	elseif (x <= 4.7e-57)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = eps * (5.0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.75e-47], N[(5.0 * N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.7e-57], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(5.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7499999999999999e-47

   1. Initial program 27.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 96.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 96.2

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

   7. Applied egg-rr 96.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 96.2

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

   10. Simplified 96.2%

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

   if -1.7499999999999999e-47 < x < 4.6999999999999998e-57

   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. pow-lowering-pow.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   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. *-lowering-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. *-lowering-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 99.6

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

   8. Simplified 99.6%

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

   if 4.6999999999999998e-57 < x

   1. Initial program 37.8%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 96.4%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 96.3

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

   7. Applied egg-rr 96.3%

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

   8. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 94.3

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

   10. Simplified 94.3%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-47}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-57}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-49}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot t\_0\right)\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-58}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(5 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -4.2e-49)
     (* 5.0 (* eps (* x t_0)))
     (if (<= x 2.6e-58)
       (* eps (* eps (* eps (* eps eps))))
       (* eps (* x (* 5.0 t_0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -4.2e-49) {
		tmp = 5.0 * (eps * (x * t_0));
	} else if (x <= 2.6e-58) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = eps * (x * (5.0 * 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 <= (-4.2d-49)) then
        tmp = 5.0d0 * (eps * (x * t_0))
    else if (x <= 2.6d-58) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = eps * (x * (5.0d0 * 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 <= -4.2e-49) {
		tmp = 5.0 * (eps * (x * t_0));
	} else if (x <= 2.6e-58) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = eps * (x * (5.0 * t_0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	tmp = 0
	if x <= -4.2e-49:
		tmp = 5.0 * (eps * (x * t_0))
	elif x <= 2.6e-58:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = eps * (x * (5.0 * t_0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -4.2e-49)
		tmp = Float64(5.0 * Float64(eps * Float64(x * t_0)));
	elseif (x <= 2.6e-58)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(eps * Float64(x * Float64(5.0 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= -4.2e-49)
		tmp = 5.0 * (eps * (x * t_0));
	elseif (x <= 2.6e-58)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = eps * (x * (5.0 * 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, -4.2e-49], N[(5.0 * N[(eps * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e-58], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(x * N[(5.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.1999999999999998e-49

   1. Initial program 27.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 96.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 96.2

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

   7. Applied egg-rr 96.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 96.2

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

   10. Simplified 96.2%

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

   if -4.1999999999999998e-49 < x < 2.60000000000000007e-58

   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. pow-lowering-pow.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   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. *-lowering-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. *-lowering-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

         \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \cdot \varepsilon\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)} \cdot \varepsilon\right) \]

      11. unpow2 N/A

          \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \varepsilon\right) \]

      12. *-lowering-*.f64 99.6

          \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \varepsilon\right) \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)} \]

   if 2.60000000000000007e-58 < x

   1. Initial program 37.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(x + 4 \cdot x\right)\right)\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]

   5. Simplified 97.4%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 5 \cdot x, \left(x \cdot x\right) \cdot 10\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot 10\right), 5 \cdot {x}^{4}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \left(5 \cdot {\varepsilon}^{3} + x \cdot \left(10 \cdot {\varepsilon}^{2} + x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \left(5 \cdot {\varepsilon}^{3} + x \cdot \left(10 \cdot {\varepsilon}^{2} + x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)\right)\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \varepsilon \cdot \left(x \cdot \left(5 \cdot {\varepsilon}^{3} + \color{blue}{\left(\left(10 \cdot {\varepsilon}^{2}\right) \cdot x + \left(x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \cdot x\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \varepsilon \cdot \left(x \cdot \left(5 \cdot {\varepsilon}^{3} + \left(\color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot x\right)} + \left(x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \cdot x\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \varepsilon \cdot \left(x \cdot \color{blue}{\left(\left(5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right) + \left(x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \cdot x\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \varepsilon \cdot \left(x \cdot \left(\left(5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right) + \color{blue}{x \cdot \left(x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \varepsilon \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) + \left(5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right)}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \varepsilon \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)}, 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(10 \cdot \varepsilon + 5 \cdot x\right)}, 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right)}, 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      11. +-rgt-identity N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{5 \cdot x + 0}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{x \cdot 5} + 0\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \color{blue}{\mathsf{fma}\left(x, 5, 0\right)}\right), 5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right) \]

   8. Simplified 97.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(10, \varepsilon, \mathsf{fma}\left(x, 5, 0\right)\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \varepsilon \cdot \left(x \cdot \color{blue}{\left(5 \cdot {x}^{3}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \color{blue}{\left(5 \cdot {x}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \left(5 \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \left(5 \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \left(5 \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \varepsilon \cdot \left(x \cdot \left(5 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       6. *-lowering-*.f64 94.3

          \[\leadsto \varepsilon \cdot \left(x \cdot \left(5 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

   11. Simplified 94.3%

       \[\leadsto \varepsilon \cdot \left(x \cdot \color{blue}{\left(5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-49}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-58}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-57}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 5.0 (* eps (* x (* x (* x x)))))))
   (if (<= x -1.7e-47)
     t_0
     (if (<= x 6e-57) (* eps (* eps (* eps (* eps eps)))) t_0))))

C

double code(double x, double eps) {
	double t_0 = 5.0 * (eps * (x * (x * (x * x))));
	double tmp;
	if (x <= -1.7e-47) {
		tmp = t_0;
	} else if (x <= 6e-57) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = 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 = 5.0d0 * (eps * (x * (x * (x * x))))
    if (x <= (-1.7d-47)) then
        tmp = t_0
    else if (x <= 6d-57) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 5.0 * (eps * (x * (x * (x * x))));
	double tmp;
	if (x <= -1.7e-47) {
		tmp = t_0;
	} else if (x <= 6e-57) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 5.0 * (eps * (x * (x * (x * x))))
	tmp = 0
	if x <= -1.7e-47:
		tmp = t_0
	elif x <= 6e-57:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(5.0 * Float64(eps * Float64(x * Float64(x * Float64(x * x)))))
	tmp = 0.0
	if (x <= -1.7e-47)
		tmp = t_0;
	elseif (x <= 6e-57)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 5.0 * (eps * (x * (x * (x * x))));
	tmp = 0.0;
	if (x <= -1.7e-47)
		tmp = t_0;
	elseif (x <= 6e-57)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(5.0 * N[(eps * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e-47], t$95$0, If[LessEqual[x, 6e-57], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7000000000000001e-47 or 6.00000000000000001e-57 < x

   1. Initial program 33.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right) + {x}^{4}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right) + {x}^{4}\right) + 4 \cdot {x}^{4}\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right) + \left({x}^{4} + 4 \cdot {x}^{4}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right) + \color{blue}{\left(4 \cdot {x}^{4} + {x}^{4}\right)}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, 4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right), 4 \cdot {x}^{4} + {x}^{4}\right)} \]

   5. Simplified 96.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \left(x \cdot \left(x \cdot x\right)\right) \cdot 10, 5 \cdot {x}^{4}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10\right) + 5 \cdot {x}^{4}\right) \cdot \varepsilon} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10\right) + 5 \cdot {x}^{4}\right) \cdot \varepsilon} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \left(x \cdot \left(x \cdot x\right)\right) \cdot 10, 5 \cdot {x}^{4}\right)} \cdot \varepsilon \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 10\right)}, 5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 10\right)}, 5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 10\right)}, 5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 10\right), 5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(\left(x \cdot x\right) \cdot 10\right), \color{blue}{5 \cdot {x}^{4}}\right) \cdot \varepsilon \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot \varepsilon \]

      10. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right) \cdot \varepsilon \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \varepsilon \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right) \cdot \varepsilon \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right) \cdot \varepsilon \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \varepsilon \]

      16. *-lowering-*.f64 96.3

          \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \cdot \varepsilon \]

   7. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, x \cdot \left(\left(x \cdot x\right) \cdot 10\right), 5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \varepsilon} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto 5 \cdot \left(\varepsilon \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \]

      4. pow-plus N/A

         \[\leadsto 5 \cdot \left(\varepsilon \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto 5 \cdot \left(\varepsilon \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right) \]

      6. cube-mult N/A

         \[\leadsto 5 \cdot \left(\varepsilon \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto 5 \cdot \left(\varepsilon \cdot \left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto 5 \cdot \left(\varepsilon \cdot \left(\color{blue}{\left(x \cdot {x}^{2}\right)} \cdot x\right)\right) \]

      9. unpow2 N/A

         \[\leadsto 5 \cdot \left(\varepsilon \cdot \left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot x\right)\right) \]

      10. *-lowering-*.f64 94.8

          \[\leadsto 5 \cdot \left(\varepsilon \cdot \left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot x\right)\right) \]

   10. Simplified 94.8%

       \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)\right)} \]

   if -1.7000000000000001e-47 < x < 6.00000000000000001e-57

   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. pow-lowering-pow.f64 99.6

         \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

      5. metadata-eval N/A

         \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(3 + 1\right)}} \]

      6. pow-plus N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{3} \cdot \varepsilon\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{3} \cdot \varepsilon\right)} \]

      8. cube-mult N/A

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \cdot \varepsilon\right) \]

      9. unpow2 N/A

         \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \cdot \varepsilon\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)} \cdot \varepsilon\right) \]

      11. unpow2 N/A

          \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \varepsilon\right) \]

      12. *-lowering-*.f64 99.6

          \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \varepsilon\right) \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-47}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-57}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 87.3% 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 83.4%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
4. Step-by-step derivation

   1. pow-lowering-pow.f64 82.0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

5. Simplified 82.0%

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
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. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

   5. metadata-eval N/A

      \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(3 + 1\right)}} \]

   6. pow-plus N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{3} \cdot \varepsilon\right)} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{3} \cdot \varepsilon\right)} \]

   8. cube-mult N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \cdot \varepsilon\right) \]

   9. unpow2 N/A

      \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \cdot \varepsilon\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)} \cdot \varepsilon\right) \]

   11. unpow2 N/A

       \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \varepsilon\right) \]

   12. *-lowering-*.f64 82.0

       \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \varepsilon\right) \]

8. Simplified 82.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)} \]

9. Final simplification 82.0%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]
10. Add Preprocessing

Alternative 17: 87.2% 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 83.4%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
4. Step-by-step derivation

   1. pow-lowering-pow.f64 82.0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

5. Simplified 82.0%

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
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. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

   5. metadata-eval N/A

      \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(3 + 1\right)}} \]

   6. pow-plus N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{3} \cdot \varepsilon\right)} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{3} \cdot \varepsilon\right)} \]

   8. cube-mult N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \cdot \varepsilon\right) \]

   9. unpow2 N/A

      \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \cdot \varepsilon\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)} \cdot \varepsilon\right) \]

   11. unpow2 N/A

       \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \varepsilon\right) \]

   12. *-lowering-*.f64 82.0

       \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \varepsilon\right) \]

8. Simplified 82.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

   5. *-lowering-*.f64 81.9

      \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

10. Applied egg-rr 81.9%

    \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(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)))