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

Percentage Accurate: 88.4% → 98.1%

Time: 10.6s

Alternatives: 17

Speedup: 5.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, x \cdot 5, \left(x \cdot x\right) \cdot 10\right), 10 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-42)
   (* (pow x 4.0) (* eps 5.0))
   (if (<= x 1.15e-45)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (*
      eps
      (fma
       eps
       (fma eps (fma eps (* x 5.0) (* (* x x) 10.0)) (* 10.0 (* x (* x x))))
       (* 5.0 (pow x 4.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-42) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 1.15e-45) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = eps * fma(eps, fma(eps, fma(eps, (x * 5.0), ((x * x) * 10.0)), (10.0 * (x * (x * x)))), (5.0 * pow(x, 4.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	elseif (x <= 1.15e-45)
		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(Float64(x * x) * 10.0)), 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, -2.7e-42], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e-45], 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[(N[(x * x), $MachinePrecision] * 10.0), $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 < -2.69999999999999999e-42

   1. Initial program 25.0%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.6%

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

   if -2.69999999999999999e-42 < x < 1.14999999999999996e-45

   1. Initial program 100.0%

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

   if 1.14999999999999996e-45 < x

   1. Initial program 48.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. Applied rewrites 94.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, x \cdot 5, \left(x \cdot x\right) \cdot 10\right), 10 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 5 \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, x \cdot 5, \left(x \cdot x\right) \cdot 10\right), 10 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-42)
   (* (pow x 4.0) (* eps 5.0))
   (if (<= x 5.7e-46)
     (* (pow eps 5.0) (fma 5.0 (/ x eps) 1.0))
     (*
      eps
      (fma
       eps
       (fma eps (fma eps (* x 5.0) (* (* x x) 10.0)) (* 10.0 (* x (* x x))))
       (* 5.0 (pow x 4.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-42) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 5.7e-46) {
		tmp = pow(eps, 5.0) * fma(5.0, (x / eps), 1.0);
	} else {
		tmp = eps * fma(eps, fma(eps, fma(eps, (x * 5.0), ((x * x) * 10.0)), (10.0 * (x * (x * x)))), (5.0 * pow(x, 4.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	elseif (x <= 5.7e-46)
		tmp = Float64((eps ^ 5.0) * fma(5.0, Float64(x / eps), 1.0));
	else
		tmp = Float64(eps * fma(eps, fma(eps, fma(eps, Float64(x * 5.0), Float64(Float64(x * x) * 10.0)), 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, -2.7e-42], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e-46], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(eps * N[(eps * N[(eps * N[(eps * N[(x * 5.0), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 10.0), $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 < -2.69999999999999999e-42

   1. Initial program 25.0%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.6%

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

   if -2.69999999999999999e-42 < x < 5.7000000000000003e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 5.7000000000000003e-46 < x

   1. Initial program 48.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. Applied rewrites 94.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, x \cdot 5, \left(x \cdot x\right) \cdot 10\right), 10 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 5 \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(5, {\varepsilon}^{4}, x \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x \cdot x, 10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-42)
   (* (pow x 4.0) (* eps 5.0))
   (if (<= x 5.7e-46)
     (* (pow eps 5.0) (fma 5.0 (/ x eps) 1.0))
     (*
      x
      (fma
       5.0
       (pow eps 4.0)
       (* x (* eps (fma 5.0 (* x x) (* 10.0 (* eps (+ x eps)))))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-42) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 5.7e-46) {
		tmp = pow(eps, 5.0) * fma(5.0, (x / eps), 1.0);
	} else {
		tmp = x * fma(5.0, pow(eps, 4.0), (x * (eps * fma(5.0, (x * x), (10.0 * (eps * (x + eps)))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	elseif (x <= 5.7e-46)
		tmp = Float64((eps ^ 5.0) * fma(5.0, Float64(x / eps), 1.0));
	else
		tmp = Float64(x * fma(5.0, (eps ^ 4.0), Float64(x * Float64(eps * fma(5.0, Float64(x * x), Float64(10.0 * Float64(eps * Float64(x + eps))))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-42], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e-46], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision] + N[(x * N[(eps * N[(5.0 * N[(x * x), $MachinePrecision] + N[(10.0 * N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.69999999999999999e-42

   1. Initial program 25.0%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.6%

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

   if -2.69999999999999999e-42 < x < 5.7000000000000003e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 5.7000000000000003e-46 < x

   1. Initial program 48.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. Applied rewrites 94.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 94.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(5, {\varepsilon}^{4}, x \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x \cdot x, 10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, 5, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-42)
   (* (pow x 4.0) (* eps 5.0))
   (if (<= x 5.7e-46)
     (* (pow eps 5.0) (fma 5.0 (/ x eps) 1.0))
     (*
      eps
      (*
       x
       (fma
        (* eps eps)
        (fma eps 5.0 (* x 10.0))
        (* x (* x (fma eps 10.0 (* x 5.0))))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-42) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 5.7e-46) {
		tmp = pow(eps, 5.0) * fma(5.0, (x / eps), 1.0);
	} else {
		tmp = eps * (x * fma((eps * eps), fma(eps, 5.0, (x * 10.0)), (x * (x * fma(eps, 10.0, (x * 5.0))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	elseif (x <= 5.7e-46)
		tmp = Float64((eps ^ 5.0) * fma(5.0, Float64(x / eps), 1.0));
	else
		tmp = Float64(eps * Float64(x * fma(Float64(eps * eps), fma(eps, 5.0, Float64(x * 10.0)), Float64(x * Float64(x * fma(eps, 10.0, Float64(x * 5.0)))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-42], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e-46], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(eps * N[(x * N[(N[(eps * eps), $MachinePrecision] * N[(eps * 5.0 + N[(x * 10.0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(eps * 10.0 + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.69999999999999999e-42

   1. Initial program 25.0%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.6%

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

   if -2.69999999999999999e-42 < x < 5.7000000000000003e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 5.7000000000000003e-46 < x

   1. Initial program 48.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. Applied rewrites 94.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, 5, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, 5, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-42)
   (* (pow x 4.0) (* eps 5.0))
   (if (<= x 5.7e-46)
     (* (pow eps 4.0) (fma 5.0 x eps))
     (*
      eps
      (*
       x
       (fma
        (* eps eps)
        (fma eps 5.0 (* x 10.0))
        (* x (* x (fma eps 10.0 (* x 5.0))))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-42) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 5.7e-46) {
		tmp = pow(eps, 4.0) * fma(5.0, x, eps);
	} else {
		tmp = eps * (x * fma((eps * eps), fma(eps, 5.0, (x * 10.0)), (x * (x * fma(eps, 10.0, (x * 5.0))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	elseif (x <= 5.7e-46)
		tmp = Float64((eps ^ 4.0) * fma(5.0, x, eps));
	else
		tmp = Float64(eps * Float64(x * fma(Float64(eps * eps), fma(eps, 5.0, Float64(x * 10.0)), Float64(x * Float64(x * fma(eps, 10.0, Float64(x * 5.0)))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-42], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e-46], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision], N[(eps * N[(x * N[(N[(eps * eps), $MachinePrecision] * N[(eps * 5.0 + N[(x * 10.0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(eps * 10.0 + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.69999999999999999e-42

   1. Initial program 25.0%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.6%

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

   if -2.69999999999999999e-42 < x < 5.7000000000000003e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 5.7000000000000003e-46 < x

   1. Initial program 48.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. Applied rewrites 94.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, 5, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot t\_0, 5, \varepsilon \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, 5, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* eps (* eps eps)))))
   (if (<= x -2.7e-42)
     (* (pow x 4.0) (* eps 5.0))
     (if (<= x 5.7e-46)
       (fma (* x t_0) 5.0 (* eps t_0))
       (*
        eps
        (*
         x
         (fma
          (* eps eps)
          (fma eps 5.0 (* x 10.0))
          (* x (* x (fma eps 10.0 (* x 5.0)))))))))))

C

double code(double x, double eps) {
	double t_0 = eps * (eps * (eps * eps));
	double tmp;
	if (x <= -2.7e-42) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 5.7e-46) {
		tmp = fma((x * t_0), 5.0, (eps * t_0));
	} else {
		tmp = eps * (x * fma((eps * eps), fma(eps, 5.0, (x * 10.0)), (x * (x * fma(eps, 10.0, (x * 5.0))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(eps * Float64(eps * eps)))
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	elseif (x <= 5.7e-46)
		tmp = fma(Float64(x * t_0), 5.0, Float64(eps * t_0));
	else
		tmp = Float64(eps * Float64(x * fma(Float64(eps * eps), fma(eps, 5.0, Float64(x * 10.0)), Float64(x * Float64(x * fma(eps, 10.0, Float64(x * 5.0)))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-42], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e-46], N[(N[(x * t$95$0), $MachinePrecision] * 5.0 + N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision], N[(eps * N[(x * N[(N[(eps * eps), $MachinePrecision] * N[(eps * 5.0 + N[(x * 10.0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(eps * 10.0 + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.69999999999999999e-42

   1. Initial program 25.0%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.6%

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

   if -2.69999999999999999e-42 < x < 5.7000000000000003e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   if 5.7000000000000003e-46 < x

   1. Initial program 48.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. Applied rewrites 94.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), 5, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, 5, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ t_1 := x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot t\_1\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot t\_0, 5, \varepsilon \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, 5, x \cdot 10\right), x \cdot t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* eps (* eps eps)))) (t_1 (* x (fma eps 10.0 (* x 5.0)))))
   (if (<= x -2.7e-42)
     (* (* x x) (* eps t_1))
     (if (<= x 5.7e-46)
       (fma (* x t_0) 5.0 (* eps t_0))
       (* eps (* x (fma (* eps eps) (fma eps 5.0 (* x 10.0)) (* x t_1))))))))

C

double code(double x, double eps) {
	double t_0 = eps * (eps * (eps * eps));
	double t_1 = x * fma(eps, 10.0, (x * 5.0));
	double tmp;
	if (x <= -2.7e-42) {
		tmp = (x * x) * (eps * t_1);
	} else if (x <= 5.7e-46) {
		tmp = fma((x * t_0), 5.0, (eps * t_0));
	} else {
		tmp = eps * (x * fma((eps * eps), fma(eps, 5.0, (x * 10.0)), (x * t_1)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(eps * Float64(eps * eps)))
	t_1 = Float64(x * fma(eps, 10.0, Float64(x * 5.0)))
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = Float64(Float64(x * x) * Float64(eps * t_1));
	elseif (x <= 5.7e-46)
		tmp = fma(Float64(x * t_0), 5.0, Float64(eps * t_0));
	else
		tmp = Float64(eps * Float64(x * fma(Float64(eps * eps), fma(eps, 5.0, Float64(x * 10.0)), Float64(x * t_1))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(eps * 10.0 + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-42], N[(N[(x * x), $MachinePrecision] * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e-46], N[(N[(x * t$95$0), $MachinePrecision] * 5.0 + N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision], N[(eps * N[(x * N[(N[(eps * eps), $MachinePrecision] * N[(eps * 5.0 + N[(x * 10.0), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.69999999999999999e-42

   1. Initial program 25.0%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 99.6%

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

   if -2.69999999999999999e-42 < x < 5.7000000000000003e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   if 5.7000000000000003e-46 < x

   1. Initial program 48.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. Applied rewrites 94.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), 5, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, 5, x \cdot 10\right), x \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot t\_0, 5, \varepsilon \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(10, \frac{\varepsilon \cdot \varepsilon}{x}, \varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* eps (* eps eps)))))
   (if (<= x -2.7e-42)
     (* (* x x) (* eps (* x (fma eps 10.0 (* x 5.0)))))
     (if (<= x 5.7e-46)
       (fma (* x t_0) 5.0 (* eps t_0))
       (* (fma 10.0 (/ (* eps eps) x) (* eps 5.0)) (* x (* x (* x x))))))))

C

double code(double x, double eps) {
	double t_0 = eps * (eps * (eps * eps));
	double tmp;
	if (x <= -2.7e-42) {
		tmp = (x * x) * (eps * (x * fma(eps, 10.0, (x * 5.0))));
	} else if (x <= 5.7e-46) {
		tmp = fma((x * t_0), 5.0, (eps * t_0));
	} else {
		tmp = fma(10.0, ((eps * eps) / x), (eps * 5.0)) * (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(eps * Float64(eps * eps)))
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = Float64(Float64(x * x) * Float64(eps * Float64(x * fma(eps, 10.0, Float64(x * 5.0)))));
	elseif (x <= 5.7e-46)
		tmp = fma(Float64(x * t_0), 5.0, Float64(eps * t_0));
	else
		tmp = Float64(fma(10.0, Float64(Float64(eps * eps) / x), Float64(eps * 5.0)) * Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-42], N[(N[(x * x), $MachinePrecision] * N[(eps * N[(x * N[(eps * 10.0 + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e-46], N[(N[(x * t$95$0), $MachinePrecision] * 5.0 + N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(10.0 * N[(N[(eps * eps), $MachinePrecision] / x), $MachinePrecision] + N[(eps * 5.0), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.69999999999999999e-42

   1. Initial program 25.0%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 99.6%

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

   if -2.69999999999999999e-42 < x < 5.7000000000000003e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   if 5.7000000000000003e-46 < x

   1. Initial program 48.7%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 92.1%

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

   7. Applied rewrites 92.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), 5, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(10, \frac{\varepsilon \cdot \varepsilon}{x}, \varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ t_1 := \left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot t\_0, 5, \varepsilon \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* eps (* eps eps))))
        (t_1 (* (* x x) (* eps (* x (fma eps 10.0 (* x 5.0)))))))
   (if (<= x -2.7e-42)
     t_1
     (if (<= x 5.7e-46) (fma (* x t_0) 5.0 (* eps t_0)) t_1))))

C

double code(double x, double eps) {
	double t_0 = eps * (eps * (eps * eps));
	double t_1 = (x * x) * (eps * (x * fma(eps, 10.0, (x * 5.0))));
	double tmp;
	if (x <= -2.7e-42) {
		tmp = t_1;
	} else if (x <= 5.7e-46) {
		tmp = fma((x * t_0), 5.0, (eps * t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(eps * Float64(eps * eps)))
	t_1 = Float64(Float64(x * x) * Float64(eps * Float64(x * fma(eps, 10.0, Float64(x * 5.0)))))
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = t_1;
	elseif (x <= 5.7e-46)
		tmp = fma(Float64(x * t_0), 5.0, Float64(eps * t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * N[(eps * N[(x * N[(eps * 10.0 + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-42], t$95$1, If[LessEqual[x, 5.7e-46], N[(N[(x * t$95$0), $MachinePrecision] * 5.0 + N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999999e-42 or 5.7000000000000003e-46 < x

   1. Initial program 35.2%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 96.4%

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

   7. Applied rewrites 96.2%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 96.3%

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

   if -2.69999999999999999e-42 < x < 5.7000000000000003e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), 5, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) (* eps (* x (fma eps 10.0 (* x 5.0)))))))
   (if (<= x -2.7e-42)
     t_0
     (if (<= x 5.7e-46) (* (* eps (* eps (* eps eps))) (fma x 5.0 eps)) t_0))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * (eps * (x * fma(eps, 10.0, (x * 5.0))));
	double tmp;
	if (x <= -2.7e-42) {
		tmp = t_0;
	} else if (x <= 5.7e-46) {
		tmp = (eps * (eps * (eps * eps))) * fma(x, 5.0, eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * Float64(eps * Float64(x * fma(eps, 10.0, Float64(x * 5.0)))))
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = t_0;
	elseif (x <= 5.7e-46)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) * fma(x, 5.0, eps));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(eps * N[(x * N[(eps * 10.0 + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-42], t$95$0, If[LessEqual[x, 5.7e-46], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * 5.0 + eps), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999999e-42 or 5.7000000000000003e-46 < x

   1. Initial program 35.2%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 96.4%

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

   7. Applied rewrites 96.2%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 96.3%

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

   if -2.69999999999999999e-42 < x < 5.7000000000000003e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-42)
   (* (* eps 5.0) (* x (* x (* x x))))
   (if (<= x 5.7e-46)
     (* (* eps (* eps (* eps eps))) (fma x 5.0 eps))
     (* (* x x) (* x (* eps (fma eps 10.0 (* x 5.0))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-42) {
		tmp = (eps * 5.0) * (x * (x * (x * x)));
	} else if (x <= 5.7e-46) {
		tmp = (eps * (eps * (eps * eps))) * fma(x, 5.0, eps);
	} else {
		tmp = (x * x) * (x * (eps * fma(eps, 10.0, (x * 5.0))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = Float64(Float64(eps * 5.0) * Float64(x * Float64(x * Float64(x * x))));
	elseif (x <= 5.7e-46)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) * fma(x, 5.0, eps));
	else
		tmp = Float64(Float64(x * x) * Float64(x * Float64(eps * fma(eps, 10.0, Float64(x * 5.0)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-42], N[(N[(eps * 5.0), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e-46], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * 5.0 + eps), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(eps * N[(eps * 10.0 + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.69999999999999999e-42

   1. Initial program 25.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

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

   if -2.69999999999999999e-42 < x < 5.7000000000000003e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   if 5.7000000000000003e-46 < x

   1. Initial program 48.7%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 91.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* eps 5.0) (* x (* x (* x x))))))
   (if (<= x -2.7e-42)
     t_0
     (if (<= x 5.7e-46) (* (* eps (* eps (* eps eps))) (fma x 5.0 eps)) t_0))))

C

double code(double x, double eps) {
	double t_0 = (eps * 5.0) * (x * (x * (x * x)));
	double tmp;
	if (x <= -2.7e-42) {
		tmp = t_0;
	} else if (x <= 5.7e-46) {
		tmp = (eps * (eps * (eps * eps))) * fma(x, 5.0, eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(eps * 5.0) * Float64(x * Float64(x * Float64(x * x))))
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = t_0;
	elseif (x <= 5.7e-46)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) * fma(x, 5.0, eps));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps * 5.0), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-42], t$95$0, If[LessEqual[x, 5.7e-46], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * 5.0 + eps), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999999e-42 or 5.7000000000000003e-46 < x

   1. Initial program 35.2%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 94.9

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

   5. Applied rewrites 94.9%

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

   7. Applied rewrites 94.8%

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

   if -2.69999999999999999e-42 < x < 5.7000000000000003e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* eps 5.0) (* x (* x (* x x))))))
   (if (<= x -2.7e-42)
     t_0
     (if (<= x 5.7e-46) (* (* eps (* eps eps)) (* eps (fma x 5.0 eps))) t_0))))

C

double code(double x, double eps) {
	double t_0 = (eps * 5.0) * (x * (x * (x * x)));
	double tmp;
	if (x <= -2.7e-42) {
		tmp = t_0;
	} else if (x <= 5.7e-46) {
		tmp = (eps * (eps * eps)) * (eps * fma(x, 5.0, eps));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(eps * 5.0) * Float64(x * Float64(x * Float64(x * x))))
	tmp = 0.0
	if (x <= -2.7e-42)
		tmp = t_0;
	elseif (x <= 5.7e-46)
		tmp = Float64(Float64(eps * Float64(eps * eps)) * Float64(eps * fma(x, 5.0, eps)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps * 5.0), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-42], t$95$0, If[LessEqual[x, 5.7e-46], N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(x * 5.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999999e-42 or 5.7000000000000003e-46 < x

   1. Initial program 35.2%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 94.9

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

   5. Applied rewrites 94.9%

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

   7. Applied rewrites 94.8%

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

   if -2.69999999999999999e-42 < x < 5.7000000000000003e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   12. Applied rewrites 99.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-42}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-46}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.4% accurate, 8.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* (* eps 5.0) (* x (* x (* x x)))))

C

double code(double x, double eps) {
	return (eps * 5.0) * (x * (x * (x * x)));
}

Fortran

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

Java

public static double code(double x, double eps) {
	return (eps * 5.0) * (x * (x * (x * x)));
}

Python

def code(x, eps):
	return (eps * 5.0) * (x * (x * (x * x)))

Julia

function code(x, eps)
	return Float64(Float64(eps * 5.0) * Float64(x * Float64(x * Float64(x * x))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * 5.0) * (x * (x * (x * x)));
end

Wolfram

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

Derivation

1. Initial program 88.9%

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

3. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. lower-*.f64 N/A

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

   7. distribute-lft1-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 83.0

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

5. Applied rewrites 83.0%

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

7. Applied rewrites 82.9%

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

8. Final simplification 82.9%

   \[\leadsto \left(\varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \]
9. Add Preprocessing

Alternative 15: 82.4% accurate, 8.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* eps (* 5.0 (* x (* x (* x x))))))

C

double code(double x, double eps) {
	return eps * (5.0 * (x * (x * (x * x))));
}

Fortran

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

Java

public static double code(double x, double eps) {
	return eps * (5.0 * (x * (x * (x * x))));
}

Python

def code(x, eps):
	return eps * (5.0 * (x * (x * (x * x))))

Julia

function code(x, eps)
	return Float64(eps * Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (5.0 * (x * (x * (x * x))));
end

Wolfram

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

Derivation

1. Initial program 88.9%

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

3. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. lower-*.f64 N/A

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

   7. distribute-lft1-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 83.0

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

5. Applied rewrites 83.0%

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

7. Applied rewrites 82.9%

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

9. Applied rewrites 82.9%

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

10. Final simplification 82.9%

    \[\leadsto \varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 16: 71.0% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (* eps (* eps (* (* x x) 10.0)))))

C

double code(double x, double eps) {
	return eps * (eps * (eps * ((x * x) * 10.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (eps * (eps * ((x * x) * 10.0d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * (eps * (eps * ((x * x) * 10.0)));
}

Python

def code(x, eps):
	return eps * (eps * (eps * ((x * x) * 10.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(eps * Float64(eps * Float64(Float64(x * x) * 10.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (eps * (eps * ((x * x) * 10.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(eps * N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.9%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
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. Applied rewrites 83.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 71.9%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 71.9%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{10}\right)\right)\right) \]
12. Add Preprocessing

Alternative 17: 70.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \left(10 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* 10.0 (* x (* x x))) (* eps eps)))

C

double code(double x, double eps) {
	return (10.0 * (x * (x * x))) * (eps * eps);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (10.0d0 * (x * (x * x))) * (eps * eps)
end function

Java

public static double code(double x, double eps) {
	return (10.0 * (x * (x * x))) * (eps * eps);
}

Python

def code(x, eps):
	return (10.0 * (x * (x * x))) * (eps * eps)

Julia

function code(x, eps)
	return Float64(Float64(10.0 * Float64(x * Float64(x * x))) * Float64(eps * eps))
end

MATLAB

function tmp = code(x, eps)
	tmp = (10.0 * (x * (x * x))) * (eps * eps);
end

Wolfram

code[x_, eps_] := N[(N[(10.0 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.9%

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

3. Taylor expanded in x around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. lower--.f64 N/A

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

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

5. Applied rewrites 83.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 71.8%

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

9. Final simplification 71.8%

   \[\leadsto \left(10 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024237 
(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)))