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

Percentage Accurate: 88.5% → 98.1%

Time: 9.4s

Alternatives: 13

Speedup: 5.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(5, x \cdot {\varepsilon}^{4}, {\varepsilon}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, \frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(-10\right)}{x}\right)}{-x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-43)
   (* (pow x 4.0) (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)))
   (if (<= x 1.22e-52)
     (fma 5.0 (* x (pow eps 4.0)) (pow eps 5.0))
     (*
      (* x (* x (* x x)))
      (fma
       eps
       5.0
       (/
        (fma (* eps eps) -10.0 (/ (* (* eps (* eps eps)) (- 10.0)) x))
        (- x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-43) {
		tmp = pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} else if (x <= 1.22e-52) {
		tmp = fma(5.0, (x * pow(eps, 4.0)), pow(eps, 5.0));
	} else {
		tmp = (x * (x * (x * x))) * fma(eps, 5.0, (fma((eps * eps), -10.0, (((eps * (eps * eps)) * -10.0) / x)) / -x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)));
	elseif (x <= 1.22e-52)
		tmp = fma(5.0, Float64(x * (eps ^ 4.0)), (eps ^ 5.0));
	else
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) * fma(eps, 5.0, Float64(fma(Float64(eps * eps), -10.0, Float64(Float64(Float64(eps * Float64(eps * eps)) * Float64(-10.0)) / x)) / Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-43], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e-52], N[(5.0 * N[(x * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * -10.0 + N[(N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * (-10.0)), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999985e-43

   1. Initial program 47.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 100.0%

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

   if -3.19999999999999985e-43 < x < 1.22e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 1.22e-52 < x

   1. Initial program 48.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(5, x \cdot {\varepsilon}^{4}, {\varepsilon}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, \frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(-10\right)}{x}\right)}{-x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, \frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(-10\right)}{x}\right)}{-x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-43)
   (* (pow x 4.0) (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)))
   (if (<= x 1.22e-52)
     (* (pow eps 5.0) (fma 5.0 (/ x eps) 1.0))
     (*
      (* x (* x (* x x)))
      (fma
       eps
       5.0
       (/
        (fma (* eps eps) -10.0 (/ (* (* eps (* eps eps)) (- 10.0)) x))
        (- x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-43) {
		tmp = pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} else if (x <= 1.22e-52) {
		tmp = pow(eps, 5.0) * fma(5.0, (x / eps), 1.0);
	} else {
		tmp = (x * (x * (x * x))) * fma(eps, 5.0, (fma((eps * eps), -10.0, (((eps * (eps * eps)) * -10.0) / x)) / -x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)));
	elseif (x <= 1.22e-52)
		tmp = Float64((eps ^ 5.0) * fma(5.0, Float64(x / eps), 1.0));
	else
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) * fma(eps, 5.0, Float64(fma(Float64(eps * eps), -10.0, Float64(Float64(Float64(eps * Float64(eps * eps)) * Float64(-10.0)) / x)) / Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-43], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e-52], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * -10.0 + N[(N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * (-10.0)), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999985e-43

   1. Initial program 47.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 100.0%

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

   if -3.19999999999999985e-43 < x < 1.22e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 1.22e-52 < x

   1. Initial program 48.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, \frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(-10\right)}{x}\right)}{-x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, \frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(-10\right)}{x}\right)}{-x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-43)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 1.22e-52)
     (* (pow eps 5.0) (fma 5.0 (/ x eps) 1.0))
     (*
      (* x (* x (* x x)))
      (fma
       eps
       5.0
       (/
        (fma (* eps eps) -10.0 (/ (* (* eps (* eps eps)) (- 10.0)) x))
        (- x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-43) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 1.22e-52) {
		tmp = pow(eps, 5.0) * fma(5.0, (x / eps), 1.0);
	} else {
		tmp = (x * (x * (x * x))) * fma(eps, 5.0, (fma((eps * eps), -10.0, (((eps * (eps * eps)) * -10.0) / x)) / -x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 1.22e-52)
		tmp = Float64((eps ^ 5.0) * fma(5.0, Float64(x / eps), 1.0));
	else
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) * fma(eps, 5.0, Float64(fma(Float64(eps * eps), -10.0, Float64(Float64(Float64(eps * Float64(eps * eps)) * Float64(-10.0)) / x)) / Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-43], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e-52], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * -10.0 + N[(N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * (-10.0)), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999985e-43

   1. Initial program 47.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -3.19999999999999985e-43 < x < 1.22e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 1.22e-52 < x

   1. Initial program 48.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, \frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(-10\right)}{x}\right)}{-x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, \frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(-10\right)}{x}\right)}{-x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-43)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 1.22e-52)
     (pow eps 5.0)
     (*
      (* x (* x (* x x)))
      (fma
       eps
       5.0
       (/
        (fma (* eps eps) -10.0 (/ (* (* eps (* eps eps)) (- 10.0)) x))
        (- x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-43) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 1.22e-52) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (x * (x * (x * x))) * fma(eps, 5.0, (fma((eps * eps), -10.0, (((eps * (eps * eps)) * -10.0) / x)) / -x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 1.22e-52)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) * fma(eps, 5.0, Float64(fma(Float64(eps * eps), -10.0, Float64(Float64(Float64(eps * Float64(eps * eps)) * Float64(-10.0)) / x)) / Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-43], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e-52], N[Power[eps, 5.0], $MachinePrecision], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * -10.0 + N[(N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * (-10.0)), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999985e-43

   1. Initial program 47.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -3.19999999999999985e-43 < x < 1.22e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 1.22e-52 < x

   1. Initial program 48.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, \frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(-10\right)}{x}\right)}{-x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, \frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(-10\right)}{x}\right)}{-x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-43)
   (* eps (* 5.0 (* (* x x) (* x x))))
   (if (<= x 1.22e-52)
     (pow eps 5.0)
     (*
      (* x (* x (* x x)))
      (fma
       eps
       5.0
       (/
        (fma (* eps eps) -10.0 (/ (* (* eps (* eps eps)) (- 10.0)) x))
        (- x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-43) {
		tmp = eps * (5.0 * ((x * x) * (x * x)));
	} else if (x <= 1.22e-52) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (x * (x * (x * x))) * fma(eps, 5.0, (fma((eps * eps), -10.0, (((eps * (eps * eps)) * -10.0) / x)) / -x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))));
	elseif (x <= 1.22e-52)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) * fma(eps, 5.0, Float64(fma(Float64(eps * eps), -10.0, Float64(Float64(Float64(eps * Float64(eps * eps)) * Float64(-10.0)) / x)) / Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-43], N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e-52], N[Power[eps, 5.0], $MachinePrecision], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * -10.0 + N[(N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * (-10.0)), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999985e-43

   1. Initial program 47.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.7%

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

   if -3.19999999999999985e-43 < x < 1.22e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 1.22e-52 < x

   1. Initial program 48.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, \frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(-10\right)}{x}\right)}{-x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\varepsilon \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, \frac{t\_0 \cdot \left(-10\right)}{x}\right)}{-x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* eps eps))))
   (if (<= x -3.2e-43)
     (* eps (* 5.0 (* (* x x) (* x x))))
     (if (<= x 1.22e-52)
       (* (fma x 5.0 eps) (* eps t_0))
       (*
        (* x (* x (* x x)))
        (fma
         eps
         5.0
         (/ (fma (* eps eps) -10.0 (/ (* t_0 (- 10.0)) x)) (- x))))))))

C

double code(double x, double eps) {
	double t_0 = eps * (eps * eps);
	double tmp;
	if (x <= -3.2e-43) {
		tmp = eps * (5.0 * ((x * x) * (x * x)));
	} else if (x <= 1.22e-52) {
		tmp = fma(x, 5.0, eps) * (eps * t_0);
	} else {
		tmp = (x * (x * (x * x))) * fma(eps, 5.0, (fma((eps * eps), -10.0, ((t_0 * -10.0) / x)) / -x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(eps * eps))
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))));
	elseif (x <= 1.22e-52)
		tmp = Float64(fma(x, 5.0, eps) * Float64(eps * t_0));
	else
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) * fma(eps, 5.0, Float64(fma(Float64(eps * eps), -10.0, Float64(Float64(t_0 * Float64(-10.0)) / x)) / Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-43], N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e-52], N[(N[(x * 5.0 + eps), $MachinePrecision] * N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * -10.0 + N[(N[(t$95$0 * (-10.0)), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999985e-43

   1. Initial program 47.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.7%

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

   if -3.19999999999999985e-43 < x < 1.22e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-in N/A

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

      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 99.6

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.5%

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

   12. Applied rewrites 99.5%

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

   if 1.22e-52 < x

   1. Initial program 48.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\varepsilon, 5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, \frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(-10\right)}{x}\right)}{-x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 10\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, t\_0, \varepsilon \cdot t\_0\right), 5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) 10.0)))
   (if (<= x -3.2e-43)
     (* eps (* 5.0 (* (* x x) (* x x))))
     (if (<= x 1.22e-52)
       (* (fma x 5.0 eps) (* eps (* eps (* eps eps))))
       (*
        eps
        (fma eps (fma x t_0 (* eps t_0)) (* 5.0 (* x (* x (* x x))))))))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * 10.0;
	double tmp;
	if (x <= -3.2e-43) {
		tmp = eps * (5.0 * ((x * x) * (x * x)));
	} else if (x <= 1.22e-52) {
		tmp = fma(x, 5.0, eps) * (eps * (eps * (eps * eps)));
	} else {
		tmp = eps * fma(eps, fma(x, t_0, (eps * t_0)), (5.0 * (x * (x * (x * x)))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * 10.0)
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))));
	elseif (x <= 1.22e-52)
		tmp = Float64(fma(x, 5.0, eps) * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(eps * fma(eps, fma(x, t_0, Float64(eps * t_0)), Float64(5.0 * Float64(x * Float64(x * Float64(x * x))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]}, If[LessEqual[x, -3.2e-43], N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e-52], N[(N[(x * 5.0 + eps), $MachinePrecision] * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(eps * N[(x * t$95$0 + N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999985e-43

   1. Initial program 47.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.7%

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

   if -3.19999999999999985e-43 < x < 1.22e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-in N/A

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

      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 99.6

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.5%

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

   12. Applied rewrites 99.5%

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

   if 1.22e-52 < x

   1. Initial program 48.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 10, \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right), 5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-43)
   (* eps (* 5.0 (* (* x x) (* x x))))
   (if (<= x 1.9e-52)
     (* (fma x 5.0 eps) (* eps (* eps (* eps eps))))
     (* (* x x) (* (* eps 5.0) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-43) {
		tmp = eps * (5.0 * ((x * x) * (x * x)));
	} else if (x <= 1.9e-52) {
		tmp = fma(x, 5.0, eps) * (eps * (eps * (eps * eps)));
	} else {
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))));
	elseif (x <= 1.9e-52)
		tmp = Float64(fma(x, 5.0, eps) * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(eps * 5.0) * Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-43], N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-52], N[(N[(x * 5.0 + eps), $MachinePrecision] * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999985e-43

   1. Initial program 47.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.7%

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

   if -3.19999999999999985e-43 < x < 1.9000000000000002e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-in N/A

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

      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 99.6

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.5%

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

   12. Applied rewrites 99.5%

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

   if 1.9000000000000002e-52 < x

   1. Initial program 48.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-43)
   (* eps (* 5.0 (* (* x x) (* x x))))
   (if (<= x 1.9e-52)
     (* (fma x 5.0 eps) (* (* eps eps) (* eps eps)))
     (* (* x x) (* (* eps 5.0) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-43) {
		tmp = eps * (5.0 * ((x * x) * (x * x)));
	} else if (x <= 1.9e-52) {
		tmp = fma(x, 5.0, eps) * ((eps * eps) * (eps * eps));
	} else {
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))));
	elseif (x <= 1.9e-52)
		tmp = Float64(fma(x, 5.0, eps) * Float64(Float64(eps * eps) * Float64(eps * eps)));
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(eps * 5.0) * Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-43], N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-52], N[(N[(x * 5.0 + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999985e-43

   1. Initial program 47.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.7%

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

   if -3.19999999999999985e-43 < x < 1.9000000000000002e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-in N/A

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

      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 99.6

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.5%

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

   if 1.9000000000000002e-52 < x

   1. Initial program 48.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(x, 5, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-43)
   (* eps (* 5.0 (* (* x x) (* x x))))
   (if (<= x 1.9e-52)
     (* (* eps eps) (* (* eps eps) (fma x 5.0 eps)))
     (* (* x x) (* (* eps 5.0) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-43) {
		tmp = eps * (5.0 * ((x * x) * (x * x)));
	} else if (x <= 1.9e-52) {
		tmp = (eps * eps) * ((eps * eps) * fma(x, 5.0, eps));
	} else {
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))));
	elseif (x <= 1.9e-52)
		tmp = Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * fma(x, 5.0, eps)));
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(eps * 5.0) * Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-43], N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-52], N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(x * 5.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999985e-43

   1. Initial program 47.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.7%

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

   if -3.19999999999999985e-43 < x < 1.9000000000000002e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-in N/A

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

      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 99.6

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.5%

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

   if 1.9000000000000002e-52 < x

   1. Initial program 48.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-43)
   (* eps (* 5.0 (* (* x x) (* x x))))
   (if (<= x 1.9e-52)
     (* (* eps eps) (* eps (* eps eps)))
     (* (* x x) (* (* eps 5.0) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-43) {
		tmp = eps * (5.0 * ((x * x) * (x * x)));
	} else if (x <= 1.9e-52) {
		tmp = (eps * eps) * (eps * (eps * eps));
	} else {
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.2d-43)) then
        tmp = eps * (5.0d0 * ((x * x) * (x * x)))
    else if (x <= 1.9d-52) then
        tmp = (eps * eps) * (eps * (eps * eps))
    else
        tmp = (x * x) * ((eps * 5.0d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-43) {
		tmp = eps * (5.0 * ((x * x) * (x * x)));
	} else if (x <= 1.9e-52) {
		tmp = (eps * eps) * (eps * (eps * eps));
	} else {
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.2e-43:
		tmp = eps * (5.0 * ((x * x) * (x * x)))
	elif x <= 1.9e-52:
		tmp = (eps * eps) * (eps * (eps * eps))
	else:
		tmp = (x * x) * ((eps * 5.0) * (x * x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))));
	elseif (x <= 1.9e-52)
		tmp = Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)));
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(eps * 5.0) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.2e-43)
		tmp = eps * (5.0 * ((x * x) * (x * x)));
	elseif (x <= 1.9e-52)
		tmp = (eps * eps) * (eps * (eps * eps));
	else
		tmp = (x * x) * ((eps * 5.0) * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-43], N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-52], N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999985e-43

   1. Initial program 47.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.7%

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

   if -3.19999999999999985e-43 < x < 1.9000000000000002e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-in N/A

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

      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 99.6

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.5%

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

   if 1.9000000000000002e-52 < x

   1. Initial program 48.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* 5.0 (* (* x x) (* x x))))))
   (if (<= x -3.2e-43)
     t_0
     (if (<= x 1.9e-52) (* (* eps eps) (* eps (* eps eps))) t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (5.0 * ((x * x) * (x * x)));
	double tmp;
	if (x <= -3.2e-43) {
		tmp = t_0;
	} else if (x <= 1.9e-52) {
		tmp = (eps * eps) * (eps * (eps * eps));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * (5.0d0 * ((x * x) * (x * x)))
    if (x <= (-3.2d-43)) then
        tmp = t_0
    else if (x <= 1.9d-52) then
        tmp = (eps * eps) * (eps * (eps * eps))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (5.0 * ((x * x) * (x * x)));
	double tmp;
	if (x <= -3.2e-43) {
		tmp = t_0;
	} else if (x <= 1.9e-52) {
		tmp = (eps * eps) * (eps * (eps * eps));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (5.0 * ((x * x) * (x * x)))
	tmp = 0
	if x <= -3.2e-43:
		tmp = t_0
	elif x <= 1.9e-52:
		tmp = (eps * eps) * (eps * (eps * eps))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))))
	tmp = 0.0
	if (x <= -3.2e-43)
		tmp = t_0;
	elseif (x <= 1.9e-52)
		tmp = Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * (5.0 * ((x * x) * (x * x)));
	tmp = 0.0;
	if (x <= -3.2e-43)
		tmp = t_0;
	elseif (x <= 1.9e-52)
		tmp = (eps * eps) * (eps * (eps * eps));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-43], t$95$0, If[LessEqual[x, 1.9e-52], N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999985e-43 or 1.9000000000000002e-52 < x

   1. Initial program 48.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 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 99.6%

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

   if -3.19999999999999985e-43 < x < 1.9000000000000002e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-in N/A

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

      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 99.6

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.4% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 93.0

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

5. Applied rewrites 93.0%

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

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. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. pow-plus N/A

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

   4. distribute-lft1-in N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-lft-in N/A

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

   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 92.9

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

8. Applied rewrites 92.9%

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

10. Applied rewrites 92.8%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 92.8%

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

14. Final simplification 92.8%

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

Reproduce

herbie shell --seed 2024221 
(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)))