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

Percentage Accurate: 88.2% → 99.5%

Time: 10.7s

Alternatives: 11

Speedup: 9.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-322}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-322)
     t_0
     (if (<= t_0 0.0) (* eps (* 5.0 (pow x 4.0))) t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} 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 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-2d-322)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -2e-322:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -2e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-322], t$95$0, If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.97626e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.8%

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

   if -1.97626e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.3%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(4 \cdot {x}^{4} + {x}^{4}\right)}\right) \]

      7. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(4 + 1\right) \cdot \color{blue}{{x}^{4}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(5 \cdot {\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{\left({x}^{4}\right)}\right)\right) \]

      10. pow-lowering-pow.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \mathsf{pow.f64}\left(x, \color{blue}{4}\right)\right)\right) \]

   5. Simplified 99.9%

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

Alternative 2: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-54}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6e-54)
   (* (* x (* x (* x x))) (* eps 5.0))
   (if (<= x 2.2e-65)
     (pow eps 5.0)
     (* (pow x 4.0) (- (* eps 5.0) (/ (* (* eps eps) -10.0) x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6e-54) {
		tmp = (x * (x * (x * x))) * (eps * 5.0);
	} else if (x <= 2.2e-65) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / 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 <= (-6d-54)) then
        tmp = (x * (x * (x * x))) * (eps * 5.0d0)
    else if (x <= 2.2d-65) then
        tmp = eps ** 5.0d0
    else
        tmp = (x ** 4.0d0) * ((eps * 5.0d0) - (((eps * eps) * (-10.0d0)) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6e-54) {
		tmp = (x * (x * (x * x))) * (eps * 5.0);
	} else if (x <= 2.2e-65) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = Math.pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6e-54:
		tmp = (x * (x * (x * x))) * (eps * 5.0)
	elif x <= 2.2e-65:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = math.pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6e-54)
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) * Float64(eps * 5.0));
	elseif (x <= 2.2e-65)
		tmp = eps ^ 5.0;
	else
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6e-54)
		tmp = (x * (x * (x * x))) * (eps * 5.0);
	elseif (x <= 2.2e-65)
		tmp = eps ^ 5.0;
	else
		tmp = (x ^ 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6e-54], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-65], N[Power[eps, 5.0], $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.00000000000000018e-54

   1. Initial program 37.7%

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

   3. Taylor expanded in x around -inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 94.4%

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

   6. Taylor expanded in eps around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \color{blue}{\left(5 \cdot \varepsilon\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 94.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(5, \color{blue}{\varepsilon}\right)\right) \]

   8. Simplified 94.4%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(3 + 1\right)}\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{3} \cdot x\right), \mathsf{*.f64}\left(\color{blue}{5}, \varepsilon\right)\right) \]

      3. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right), \mathsf{*.f64}\left(\color{blue}{5}, \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

      6. *-lowering-*.f64 94.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

   10. Applied egg-rr 94.6%

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

   if -6.00000000000000018e-54 < x < 2.20000000000000021e-65

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. pow-lowering-pow.f64 100.0%

         \[\leadsto \mathsf{pow.f64}\left(\varepsilon, \color{blue}{5}\right) \]

   5. Simplified 100.0%

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

   if 2.20000000000000021e-65 < x

   1. Initial program 59.6%

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 95.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-54}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-54}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-73}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.2e-54)
   (* (* x (* x (* x x))) (* eps 5.0))
   (if (<= x 5e-73)
     (pow eps 5.0)
     (* (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)) (* (* x x) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.2e-54) {
		tmp = (x * (x * (x * x))) * (eps * 5.0);
	} else if (x <= 5e-73) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (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 <= (-2.2d-54)) then
        tmp = (x * (x * (x * x))) * (eps * 5.0d0)
    else if (x <= 5d-73) then
        tmp = eps ** 5.0d0
    else
        tmp = ((eps * 5.0d0) - (((eps * eps) * (-10.0d0)) / x)) * ((x * x) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.2e-54) {
		tmp = (x * (x * (x * x))) * (eps * 5.0);
	} else if (x <= 5e-73) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.2e-54:
		tmp = (x * (x * (x * x))) * (eps * 5.0)
	elif x <= 5e-73:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.2e-54)
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) * Float64(eps * 5.0));
	elseif (x <= 5e-73)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)) * Float64(Float64(x * x) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.2e-54)
		tmp = (x * (x * (x * x))) * (eps * 5.0);
	elseif (x <= 5e-73)
		tmp = eps ^ 5.0;
	else
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.2e-54], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-73], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2e-54

   1. Initial program 37.7%

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

   3. Taylor expanded in x around -inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 94.4%

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

   6. Taylor expanded in eps around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \color{blue}{\left(5 \cdot \varepsilon\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 94.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(5, \color{blue}{\varepsilon}\right)\right) \]

   8. Simplified 94.4%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(3 + 1\right)}\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{3} \cdot x\right), \mathsf{*.f64}\left(\color{blue}{5}, \varepsilon\right)\right) \]

      3. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right), \mathsf{*.f64}\left(\color{blue}{5}, \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

      6. *-lowering-*.f64 94.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

   10. Applied egg-rr 94.6%

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

   if -2.2e-54 < x < 4.9999999999999998e-73

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. pow-lowering-pow.f64 100.0%

         \[\leadsto \mathsf{pow.f64}\left(\varepsilon, \color{blue}{5}\right) \]

   5. Simplified 100.0%

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

   if 4.9999999999999998e-73 < x

   1. Initial program 59.6%

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 95.0%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \color{blue}{5}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\varepsilon, 5\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot x\right)}^{2}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\varepsilon, 5\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\varepsilon, 5\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot x\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\varepsilon, 5\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\varepsilon}, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

      7. *-lowering-*.f64 94.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \color{blue}{5}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

   7. Applied egg-rr 94.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-54}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-73}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-53}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.6e-53)
   (* (* x (* x (* x x))) (* eps 5.0))
   (if (<= x 2.2e-65)
     (* eps (* eps (* eps (* eps eps))))
     (* (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)) (* (* x x) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.6e-53) {
		tmp = (x * (x * (x * x))) * (eps * 5.0);
	} else if (x <= 2.2e-65) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (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 <= (-4.6d-53)) then
        tmp = (x * (x * (x * x))) * (eps * 5.0d0)
    else if (x <= 2.2d-65) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = ((eps * 5.0d0) - (((eps * eps) * (-10.0d0)) / x)) * ((x * x) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -4.6e-53) {
		tmp = (x * (x * (x * x))) * (eps * 5.0);
	} else if (x <= 2.2e-65) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4.6e-53:
		tmp = (x * (x * (x * x))) * (eps * 5.0)
	elif x <= 2.2e-65:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.6e-53)
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) * Float64(eps * 5.0));
	elseif (x <= 2.2e-65)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)) * Float64(Float64(x * x) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4.6e-53)
		tmp = (x * (x * (x * x))) * (eps * 5.0);
	elseif (x <= 2.2e-65)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.6e-53], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-65], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6000000000000003e-53

   1. Initial program 37.7%

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

   3. Taylor expanded in x around -inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 94.4%

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

   6. Taylor expanded in eps around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \color{blue}{\left(5 \cdot \varepsilon\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 94.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(5, \color{blue}{\varepsilon}\right)\right) \]

   8. Simplified 94.4%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(3 + 1\right)}\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{3} \cdot x\right), \mathsf{*.f64}\left(\color{blue}{5}, \varepsilon\right)\right) \]

      3. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right), \mathsf{*.f64}\left(\color{blue}{5}, \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

      6. *-lowering-*.f64 94.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

   10. Applied egg-rr 94.6%

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

   if -4.6000000000000003e-53 < x < 2.20000000000000021e-65

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{5}\right), \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(\color{blue}{1} + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \left(4 + 1\right) \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + 5 \cdot \frac{\color{blue}{x}}{\varepsilon}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{5 \cdot x}{\color{blue}{\varepsilon}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{x + 4 \cdot x}{\varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x + 4 \cdot x}{\varepsilon}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + 4 \cdot x\right), \color{blue}{\varepsilon}\right)\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 + 1\right) \cdot x\right), \varepsilon\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(5 \cdot x\right), \varepsilon\right)\right)\right) \]

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(5, x\right), \varepsilon\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{4}\right), \color{blue}{\varepsilon}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \varepsilon\right) \]

      5. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \varepsilon\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \varepsilon\right), \varepsilon\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \varepsilon\right), \varepsilon\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \varepsilon\right), \varepsilon\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

      11. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

   8. Simplified 99.9%

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

   if 2.20000000000000021e-65 < x

   1. Initial program 59.6%

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 95.0%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \color{blue}{5}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\varepsilon, 5\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot x\right)}^{2}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\varepsilon, 5\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\varepsilon, 5\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot x\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\varepsilon, 5\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\varepsilon}, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

      7. *-lowering-*.f64 94.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \color{blue}{5}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), -10\right), x\right)\right)\right) \]

   7. Applied egg-rr 94.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-53}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-53}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.3e-53)
   (* (* x (* x (* x x))) (* eps 5.0))
   (if (<= x 2.2e-65)
     (* eps (* eps (* eps (* eps eps))))
     (* x (* (* x x) (* eps (+ (* x 5.0) (* eps 10.0))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.3e-53) {
		tmp = (x * (x * (x * x))) * (eps * 5.0);
	} else if (x <= 2.2e-65) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = x * ((x * x) * (eps * ((x * 5.0) + (eps * 10.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.3d-53)) then
        tmp = (x * (x * (x * x))) * (eps * 5.0d0)
    else if (x <= 2.2d-65) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = x * ((x * x) * (eps * ((x * 5.0d0) + (eps * 10.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.3e-53) {
		tmp = (x * (x * (x * x))) * (eps * 5.0);
	} else if (x <= 2.2e-65) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = x * ((x * x) * (eps * ((x * 5.0) + (eps * 10.0))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.3e-53:
		tmp = (x * (x * (x * x))) * (eps * 5.0)
	elif x <= 2.2e-65:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = x * ((x * x) * (eps * ((x * 5.0) + (eps * 10.0))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.3e-53)
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) * Float64(eps * 5.0));
	elseif (x <= 2.2e-65)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(x * Float64(Float64(x * x) * Float64(eps * Float64(Float64(x * 5.0) + Float64(eps * 10.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.3e-53)
		tmp = (x * (x * (x * x))) * (eps * 5.0);
	elseif (x <= 2.2e-65)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = x * ((x * x) * (eps * ((x * 5.0) + (eps * 10.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.3e-53], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-65], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * x), $MachinePrecision] * N[(eps * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3000000000000001e-53

   1. Initial program 37.7%

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

   3. Taylor expanded in x around -inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 94.4%

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

   6. Taylor expanded in eps around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \color{blue}{\left(5 \cdot \varepsilon\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 94.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(5, \color{blue}{\varepsilon}\right)\right) \]

   8. Simplified 94.4%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(3 + 1\right)}\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{3} \cdot x\right), \mathsf{*.f64}\left(\color{blue}{5}, \varepsilon\right)\right) \]

      3. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right), \mathsf{*.f64}\left(\color{blue}{5}, \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

      6. *-lowering-*.f64 94.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

   10. Applied egg-rr 94.6%

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

   if -2.3000000000000001e-53 < x < 2.20000000000000021e-65

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{5}\right), \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(\color{blue}{1} + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \left(4 + 1\right) \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + 5 \cdot \frac{\color{blue}{x}}{\varepsilon}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{5 \cdot x}{\color{blue}{\varepsilon}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{x + 4 \cdot x}{\varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x + 4 \cdot x}{\varepsilon}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + 4 \cdot x\right), \color{blue}{\varepsilon}\right)\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 + 1\right) \cdot x\right), \varepsilon\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(5 \cdot x\right), \varepsilon\right)\right)\right) \]

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(5, x\right), \varepsilon\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{4}\right), \color{blue}{\varepsilon}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \varepsilon\right) \]

      5. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \varepsilon\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \varepsilon\right), \varepsilon\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \varepsilon\right), \varepsilon\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \varepsilon\right), \varepsilon\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

      11. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

   8. Simplified 99.9%

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

   if 2.20000000000000021e-65 < x

   1. Initial program 59.6%

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 95.0%

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

   6. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\color{blue}{5 \cdot \left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), \left(\color{blue}{5 \cdot \left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right) \cdot {\varepsilon}^{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\color{blue}{\varepsilon}}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(5 \cdot \left(\varepsilon \cdot x\right)\right), \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right)}\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(\left(5 \cdot \varepsilon\right) \cdot x\right), \left(\color{blue}{10} \cdot {\varepsilon}^{2}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(\left(\left(4 + 1\right) \cdot \varepsilon\right) \cdot x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

      14. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(\left(\varepsilon + 4 \cdot \varepsilon\right) \cdot x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), x\right), \left(\color{blue}{10} \cdot {\varepsilon}^{2}\right)\right)\right) \]

      16. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(5 \cdot \varepsilon\right), x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \varepsilon\right), x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \varepsilon\right), x\right), \mathsf{*.f64}\left(10, \color{blue}{\left({\varepsilon}^{2}\right)}\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \varepsilon\right), x\right), \mathsf{*.f64}\left(10, \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 94.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \varepsilon\right), x\right), \mathsf{*.f64}\left(10, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right)\right)\right)\right) \]

   8. Simplified 94.9%

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

   9. Taylor expanded in x around 0

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{5 \cdot \left(\varepsilon \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{5 \cdot \left(\varepsilon \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\varepsilon \cdot x\right) \cdot 5 + \color{blue}{10} \cdot {\varepsilon}^{2}\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\varepsilon \cdot \left(x \cdot 5\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\varepsilon \cdot \left(5 \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\varepsilon \cdot \left(5 \cdot x\right) + {\varepsilon}^{2} \cdot \color{blue}{10}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\varepsilon \cdot \left(5 \cdot x\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\right)\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\varepsilon \cdot \left(5 \cdot x\right) + \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot 10\right)}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\varepsilon \cdot \left(5 \cdot x\right) + \varepsilon \cdot \left(10 \cdot \color{blue}{\varepsilon}\right)\right)\right)\right) \]

       15. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\varepsilon \cdot \color{blue}{\left(5 \cdot x + 10 \cdot \varepsilon\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(5 \cdot x + 10 \cdot \varepsilon\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(5 \cdot x\right), \color{blue}{\left(10 \cdot \varepsilon\right)}\right)\right)\right)\right) \]

       18. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(x \cdot 5\right), \left(\color{blue}{10} \cdot \varepsilon\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \left(\color{blue}{10} \cdot \varepsilon\right)\right)\right)\right)\right) \]

       20. *-lowering-*.f64 94.9%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \color{blue}{\varepsilon}\right)\right)\right)\right)\right) \]

   11. Simplified 94.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-53}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{-50}:\\ \;\;\;\;\left(x \cdot t\_0\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 10^{-65}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -8.6e-50)
     (* (* x t_0) (* eps 5.0))
     (if (<= x 1e-65)
       (* eps (* eps (* eps (* eps eps))))
       (* t_0 (* eps (* x 5.0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -8.6e-50) {
		tmp = (x * t_0) * (eps * 5.0);
	} else if (x <= 1e-65) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (eps * (x * 5.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= (-8.6d-50)) then
        tmp = (x * t_0) * (eps * 5.0d0)
    else if (x <= 1d-65) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0 * (eps * (x * 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -8.6e-50) {
		tmp = (x * t_0) * (eps * 5.0);
	} else if (x <= 1e-65) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (eps * (x * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	tmp = 0
	if x <= -8.6e-50:
		tmp = (x * t_0) * (eps * 5.0)
	elif x <= 1e-65:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0 * (eps * (x * 5.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -8.6e-50)
		tmp = Float64(Float64(x * t_0) * Float64(eps * 5.0));
	elseif (x <= 1e-65)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(t_0 * Float64(eps * Float64(x * 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= -8.6e-50)
		tmp = (x * t_0) * (eps * 5.0);
	elseif (x <= 1e-65)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0 * (eps * (x * 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.6e-50], N[(N[(x * t$95$0), $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-65], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.59999999999999995e-50

   1. Initial program 37.7%

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

   3. Taylor expanded in x around -inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 94.4%

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

   6. Taylor expanded in eps around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \color{blue}{\left(5 \cdot \varepsilon\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 94.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(5, \color{blue}{\varepsilon}\right)\right) \]

   8. Simplified 94.4%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(3 + 1\right)}\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{3} \cdot x\right), \mathsf{*.f64}\left(\color{blue}{5}, \varepsilon\right)\right) \]

      3. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right), \mathsf{*.f64}\left(\color{blue}{5}, \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

      6. *-lowering-*.f64 94.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{*.f64}\left(5, \varepsilon\right)\right) \]

   10. Applied egg-rr 94.6%

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

   if -8.59999999999999995e-50 < x < 9.99999999999999923e-66

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{5}\right), \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(\color{blue}{1} + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \left(4 + 1\right) \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + 5 \cdot \frac{\color{blue}{x}}{\varepsilon}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{5 \cdot x}{\color{blue}{\varepsilon}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{x + 4 \cdot x}{\varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x + 4 \cdot x}{\varepsilon}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + 4 \cdot x\right), \color{blue}{\varepsilon}\right)\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 + 1\right) \cdot x\right), \varepsilon\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(5 \cdot x\right), \varepsilon\right)\right)\right) \]

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(5, x\right), \varepsilon\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{4}\right), \color{blue}{\varepsilon}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \varepsilon\right) \]

      5. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \varepsilon\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \varepsilon\right), \varepsilon\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \varepsilon\right), \varepsilon\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \varepsilon\right), \varepsilon\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

      11. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

   8. Simplified 99.9%

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

   if 9.99999999999999923e-66 < x

   1. Initial program 59.6%

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 95.0%

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

   6. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\color{blue}{5 \cdot \left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), \left(\color{blue}{5 \cdot \left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right) \cdot {\varepsilon}^{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\color{blue}{\varepsilon}}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(5 \cdot \left(\varepsilon \cdot x\right)\right), \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right)}\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(\left(5 \cdot \varepsilon\right) \cdot x\right), \left(\color{blue}{10} \cdot {\varepsilon}^{2}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(\left(\left(4 + 1\right) \cdot \varepsilon\right) \cdot x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

      14. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(\left(\varepsilon + 4 \cdot \varepsilon\right) \cdot x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), x\right), \left(\color{blue}{10} \cdot {\varepsilon}^{2}\right)\right)\right) \]

      16. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(5 \cdot \varepsilon\right), x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \varepsilon\right), x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \varepsilon\right), x\right), \mathsf{*.f64}\left(10, \color{blue}{\left({\varepsilon}^{2}\right)}\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \varepsilon\right), x\right), \mathsf{*.f64}\left(10, \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 94.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \varepsilon\right), x\right), \mathsf{*.f64}\left(10, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right)\right)\right)\right) \]

   8. Simplified 94.9%

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

   9. Taylor expanded in eps around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(\varepsilon \cdot x\right) \cdot \color{blue}{5}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \color{blue}{\left(x \cdot 5\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \left(5 \cdot \color{blue}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(5 \cdot x\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \color{blue}{5}\right)\right)\right) \]

       6. *-lowering-*.f64 94.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{5}\right)\right)\right) \]

   11. Simplified 94.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-50}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 10^{-65}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{-50}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot t\_0\right)\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -7.2e-50)
     (* 5.0 (* eps (* x t_0)))
     (if (<= x 2.2e-65)
       (* eps (* eps (* eps (* eps eps))))
       (* t_0 (* eps (* x 5.0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -7.2e-50) {
		tmp = 5.0 * (eps * (x * t_0));
	} else if (x <= 2.2e-65) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (eps * (x * 5.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= (-7.2d-50)) then
        tmp = 5.0d0 * (eps * (x * t_0))
    else if (x <= 2.2d-65) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0 * (eps * (x * 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -7.2e-50) {
		tmp = 5.0 * (eps * (x * t_0));
	} else if (x <= 2.2e-65) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (eps * (x * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	tmp = 0
	if x <= -7.2e-50:
		tmp = 5.0 * (eps * (x * t_0))
	elif x <= 2.2e-65:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0 * (eps * (x * 5.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -7.2e-50)
		tmp = Float64(5.0 * Float64(eps * Float64(x * t_0)));
	elseif (x <= 2.2e-65)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(t_0 * Float64(eps * Float64(x * 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= -7.2e-50)
		tmp = 5.0 * (eps * (x * t_0));
	elseif (x <= 2.2e-65)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0 * (eps * (x * 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.2e-50], N[(5.0 * N[(eps * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-65], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.19999999999999958e-50

   1. Initial program 37.7%

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

   3. Taylor expanded in x around -inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 94.4%

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

   6. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(5, \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({x}^{4}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right) \]

      4. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{x}\right)\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), x\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right)\right)\right) \]

      10. *-lowering-*.f64 94.5%

          \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right)\right)\right) \]

   8. Simplified 94.5%

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

   if -7.19999999999999958e-50 < x < 2.20000000000000021e-65

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{5}\right), \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(\color{blue}{1} + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \left(4 + 1\right) \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + 5 \cdot \frac{\color{blue}{x}}{\varepsilon}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{5 \cdot x}{\color{blue}{\varepsilon}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{x + 4 \cdot x}{\varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x + 4 \cdot x}{\varepsilon}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + 4 \cdot x\right), \color{blue}{\varepsilon}\right)\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 + 1\right) \cdot x\right), \varepsilon\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(5 \cdot x\right), \varepsilon\right)\right)\right) \]

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(5, x\right), \varepsilon\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{4}\right), \color{blue}{\varepsilon}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \varepsilon\right) \]

      5. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \varepsilon\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \varepsilon\right), \varepsilon\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \varepsilon\right), \varepsilon\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \varepsilon\right), \varepsilon\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

      11. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

   8. Simplified 99.9%

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

   if 2.20000000000000021e-65 < x

   1. Initial program 59.6%

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 95.0%

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

   6. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\color{blue}{5 \cdot \left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), \left(\color{blue}{5 \cdot \left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)} - -10 \cdot {\varepsilon}^{2}\right)\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right) \cdot {\varepsilon}^{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\color{blue}{\varepsilon}}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(5 \cdot \left(\varepsilon \cdot x\right)\right), \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right)}\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(\left(5 \cdot \varepsilon\right) \cdot x\right), \left(\color{blue}{10} \cdot {\varepsilon}^{2}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(\left(\left(4 + 1\right) \cdot \varepsilon\right) \cdot x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

      14. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(\left(\varepsilon + 4 \cdot \varepsilon\right) \cdot x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), x\right), \left(\color{blue}{10} \cdot {\varepsilon}^{2}\right)\right)\right) \]

      16. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(5 \cdot \varepsilon\right), x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \varepsilon\right), x\right), \left(10 \cdot {\varepsilon}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \varepsilon\right), x\right), \mathsf{*.f64}\left(10, \color{blue}{\left({\varepsilon}^{2}\right)}\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \varepsilon\right), x\right), \mathsf{*.f64}\left(10, \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 94.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \varepsilon\right), x\right), \mathsf{*.f64}\left(10, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right)\right)\right)\right) \]

   8. Simplified 94.9%

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

   9. Taylor expanded in eps around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(\varepsilon \cdot x\right) \cdot \color{blue}{5}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \color{blue}{\left(x \cdot 5\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \left(5 \cdot \color{blue}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(5 \cdot x\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \color{blue}{5}\right)\right)\right) \]

       6. *-lowering-*.f64 94.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{5}\right)\right)\right) \]

   11. Simplified 94.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-50}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.6% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{-51}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot t\_0\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x -3e-51)
     (* 5.0 (* eps t_0))
     (if (<= x 2.2e-65)
       (* eps (* eps (* eps (* eps eps))))
       (* eps (* 5.0 t_0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -3e-51) {
		tmp = 5.0 * (eps * t_0);
	} else if (x <= 2.2e-65) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = eps * (5.0 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    if (x <= (-3d-51)) then
        tmp = 5.0d0 * (eps * t_0)
    else if (x <= 2.2d-65) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = eps * (5.0d0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -3e-51) {
		tmp = 5.0 * (eps * t_0);
	} else if (x <= 2.2e-65) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = eps * (5.0 * t_0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * (x * x))
	tmp = 0
	if x <= -3e-51:
		tmp = 5.0 * (eps * t_0)
	elif x <= 2.2e-65:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = eps * (5.0 * t_0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= -3e-51)
		tmp = Float64(5.0 * Float64(eps * t_0));
	elseif (x <= 2.2e-65)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(eps * Float64(5.0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * (x * x));
	tmp = 0.0;
	if (x <= -3e-51)
		tmp = 5.0 * (eps * t_0);
	elseif (x <= 2.2e-65)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = eps * (5.0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e-51], N[(5.0 * N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-65], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(5.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.00000000000000002e-51

   1. Initial program 37.7%

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

   3. Taylor expanded in x around -inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 94.4%

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

   6. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(5, \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({x}^{4}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right) \]

      4. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{x}\right)\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), x\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right)\right)\right) \]

      10. *-lowering-*.f64 94.5%

          \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right)\right)\right) \]

   8. Simplified 94.5%

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

   if -3.00000000000000002e-51 < x < 2.20000000000000021e-65

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{5}\right), \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(\color{blue}{1} + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \left(4 + 1\right) \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + 5 \cdot \frac{\color{blue}{x}}{\varepsilon}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{5 \cdot x}{\color{blue}{\varepsilon}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{x + 4 \cdot x}{\varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x + 4 \cdot x}{\varepsilon}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + 4 \cdot x\right), \color{blue}{\varepsilon}\right)\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 + 1\right) \cdot x\right), \varepsilon\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(5 \cdot x\right), \varepsilon\right)\right)\right) \]

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(5, x\right), \varepsilon\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{4}\right), \color{blue}{\varepsilon}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \varepsilon\right) \]

      5. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \varepsilon\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \varepsilon\right), \varepsilon\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \varepsilon\right), \varepsilon\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \varepsilon\right), \varepsilon\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

      11. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

   8. Simplified 99.9%

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

   if 2.20000000000000021e-65 < x

   1. Initial program 59.6%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right), \mathsf{pow.f64}\left(\color{blue}{x}, 5\right)\right) \]

      2. sqr-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left({\left(x + \varepsilon\right)}^{\left(\frac{\frac{5}{2}}{2}\right)} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{\frac{5}{2}}{2}\right)}\right) \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({\left(x + \varepsilon\right)}^{\left(\frac{\frac{5}{2}}{2}\right)} \cdot \left({\left(x + \varepsilon\right)}^{\left(\frac{\frac{5}{2}}{2}\right)} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, 5\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({\left(x + \varepsilon\right)}^{\left(\frac{\frac{5}{2}}{2}\right)}\right), \left({\left(x + \varepsilon\right)}^{\left(\frac{\frac{5}{2}}{2}\right)} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, 5\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x + \varepsilon\right), \left(\frac{\frac{5}{2}}{2}\right)\right), \left({\left(x + \varepsilon\right)}^{\left(\frac{\frac{5}{2}}{2}\right)} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right)\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \left(\frac{\frac{5}{2}}{2}\right)\right), \left({\left(x + \varepsilon\right)}^{\left(\frac{\frac{5}{2}}{2}\right)} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right)\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \left(\frac{\frac{5}{2}}{2}\right)\right), \left({\left(x + \varepsilon\right)}^{\left(\frac{\frac{5}{2}}{2}\right)} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right)\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \left({\left(x + \varepsilon\right)}^{\left(\frac{\frac{5}{2}}{2}\right)} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right)\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \mathsf{*.f64}\left(\left({\left(x + \varepsilon\right)}^{\left(\frac{\frac{5}{2}}{2}\right)}\right), \left({\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right)\right)\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x + \varepsilon\right), \left(\frac{\frac{5}{2}}{2}\right)\right), \left({\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right)\right)\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \left(\frac{\frac{5}{2}}{2}\right)\right), \left({\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right)\right)\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \left(\frac{\frac{5}{2}}{2}\right)\right), \left({\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right)\right)\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \left({\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right)\right)\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \mathsf{pow.f64}\left(\left(x + \varepsilon\right), \left(\frac{5}{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \left(\frac{5}{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

      16. metadata-eval 28.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{4}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right), \frac{5}{2}\right)\right)\right), \mathsf{pow.f64}\left(x, 5\right)\right) \]

   4. Applied egg-rr 28.7%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{1.25} \cdot \left({\left(x + \varepsilon\right)}^{1.25} \cdot {\left(x + \varepsilon\right)}^{2.5}\right)} - {x}^{5} \]

   5. Taylor expanded in eps around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(4 \cdot {x}^{4} + {x}^{4}\right)}\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(4 + 1\right) \cdot \color{blue}{{x}^{4}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(5 \cdot {\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{\left({x}^{4}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right) \]

      6. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{x}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), x\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), x\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right)\right)\right) \]

      12. *-lowering-*.f64 94.4%

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right)\right)\right) \]

   7. Simplified 94.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-51}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{-51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 5.0 (* eps (* x (* x (* x x)))))))
   (if (<= x -7.6e-51)
     t_0
     (if (<= x 2.2e-65) (* eps (* eps (* eps (* eps eps)))) t_0))))

C

double code(double x, double eps) {
	double t_0 = 5.0 * (eps * (x * (x * (x * x))));
	double tmp;
	if (x <= -7.6e-51) {
		tmp = t_0;
	} else if (x <= 2.2e-65) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 5.0d0 * (eps * (x * (x * (x * x))))
    if (x <= (-7.6d-51)) then
        tmp = t_0
    else if (x <= 2.2d-65) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 5.0 * (eps * (x * (x * (x * x))));
	double tmp;
	if (x <= -7.6e-51) {
		tmp = t_0;
	} else if (x <= 2.2e-65) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 5.0 * (eps * (x * (x * (x * x))))
	tmp = 0
	if x <= -7.6e-51:
		tmp = t_0
	elif x <= 2.2e-65:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(5.0 * Float64(eps * Float64(x * Float64(x * Float64(x * x)))))
	tmp = 0.0
	if (x <= -7.6e-51)
		tmp = t_0;
	elseif (x <= 2.2e-65)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 5.0 * (eps * (x * (x * (x * x))));
	tmp = 0.0;
	if (x <= -7.6e-51)
		tmp = t_0;
	elseif (x <= 2.2e-65)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(5.0 * N[(eps * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.6e-51], t$95$0, If[LessEqual[x, 2.2e-65], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.60000000000000006e-51 or 2.20000000000000021e-65 < x

   1. Initial program 53.1%

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

   3. Taylor expanded in x around -inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 94.8%

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

   6. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(5, \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({x}^{4}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right) \]

      4. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{x}\right)\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), x\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right)\right)\right) \]

      10. *-lowering-*.f64 94.2%

          \[\leadsto \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right)\right)\right) \]

   8. Simplified 94.2%

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

   if -7.60000000000000006e-51 < x < 2.20000000000000021e-65

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{5}\right), \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(\color{blue}{1} + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \left(4 + 1\right) \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + 5 \cdot \frac{\color{blue}{x}}{\varepsilon}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{5 \cdot x}{\color{blue}{\varepsilon}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{x + 4 \cdot x}{\varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x + 4 \cdot x}{\varepsilon}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + 4 \cdot x\right), \color{blue}{\varepsilon}\right)\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 + 1\right) \cdot x\right), \varepsilon\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(5 \cdot x\right), \varepsilon\right)\right)\right) \]

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(5, x\right), \varepsilon\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{4}\right), \color{blue}{\varepsilon}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \varepsilon\right) \]

      5. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \varepsilon\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \varepsilon\right), \varepsilon\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \varepsilon\right), \varepsilon\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \varepsilon\right), \varepsilon\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

      11. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

   8. Simplified 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-51}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.1% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.5%

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

3. Taylor expanded in eps around inf

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

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

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{5}\right), \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(\color{blue}{1} + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)\right) \]

   3. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \left(4 + 1\right) \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + 5 \cdot \frac{\color{blue}{x}}{\varepsilon}\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{5 \cdot x}{\color{blue}{\varepsilon}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right) \]

   7. distribute-rgt1-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{x + 4 \cdot x}{\varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x + 4 \cdot x}{\varepsilon}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + 4 \cdot x\right), \color{blue}{\varepsilon}\right)\right)\right) \]

   10. distribute-rgt1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 + 1\right) \cdot x\right), \varepsilon\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(5 \cdot x\right), \varepsilon\right)\right)\right) \]

   12. *-lowering-*.f64 89.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(5, x\right), \varepsilon\right)\right)\right) \]

5. Simplified 89.2%

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

6. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{4}\right), \color{blue}{\varepsilon}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \varepsilon\right) \]

   5. pow-plus N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \varepsilon\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \varepsilon\right), \varepsilon\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \varepsilon\right), \varepsilon\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \varepsilon\right), \varepsilon\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

   11. *-lowering-*.f64 89.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

8. Simplified 89.0%

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

9. Final simplification 89.0%

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

Alternative 11: 87.0% accurate, 23.0× 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 89.5%

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

3. Taylor expanded in eps around inf

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

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

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{5}\right), \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(\color{blue}{1} + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)\right) \]

   3. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \left(4 + 1\right) \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + 5 \cdot \frac{\color{blue}{x}}{\varepsilon}\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{5 \cdot x}{\color{blue}{\varepsilon}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right) \]

   7. distribute-rgt1-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{x + 4 \cdot x}{\varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x + 4 \cdot x}{\varepsilon}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x + 4 \cdot x\right), \color{blue}{\varepsilon}\right)\right)\right) \]

   10. distribute-rgt1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 + 1\right) \cdot x\right), \varepsilon\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(5 \cdot x\right), \varepsilon\right)\right)\right) \]

   12. *-lowering-*.f64 89.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(5, x\right), \varepsilon\right)\right)\right) \]

5. Simplified 89.2%

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

6. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{4}\right), \color{blue}{\varepsilon}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \varepsilon\right) \]

   5. pow-plus N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \varepsilon\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \varepsilon\right), \varepsilon\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \varepsilon\right), \varepsilon\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \varepsilon\right), \varepsilon\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

   11. *-lowering-*.f64 89.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

8. Simplified 89.0%

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

   1. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{\varepsilon} \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\varepsilon \cdot \varepsilon\right)\right) \]

   5. *-lowering-*.f64 88.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right)\right) \]

10. Applied egg-rr 88.9%

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

11. Final simplification 88.9%

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

Reproduce

herbie shell --seed 2024155 
(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)))