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

Percentage Accurate: 87.8% → 98.8%

Time: 10.5s

Alternatives: 15

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: 87.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-286)
     t_0
     (if (<= t_0 0.0)
       (* (pow x 4.0) (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)))
       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 <= -1e-286) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} 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 <= (-1d-286)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (x ** 4.0d0) * ((eps * 5.0d0) - (((eps * eps) * (-10.0d0)) / x))
    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 <= -1e-286) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = Math.pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} 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 <= -1e-286:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = math.pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x))
	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 <= -1e-286)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)));
	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 <= -1e-286)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (x ^ 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	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, -1e-286], t$95$0, If[LessEqual[t$95$0, 0.0], 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], 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))) < -1.00000000000000005e-286 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.0%

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

   if -1.00000000000000005e-286 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.7%

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

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

Alternative 2: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + \varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 + \frac{\frac{10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{x} - \left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5e-52)
   (*
    eps
    (+
     (* 5.0 (pow x 4.0))
     (*
      eps
      (+
       (* (* x (* x x)) 10.0)
       (* eps (+ (* (* x x) 10.0) (* eps (* x 5.0))))))))
   (if (<= x 1.45e-61)
     (pow eps 5.0)
     (*
      (pow x 4.0)
      (+
       (* eps 5.0)
       (/ (- (/ (* 10.0 (* eps (* eps eps))) x) (* (* eps eps) -10.0)) x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5e-52) {
		tmp = eps * ((5.0 * pow(x, 4.0)) + (eps * (((x * (x * x)) * 10.0) + (eps * (((x * x) * 10.0) + (eps * (x * 5.0)))))));
	} else if (x <= 1.45e-61) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = pow(x, 4.0) * ((eps * 5.0) + ((((10.0 * (eps * (eps * eps))) / x) - ((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 <= (-5d-52)) then
        tmp = eps * ((5.0d0 * (x ** 4.0d0)) + (eps * (((x * (x * x)) * 10.0d0) + (eps * (((x * x) * 10.0d0) + (eps * (x * 5.0d0)))))))
    else if (x <= 1.45d-61) then
        tmp = eps ** 5.0d0
    else
        tmp = (x ** 4.0d0) * ((eps * 5.0d0) + ((((10.0d0 * (eps * (eps * eps))) / x) - ((eps * eps) * (-10.0d0))) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5e-52) {
		tmp = eps * ((5.0 * Math.pow(x, 4.0)) + (eps * (((x * (x * x)) * 10.0) + (eps * (((x * x) * 10.0) + (eps * (x * 5.0)))))));
	} else if (x <= 1.45e-61) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = Math.pow(x, 4.0) * ((eps * 5.0) + ((((10.0 * (eps * (eps * eps))) / x) - ((eps * eps) * -10.0)) / x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5e-52:
		tmp = eps * ((5.0 * math.pow(x, 4.0)) + (eps * (((x * (x * x)) * 10.0) + (eps * (((x * x) * 10.0) + (eps * (x * 5.0)))))))
	elif x <= 1.45e-61:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = math.pow(x, 4.0) * ((eps * 5.0) + ((((10.0 * (eps * (eps * eps))) / x) - ((eps * eps) * -10.0)) / x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5e-52)
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(eps * Float64(Float64(Float64(x * Float64(x * x)) * 10.0) + Float64(eps * Float64(Float64(Float64(x * x) * 10.0) + Float64(eps * Float64(x * 5.0))))))));
	elseif (x <= 1.45e-61)
		tmp = eps ^ 5.0;
	else
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) + Float64(Float64(Float64(Float64(10.0 * Float64(eps * Float64(eps * eps))) / x) - Float64(Float64(eps * eps) * -10.0)) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5e-52)
		tmp = eps * ((5.0 * (x ^ 4.0)) + (eps * (((x * (x * x)) * 10.0) + (eps * (((x * x) * 10.0) + (eps * (x * 5.0)))))));
	elseif (x <= 1.45e-61)
		tmp = eps ^ 5.0;
	else
		tmp = (x ^ 4.0) * ((eps * 5.0) + ((((10.0 * (eps * (eps * eps))) / x) - ((eps * eps) * -10.0)) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5e-52], N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision] + N[(eps * N[(N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision] + N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-61], N[Power[eps, 5.0], $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] + N[(N[(N[(N[(10.0 * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5e-52

   1. Initial program 48.1%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 92.8%

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

   if -5e-52 < x < 1.45e-61

   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 1.45e-61 < x

   1. Initial program 44.3%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + \varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 + \frac{\frac{10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{x} - \left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{4} \cdot \left(\varepsilon \cdot 5 + \frac{\frac{10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{x} - \left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          (pow x 4.0)
          (+
           (* eps 5.0)
           (/
            (- (/ (* 10.0 (* eps (* eps eps))) x) (* (* eps eps) -10.0))
            x)))))
   (if (<= x -4.8e-52) t_0 (if (<= x 1.45e-61) (pow eps 5.0) t_0))))

C

double code(double x, double eps) {
	double t_0 = pow(x, 4.0) * ((eps * 5.0) + ((((10.0 * (eps * (eps * eps))) / x) - ((eps * eps) * -10.0)) / x));
	double tmp;
	if (x <= -4.8e-52) {
		tmp = t_0;
	} else if (x <= 1.45e-61) {
		tmp = pow(eps, 5.0);
	} 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 ** 4.0d0) * ((eps * 5.0d0) + ((((10.0d0 * (eps * (eps * eps))) / x) - ((eps * eps) * (-10.0d0))) / x))
    if (x <= (-4.8d-52)) then
        tmp = t_0
    else if (x <= 1.45d-61) then
        tmp = eps ** 5.0d0
    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, 4.0) * ((eps * 5.0) + ((((10.0 * (eps * (eps * eps))) / x) - ((eps * eps) * -10.0)) / x));
	double tmp;
	if (x <= -4.8e-52) {
		tmp = t_0;
	} else if (x <= 1.45e-61) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow(x, 4.0) * ((eps * 5.0) + ((((10.0 * (eps * (eps * eps))) / x) - ((eps * eps) * -10.0)) / x))
	tmp = 0
	if x <= -4.8e-52:
		tmp = t_0
	elif x <= 1.45e-61:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) + Float64(Float64(Float64(Float64(10.0 * Float64(eps * Float64(eps * eps))) / x) - Float64(Float64(eps * eps) * -10.0)) / x)))
	tmp = 0.0
	if (x <= -4.8e-52)
		tmp = t_0;
	elseif (x <= 1.45e-61)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x ^ 4.0) * ((eps * 5.0) + ((((10.0 * (eps * (eps * eps))) / x) - ((eps * eps) * -10.0)) / x));
	tmp = 0.0;
	if (x <= -4.8e-52)
		tmp = t_0;
	elseif (x <= 1.45e-61)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] + N[(N[(N[(N[(10.0 * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e-52], t$95$0, If[LessEqual[x, 1.45e-61], N[Power[eps, 5.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8000000000000003e-52 or 1.45e-61 < x

   1. Initial program 46.4%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 95.6%

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

   if -4.8000000000000003e-52 < x < 1.45e-61

   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%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 + \frac{\frac{10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{x} - \left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 + \frac{\frac{10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{x} - \left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-52}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.4e-52)
   (* (pow x 4.0) (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)))
   (if (<= x 1.15e-61)
     (pow eps 5.0)
     (* (pow x 4.0) (* eps (+ 5.0 (/ (* eps 10.0) x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.4e-52) {
		tmp = pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} else if (x <= 1.15e-61) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = pow(x, 4.0) * (eps * (5.0 + ((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 <= (-5.4d-52)) then
        tmp = (x ** 4.0d0) * ((eps * 5.0d0) - (((eps * eps) * (-10.0d0)) / x))
    else if (x <= 1.15d-61) then
        tmp = eps ** 5.0d0
    else
        tmp = (x ** 4.0d0) * (eps * (5.0d0 + ((eps * 10.0d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.4e-52) {
		tmp = Math.pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} else if (x <= 1.15e-61) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = Math.pow(x, 4.0) * (eps * (5.0 + ((eps * 10.0) / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.4e-52:
		tmp = math.pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x))
	elif x <= 1.15e-61:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = math.pow(x, 4.0) * (eps * (5.0 + ((eps * 10.0) / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.4e-52)
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)));
	elseif (x <= 1.15e-61)
		tmp = eps ^ 5.0;
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(Float64(eps * 10.0) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.4e-52)
		tmp = (x ^ 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	elseif (x <= 1.15e-61)
		tmp = eps ^ 5.0;
	else
		tmp = (x ^ 4.0) * (eps * (5.0 + ((eps * 10.0) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.4e-52], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e-61], N[Power[eps, 5.0], $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(N[(eps * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.40000000000000019e-52

   1. Initial program 48.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 90.0%

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

   if -5.40000000000000019e-52 < x < 1.14999999999999996e-61

   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 1.14999999999999996e-61 < x

   1. Initial program 44.3%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 99.1%

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

   6. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(5 \cdot \varepsilon + 10 \cdot \frac{{\varepsilon}^{2}}{x}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{5 \cdot \varepsilon} + 10 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(5 \cdot \varepsilon + 10 \cdot \frac{\varepsilon \cdot \varepsilon}{x}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(5 \cdot \varepsilon + 10 \cdot \left(\frac{\varepsilon}{x} \cdot \color{blue}{\varepsilon}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(5 \cdot \varepsilon + \left(10 \cdot \frac{\varepsilon}{x}\right) \cdot \color{blue}{\varepsilon}\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon \cdot \color{blue}{\left(5 + 10 \cdot \frac{\varepsilon}{x}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(5 + 10 \cdot \frac{\varepsilon}{x}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(5, \color{blue}{\left(10 \cdot \frac{\varepsilon}{x}\right)}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(5, \left(\frac{10 \cdot \varepsilon}{\color{blue}{x}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(5, \mathsf{/.f64}\left(\left(10 \cdot \varepsilon\right), \color{blue}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(5, \mathsf{/.f64}\left(\left(\varepsilon \cdot 10\right), x\right)\right)\right)\right) \]

      12. *-lowering-*.f64 99.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(5, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, 10\right), x\right)\right)\right)\right) \]

   8. Simplified 99.1%

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

Alternative 5: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{4} \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (pow x 4.0) (* eps (+ 5.0 (/ (* eps 10.0) x))))))
   (if (<= x -5e-52) t_0 (if (<= x 1.3e-61) (pow eps 5.0) t_0))))

C

double code(double x, double eps) {
	double t_0 = pow(x, 4.0) * (eps * (5.0 + ((eps * 10.0) / x)));
	double tmp;
	if (x <= -5e-52) {
		tmp = t_0;
	} else if (x <= 1.3e-61) {
		tmp = pow(eps, 5.0);
	} 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 ** 4.0d0) * (eps * (5.0d0 + ((eps * 10.0d0) / x)))
    if (x <= (-5d-52)) then
        tmp = t_0
    else if (x <= 1.3d-61) then
        tmp = eps ** 5.0d0
    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, 4.0) * (eps * (5.0 + ((eps * 10.0) / x)));
	double tmp;
	if (x <= -5e-52) {
		tmp = t_0;
	} else if (x <= 1.3e-61) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow(x, 4.0) * (eps * (5.0 + ((eps * 10.0) / x)))
	tmp = 0
	if x <= -5e-52:
		tmp = t_0
	elif x <= 1.3e-61:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(Float64(eps * 10.0) / x))))
	tmp = 0.0
	if (x <= -5e-52)
		tmp = t_0;
	elseif (x <= 1.3e-61)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x ^ 4.0) * (eps * (5.0 + ((eps * 10.0) / x)));
	tmp = 0.0;
	if (x <= -5e-52)
		tmp = t_0;
	elseif (x <= 1.3e-61)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(N[(eps * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-52], t$95$0, If[LessEqual[x, 1.3e-61], N[Power[eps, 5.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5e-52 or 1.30000000000000005e-61 < x

   1. Initial program 46.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 94.0%

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

   6. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(5 \cdot \varepsilon + 10 \cdot \frac{{\varepsilon}^{2}}{x}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{5 \cdot \varepsilon} + 10 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(5 \cdot \varepsilon + 10 \cdot \frac{\varepsilon \cdot \varepsilon}{x}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(5 \cdot \varepsilon + 10 \cdot \left(\frac{\varepsilon}{x} \cdot \color{blue}{\varepsilon}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(5 \cdot \varepsilon + \left(10 \cdot \frac{\varepsilon}{x}\right) \cdot \color{blue}{\varepsilon}\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon \cdot \color{blue}{\left(5 + 10 \cdot \frac{\varepsilon}{x}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(5 + 10 \cdot \frac{\varepsilon}{x}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(5, \color{blue}{\left(10 \cdot \frac{\varepsilon}{x}\right)}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(5, \left(\frac{10 \cdot \varepsilon}{\color{blue}{x}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(5, \mathsf{/.f64}\left(\left(10 \cdot \varepsilon\right), \color{blue}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(5, \mathsf{/.f64}\left(\left(\varepsilon \cdot 10\right), x\right)\right)\right)\right) \]

      12. *-lowering-*.f64 93.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(5, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, 10\right), x\right)\right)\right)\right) \]

   8. Simplified 93.9%

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

   if -5e-52 < x < 1.30000000000000005e-61

   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%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -4.9 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x -4.9e-52)
     (* (+ (* eps 5.0) (/ (* (* eps eps) 10.0) x)) t_0)
     (if (<= x 1.45e-61)
       (pow eps 5.0)
       (* t_0 (* eps (+ 5.0 (/ (* eps 10.0) x))))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -4.9e-52) {
		tmp = ((eps * 5.0) + (((eps * eps) * 10.0) / x)) * t_0;
	} else if (x <= 1.45e-61) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = t_0 * (eps * (5.0 + ((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) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    if (x <= (-4.9d-52)) then
        tmp = ((eps * 5.0d0) + (((eps * eps) * 10.0d0) / x)) * t_0
    else if (x <= 1.45d-61) then
        tmp = eps ** 5.0d0
    else
        tmp = t_0 * (eps * (5.0d0 + ((eps * 10.0d0) / x)))
    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 <= -4.9e-52) {
		tmp = ((eps * 5.0) + (((eps * eps) * 10.0) / x)) * t_0;
	} else if (x <= 1.45e-61) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = t_0 * (eps * (5.0 + ((eps * 10.0) / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * (x * x))
	tmp = 0
	if x <= -4.9e-52:
		tmp = ((eps * 5.0) + (((eps * eps) * 10.0) / x)) * t_0
	elif x <= 1.45e-61:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = t_0 * (eps * (5.0 + ((eps * 10.0) / x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= -4.9e-52)
		tmp = Float64(Float64(Float64(eps * 5.0) + Float64(Float64(Float64(eps * eps) * 10.0) / x)) * t_0);
	elseif (x <= 1.45e-61)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(t_0 * Float64(eps * Float64(5.0 + Float64(Float64(eps * 10.0) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * (x * x));
	tmp = 0.0;
	if (x <= -4.9e-52)
		tmp = ((eps * 5.0) + (((eps * eps) * 10.0) / x)) * t_0;
	elseif (x <= 1.45e-61)
		tmp = eps ^ 5.0;
	else
		tmp = t_0 * (eps * (5.0 + ((eps * 10.0) / x)));
	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, -4.9e-52], N[(N[(N[(eps * 5.0), $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 1.45e-61], N[Power[eps, 5.0], $MachinePrecision], N[(t$95$0 * N[(eps * N[(5.0 + N[(N[(eps * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.90000000000000019e-52

   1. Initial program 48.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 90.0%

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right), \color{blue}{\left({x}^{4}\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 + \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\varepsilon \cdot 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{neg}\left(-10\right)\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. *-lowering-*.f64 89.8%

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

   7. Applied egg-rr 89.8%

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

   if -4.90000000000000019e-52 < x < 1.45e-61

   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 1.45e-61 < x

   1. Initial program 44.3%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 99.1%

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right), \color{blue}{\left({x}^{4}\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 + \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\varepsilon \cdot 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{neg}\left(-10\right)\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. *-lowering-*.f64 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in eps around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-lowering-*.f64 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-62}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x -5.5e-52)
     (* (+ (* eps 5.0) (/ (* (* eps eps) 10.0) x)) t_0)
     (if (<= x 8.8e-62)
       (* eps (* eps (* eps (* eps eps))))
       (* t_0 (* eps (+ 5.0 (/ (* eps 10.0) x))))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -5.5e-52) {
		tmp = ((eps * 5.0) + (((eps * eps) * 10.0) / x)) * t_0;
	} else if (x <= 8.8e-62) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (eps * (5.0 + ((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) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    if (x <= (-5.5d-52)) then
        tmp = ((eps * 5.0d0) + (((eps * eps) * 10.0d0) / x)) * t_0
    else if (x <= 8.8d-62) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0 * (eps * (5.0d0 + ((eps * 10.0d0) / x)))
    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 <= -5.5e-52) {
		tmp = ((eps * 5.0) + (((eps * eps) * 10.0) / x)) * t_0;
	} else if (x <= 8.8e-62) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (eps * (5.0 + ((eps * 10.0) / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * (x * x))
	tmp = 0
	if x <= -5.5e-52:
		tmp = ((eps * 5.0) + (((eps * eps) * 10.0) / x)) * t_0
	elif x <= 8.8e-62:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0 * (eps * (5.0 + ((eps * 10.0) / x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= -5.5e-52)
		tmp = Float64(Float64(Float64(eps * 5.0) + Float64(Float64(Float64(eps * eps) * 10.0) / x)) * t_0);
	elseif (x <= 8.8e-62)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(t_0 * Float64(eps * Float64(5.0 + Float64(Float64(eps * 10.0) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * (x * x));
	tmp = 0.0;
	if (x <= -5.5e-52)
		tmp = ((eps * 5.0) + (((eps * eps) * 10.0) / x)) * t_0;
	elseif (x <= 8.8e-62)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0 * (eps * (5.0 + ((eps * 10.0) / x)));
	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, -5.5e-52], N[(N[(N[(eps * 5.0), $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 8.8e-62], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(eps * N[(5.0 + N[(N[(eps * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5e-52

   1. Initial program 48.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 90.0%

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right), \color{blue}{\left({x}^{4}\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 + \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\varepsilon \cdot 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{neg}\left(-10\right)\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. *-lowering-*.f64 89.8%

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

   7. Applied egg-rr 89.8%

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

   if -5.5e-52 < x < 8.80000000000000069e-62

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 91.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   10. 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. metadata-eval N/A

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

       4. pow-plus N/A

          \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

       5. associate-*r* N/A

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

       6. unpow2 N/A

          \[\leadsto {\varepsilon}^{3} \cdot {\varepsilon}^{\color{blue}{2}} \]

       7. *-commutative N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. pow-plus N/A

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

       12. metadata-eval N/A

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

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

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

       14. metadata-eval N/A

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

       15. pow-plus N/A

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

       16. *-commutative N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 99.9%

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

   11. Simplified 99.9%

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

   if 8.80000000000000069e-62 < x

   1. Initial program 44.3%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 99.1%

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right), \color{blue}{\left({x}^{4}\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 + \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\varepsilon \cdot 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{neg}\left(-10\right)\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. *-lowering-*.f64 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in eps around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-lowering-*.f64 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-62}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-62}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x (* x (* x x))) (* eps (+ 5.0 (/ (* eps 10.0) x))))))
   (if (<= x -5.5e-52)
     t_0
     (if (<= x 9e-62) (* eps (* eps (* eps (* eps eps)))) t_0))))

C

double code(double x, double eps) {
	double t_0 = (x * (x * (x * x))) * (eps * (5.0 + ((eps * 10.0) / x)));
	double tmp;
	if (x <= -5.5e-52) {
		tmp = t_0;
	} else if (x <= 9e-62) {
		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 = (x * (x * (x * x))) * (eps * (5.0d0 + ((eps * 10.0d0) / x)))
    if (x <= (-5.5d-52)) then
        tmp = t_0
    else if (x <= 9d-62) 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 = (x * (x * (x * x))) * (eps * (5.0 + ((eps * 10.0) / x)));
	double tmp;
	if (x <= -5.5e-52) {
		tmp = t_0;
	} else if (x <= 9e-62) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (x * (x * (x * x))) * (eps * (5.0 + ((eps * 10.0) / x)))
	tmp = 0
	if x <= -5.5e-52:
		tmp = t_0
	elif x <= 9e-62:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * Float64(x * Float64(x * x))) * Float64(eps * Float64(5.0 + Float64(Float64(eps * 10.0) / x))))
	tmp = 0.0
	if (x <= -5.5e-52)
		tmp = t_0;
	elseif (x <= 9e-62)
		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 = (x * (x * (x * x))) * (eps * (5.0 + ((eps * 10.0) / x)));
	tmp = 0.0;
	if (x <= -5.5e-52)
		tmp = t_0;
	elseif (x <= 9e-62)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(5.0 + N[(N[(eps * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e-52], t$95$0, If[LessEqual[x, 9e-62], 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 < -5.5e-52 or 9.00000000000000036e-62 < x

   1. Initial program 46.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 94.0%

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right), \color{blue}{\left({x}^{4}\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 + \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\varepsilon \cdot 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{neg}\left(-10\right)\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. *-lowering-*.f64 93.8%

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

   7. Applied egg-rr 93.8%

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

   8. Taylor expanded in eps around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-lowering-*.f64 93.8%

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

   10. Simplified 93.8%

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

   if -5.5e-52 < x < 9.00000000000000036e-62

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 91.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   10. 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. metadata-eval N/A

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

       4. pow-plus N/A

          \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

       5. associate-*r* N/A

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

       6. unpow2 N/A

          \[\leadsto {\varepsilon}^{3} \cdot {\varepsilon}^{\color{blue}{2}} \]

       7. *-commutative N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. pow-plus N/A

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

       12. metadata-eval N/A

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

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

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

       14. metadata-eval N/A

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

       15. pow-plus N/A

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

       16. *-commutative N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-52}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-62}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \varepsilon \cdot 10 + x \cdot 5\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(t\_0 \cdot t\_1\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-61}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (+ (* eps 10.0) (* x 5.0))))
   (if (<= x -4.8e-52)
     (* eps (* t_0 t_1))
     (if (<= x 1.45e-61)
       (* eps (* eps (* eps (* eps eps))))
       (* t_0 (* eps t_1))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double t_1 = (eps * 10.0) + (x * 5.0);
	double tmp;
	if (x <= -4.8e-52) {
		tmp = eps * (t_0 * t_1);
	} else if (x <= 1.45e-61) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (eps * t_1);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = (eps * 10.0d0) + (x * 5.0d0)
    if (x <= (-4.8d-52)) then
        tmp = eps * (t_0 * t_1)
    else if (x <= 1.45d-61) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0 * (eps * t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double t_1 = (eps * 10.0) + (x * 5.0);
	double tmp;
	if (x <= -4.8e-52) {
		tmp = eps * (t_0 * t_1);
	} else if (x <= 1.45e-61) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (eps * t_1);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	t_1 = (eps * 10.0) + (x * 5.0)
	tmp = 0
	if x <= -4.8e-52:
		tmp = eps * (t_0 * t_1)
	elif x <= 1.45e-61:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0 * (eps * t_1)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(eps * 10.0) + Float64(x * 5.0))
	tmp = 0.0
	if (x <= -4.8e-52)
		tmp = Float64(eps * Float64(t_0 * t_1));
	elseif (x <= 1.45e-61)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(t_0 * Float64(eps * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	t_1 = (eps * 10.0) + (x * 5.0);
	tmp = 0.0;
	if (x <= -4.8e-52)
		tmp = eps * (t_0 * t_1);
	elseif (x <= 1.45e-61)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0 * (eps * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * 10.0), $MachinePrecision] + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e-52], N[(eps * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-61], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8000000000000003e-52

   1. Initial program 48.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 90.0%

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right), \color{blue}{\left({x}^{4}\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 + \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\varepsilon \cdot 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{neg}\left(-10\right)\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. *-lowering-*.f64 89.8%

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

   7. Applied egg-rr 89.8%

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

   8. Taylor expanded in eps around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(5 \cdot \frac{{x}^{4}}{\varepsilon} + 10 \cdot {x}^{3}\right)\right)}\right) \]

   10. Simplified 89.5%

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

   if -4.8000000000000003e-52 < x < 1.45e-61

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 91.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   10. 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. metadata-eval N/A

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

       4. pow-plus N/A

          \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

       5. associate-*r* N/A

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

       6. unpow2 N/A

          \[\leadsto {\varepsilon}^{3} \cdot {\varepsilon}^{\color{blue}{2}} \]

       7. *-commutative N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. pow-plus N/A

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

       12. metadata-eval N/A

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

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

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

       14. metadata-eval N/A

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

       15. pow-plus N/A

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

       16. *-commutative N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 99.9%

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

   11. Simplified 99.9%

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

   if 1.45e-61 < x

   1. Initial program 44.3%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   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 99.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
   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. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. distribute-rgt-out N/A

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

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

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

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

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

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

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 98.8%

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

   8. Simplified 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-61}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.9% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-61}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* (* x (* x x)) (+ (* eps 10.0) (* x 5.0))))))
   (if (<= x -4.8e-52)
     t_0
     (if (<= x 1.05e-61) (* eps (* eps (* eps (* eps eps)))) t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * ((x * (x * x)) * ((eps * 10.0) + (x * 5.0)));
	double tmp;
	if (x <= -4.8e-52) {
		tmp = t_0;
	} else if (x <= 1.05e-61) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * ((x * (x * x)) * ((eps * 10.0d0) + (x * 5.0d0)))
    if (x <= (-4.8d-52)) then
        tmp = t_0
    else if (x <= 1.05d-61) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * ((x * (x * x)) * ((eps * 10.0) + (x * 5.0)));
	double tmp;
	if (x <= -4.8e-52) {
		tmp = t_0;
	} else if (x <= 1.05e-61) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * ((x * (x * x)) * ((eps * 10.0) + (x * 5.0)))
	tmp = 0
	if x <= -4.8e-52:
		tmp = t_0
	elif x <= 1.05e-61:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(Float64(x * Float64(x * x)) * Float64(Float64(eps * 10.0) + Float64(x * 5.0))))
	tmp = 0.0
	if (x <= -4.8e-52)
		tmp = t_0;
	elseif (x <= 1.05e-61)
		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 = eps * ((x * (x * x)) * ((eps * 10.0) + (x * 5.0)));
	tmp = 0.0;
	if (x <= -4.8e-52)
		tmp = t_0;
	elseif (x <= 1.05e-61)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * 10.0), $MachinePrecision] + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e-52], t$95$0, If[LessEqual[x, 1.05e-61], 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 < -4.8000000000000003e-52 or 1.05e-61 < x

   1. Initial program 46.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 94.0%

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right), \color{blue}{\left({x}^{4}\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot 5 + \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\varepsilon \cdot 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{neg}\left(-10\right)\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), 10\right), x\right)\right), \left({x}^{4}\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. *-lowering-*.f64 93.8%

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

   7. Applied egg-rr 93.8%

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

   8. Taylor expanded in eps around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. associate-/l* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(5 \cdot \frac{{x}^{4}}{\varepsilon} + 10 \cdot {x}^{3}\right)\right)}\right) \]

   10. Simplified 93.5%

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

   if -4.8000000000000003e-52 < x < 1.05e-61

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 91.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   10. 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. metadata-eval N/A

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

       4. pow-plus N/A

          \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

       5. associate-*r* N/A

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

       6. unpow2 N/A

          \[\leadsto {\varepsilon}^{3} \cdot {\varepsilon}^{\color{blue}{2}} \]

       7. *-commutative N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. pow-plus N/A

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

       12. metadata-eval N/A

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

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

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

       14. metadata-eval N/A

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

       15. pow-plus N/A

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

       16. *-commutative N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-61}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(x \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-61}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.8e-52)
   (* eps (* x (* x (* 5.0 (* x x)))))
   (if (<= x 1.3e-61)
     (* eps (* eps (* eps (* eps eps))))
     (* (* eps 5.0) (* x (* x (* x x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.8e-52) {
		tmp = eps * (x * (x * (5.0 * (x * x))));
	} else if (x <= 1.3e-61) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = (eps * 5.0) * (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 <= (-4.8d-52)) then
        tmp = eps * (x * (x * (5.0d0 * (x * x))))
    else if (x <= 1.3d-61) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = (eps * 5.0d0) * (x * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -4.8e-52) {
		tmp = eps * (x * (x * (5.0 * (x * x))));
	} else if (x <= 1.3e-61) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = (eps * 5.0) * (x * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4.8e-52:
		tmp = eps * (x * (x * (5.0 * (x * x))))
	elif x <= 1.3e-61:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = (eps * 5.0) * (x * (x * (x * x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.8e-52)
		tmp = Float64(eps * Float64(x * Float64(x * Float64(5.0 * Float64(x * x)))));
	elseif (x <= 1.3e-61)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(Float64(eps * 5.0) * Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4.8e-52)
		tmp = eps * (x * (x * (5.0 * (x * x))));
	elseif (x <= 1.3e-61)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = (eps * 5.0) * (x * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.8e-52], N[(eps * N[(x * N[(x * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e-61], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * 5.0), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8000000000000003e-52

   1. Initial program 48.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   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 86.8%

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

   5. Simplified 86.8%

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

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

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

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

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

      7. *-lowering-*.f64 86.6%

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

   7. Applied egg-rr 86.6%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 86.8%

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

   9. Applied egg-rr 86.8%

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

   if -4.8000000000000003e-52 < x < 1.30000000000000005e-61

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 91.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   10. 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. metadata-eval N/A

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

       4. pow-plus N/A

          \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

       5. associate-*r* N/A

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

       6. unpow2 N/A

          \[\leadsto {\varepsilon}^{3} \cdot {\varepsilon}^{\color{blue}{2}} \]

       7. *-commutative N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. pow-plus N/A

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

       12. metadata-eval N/A

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

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

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

       14. metadata-eval N/A

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

       15. pow-plus N/A

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

       16. *-commutative N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 99.9%

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

   11. Simplified 99.9%

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

   if 1.30000000000000005e-61 < x

   1. Initial program 44.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 97.4%

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

   5. Simplified 97.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 97.4%

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

   7. Applied egg-rr 97.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(x \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-61}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(x \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-61}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.2e-52)
   (* eps (* x (* x (* 5.0 (* x x)))))
   (if (<= x 1.45e-61)
     (* eps (* eps (* eps (* eps eps))))
     (* (* x x) (* eps (* x (* x 5.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-52) {
		tmp = eps * (x * (x * (5.0 * (x * x))));
	} else if (x <= 1.45e-61) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = (x * x) * (eps * (x * (x * 5.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.2d-52)) then
        tmp = eps * (x * (x * (5.0d0 * (x * x))))
    else if (x <= 1.45d-61) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = (x * x) * (eps * (x * (x * 5.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-52) {
		tmp = eps * (x * (x * (5.0 * (x * x))));
	} else if (x <= 1.45e-61) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = (x * x) * (eps * (x * (x * 5.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.2e-52:
		tmp = eps * (x * (x * (5.0 * (x * x))))
	elif x <= 1.45e-61:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = (x * x) * (eps * (x * (x * 5.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.2e-52)
		tmp = Float64(eps * Float64(x * Float64(x * Float64(5.0 * Float64(x * x)))));
	elseif (x <= 1.45e-61)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(Float64(x * x) * Float64(eps * Float64(x * Float64(x * 5.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.2e-52)
		tmp = eps * (x * (x * (5.0 * (x * x))));
	elseif (x <= 1.45e-61)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = (x * x) * (eps * (x * (x * 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.2e-52], N[(eps * N[(x * N[(x * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-61], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(eps * N[(x * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1999999999999997e-52

   1. Initial program 48.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   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 86.8%

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

   5. Simplified 86.8%

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

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

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

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

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

      7. *-lowering-*.f64 86.6%

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

   7. Applied egg-rr 86.6%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 86.8%

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

   9. Applied egg-rr 86.8%

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

   if -5.1999999999999997e-52 < x < 1.45e-61

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 91.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   10. 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. metadata-eval N/A

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

       4. pow-plus N/A

          \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

       5. associate-*r* N/A

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

       6. unpow2 N/A

          \[\leadsto {\varepsilon}^{3} \cdot {\varepsilon}^{\color{blue}{2}} \]

       7. *-commutative N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. pow-plus N/A

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

       12. metadata-eval N/A

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

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

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

       14. metadata-eval N/A

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

       15. pow-plus N/A

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

       16. *-commutative N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 99.9%

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

   11. Simplified 99.9%

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

   if 1.45e-61 < x

   1. Initial program 44.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 97.4%

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

   5. Simplified 97.4%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 97.2%

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

   7. Applied egg-rr 97.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 97.2%

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

   10. Simplified 97.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(x \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-61}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(x \cdot \left(x \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-62}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* x (* x (* 5.0 (* x x)))))))
   (if (<= x -4.8e-52)
     t_0
     (if (<= x 2.2e-62) (* eps (* eps (* eps (* eps eps)))) t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (x * (x * (5.0 * (x * x))));
	double tmp;
	if (x <= -4.8e-52) {
		tmp = t_0;
	} else if (x <= 2.2e-62) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * (x * (x * (5.0d0 * (x * x))))
    if (x <= (-4.8d-52)) then
        tmp = t_0
    else if (x <= 2.2d-62) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (x * (x * (5.0 * (x * x))));
	double tmp;
	if (x <= -4.8e-52) {
		tmp = t_0;
	} else if (x <= 2.2e-62) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (x * (x * (5.0 * (x * x))))
	tmp = 0
	if x <= -4.8e-52:
		tmp = t_0
	elif x <= 2.2e-62:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(x * Float64(x * Float64(5.0 * Float64(x * x)))))
	tmp = 0.0
	if (x <= -4.8e-52)
		tmp = t_0;
	elseif (x <= 2.2e-62)
		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 = eps * (x * (x * (5.0 * (x * x))));
	tmp = 0.0;
	if (x <= -4.8e-52)
		tmp = t_0;
	elseif (x <= 2.2e-62)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(x * N[(x * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e-52], t$95$0, If[LessEqual[x, 2.2e-62], 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 < -4.8000000000000003e-52 or 2.20000000000000017e-62 < x

   1. Initial program 46.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 91.4%

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

   5. Simplified 91.4%

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

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

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

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

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

      7. *-lowering-*.f64 91.2%

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

   7. Applied egg-rr 91.2%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 91.4%

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

   9. Applied egg-rr 91.4%

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

   if -4.8000000000000003e-52 < x < 2.20000000000000017e-62

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 91.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   10. 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. metadata-eval N/A

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

       4. pow-plus N/A

          \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

       5. associate-*r* N/A

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

       6. unpow2 N/A

          \[\leadsto {\varepsilon}^{3} \cdot {\varepsilon}^{\color{blue}{2}} \]

       7. *-commutative N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. pow-plus N/A

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

       12. metadata-eval N/A

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

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

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

       14. metadata-eval N/A

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

       15. pow-plus N/A

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

       16. *-commutative N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(x \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-62}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(x \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-62}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* 5.0 (* (* x x) (* x x))))))
   (if (<= x -4.8e-52)
     t_0
     (if (<= x 2.2e-62) (* eps (* eps (* eps (* eps eps)))) t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (5.0 * ((x * x) * (x * x)));
	double tmp;
	if (x <= -4.8e-52) {
		tmp = t_0;
	} else if (x <= 2.2e-62) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * (5.0d0 * ((x * x) * (x * x)))
    if (x <= (-4.8d-52)) then
        tmp = t_0
    else if (x <= 2.2d-62) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (5.0 * ((x * x) * (x * x)));
	double tmp;
	if (x <= -4.8e-52) {
		tmp = t_0;
	} else if (x <= 2.2e-62) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (5.0 * ((x * x) * (x * x)))
	tmp = 0
	if x <= -4.8e-52:
		tmp = t_0
	elif x <= 2.2e-62:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))))
	tmp = 0.0
	if (x <= -4.8e-52)
		tmp = t_0;
	elseif (x <= 2.2e-62)
		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 = eps * (5.0 * ((x * x) * (x * x)));
	tmp = 0.0;
	if (x <= -4.8e-52)
		tmp = t_0;
	elseif (x <= 2.2e-62)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e-52], t$95$0, If[LessEqual[x, 2.2e-62], 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 < -4.8000000000000003e-52 or 2.20000000000000017e-62 < x

   1. Initial program 46.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 91.4%

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

   5. Simplified 91.4%

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

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

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

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

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

      7. *-lowering-*.f64 91.2%

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

   7. Applied egg-rr 91.2%

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

   if -4.8000000000000003e-52 < x < 2.20000000000000017e-62

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 91.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left({x}^{2} \cdot 10\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(\left(x \cdot x\right) \cdot 10\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(x \cdot \left(x \cdot 10\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot 10\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 10\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 10\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 10\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(5 \cdot x\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 10\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right)\right)\right) \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot \left(x \cdot 10\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   10. 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. metadata-eval N/A

          \[\leadsto {\varepsilon}^{\left(3 + 1\right)} \cdot \varepsilon \]

       4. pow-plus N/A

          \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

       5. associate-*r* N/A

          \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \]

       6. unpow2 N/A

          \[\leadsto {\varepsilon}^{3} \cdot {\varepsilon}^{\color{blue}{2}} \]

       7. *-commutative N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

       8. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3} \]

       9. associate-*l* N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{3}\right)} \]

       10. *-commutative N/A

           \[\leadsto \varepsilon \cdot \left({\varepsilon}^{3} \cdot \color{blue}{\varepsilon}\right) \]

       11. pow-plus N/A

           \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(3 + 1\right)}} \]

       12. metadata-eval N/A

           \[\leadsto \varepsilon \cdot {\varepsilon}^{4} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{4}\right)}\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       15. pow-plus N/A

           \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{3} \cdot \color{blue}{\varepsilon}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{3}}\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{3}\right)}\right)\right) \]

       18. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot {\varepsilon}^{\color{blue}{2}}\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{2}\right)}\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)\right)\right) \]

       22. *-lowering-*.f64 99.9%

           \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right)\right)\right)\right) \]

   11. Simplified 99.9%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 86.7% 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 90.0%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in eps around -inf

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot {\varepsilon}^{5}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right)} \]

   3. sub-neg N/A

      \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

   5. metadata-eval N/A

      \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + -1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

   6. distribute-lft1-in N/A

      \[\leadsto \left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

   7. associate-*l* N/A

      \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

5. Simplified 77.5%

   \[\leadsto \color{blue}{\left(\frac{5 \cdot x - \frac{\left(x \cdot x\right) \cdot -10}{\varepsilon}}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

   2. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left({x}^{2} \cdot 10\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(\left(x \cdot x\right) \cdot 10\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(x \cdot \left(x \cdot 10\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot 10\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 10\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 10\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 10\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(5 \cdot x\right)}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 88.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 10\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right)\right)\right) \]

8. Simplified 88.7%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot \left(x \cdot 10\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]

9. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
10. 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. metadata-eval N/A

       \[\leadsto {\varepsilon}^{\left(3 + 1\right)} \cdot \varepsilon \]

    4. pow-plus N/A

       \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

    5. associate-*r* N/A

       \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \]

    6. unpow2 N/A

       \[\leadsto {\varepsilon}^{3} \cdot {\varepsilon}^{\color{blue}{2}} \]

    7. *-commutative N/A

       \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

    8. unpow2 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3} \]

    9. associate-*l* N/A

       \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{3}\right)} \]

    10. *-commutative N/A

        \[\leadsto \varepsilon \cdot \left({\varepsilon}^{3} \cdot \color{blue}{\varepsilon}\right) \]

    11. pow-plus N/A

        \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(3 + 1\right)}} \]

    12. metadata-eval N/A

        \[\leadsto \varepsilon \cdot {\varepsilon}^{4} \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{4}\right)}\right) \]

    14. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

    15. pow-plus N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{3} \cdot \color{blue}{\varepsilon}\right)\right) \]

    16. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{3}}\right)\right) \]

    17. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{3}\right)}\right)\right) \]

    18. cube-mult N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right)\right) \]

    19. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot {\varepsilon}^{\color{blue}{2}}\right)\right)\right) \]

    20. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{2}\right)}\right)\right)\right) \]

    21. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)\right)\right) \]

    22. *-lowering-*.f64 88.6%

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right)\right)\right)\right) \]

11. Simplified 88.6%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024156 
(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)))