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

Percentage Accurate: 88.5% → 99.5%

Time: 26.2s

Alternatives: 12

Speedup: 9.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-323}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-323)
     t_0
     (if (<= t_0 0.0) (* eps (* 5.0 (pow x 4.0))) t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-1d-323)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -1e-323:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-323)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -1e-323)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-323], t$95$0, If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.88131e-324 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.0%

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

   if -9.88131e-324 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. pow-lowering-pow.f64 99.9%

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

   5. Simplified 99.9%

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

Alternative 2: 98.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(t\_0 \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(x \cdot \varepsilon\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot t\_0\right) + \varepsilon \cdot \left(\left(x + \varepsilon\right) \cdot \left(x \cdot \left(x \cdot 10\right)\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -5.9e-39)
     (*
      eps
      (+
       (* 5.0 (pow x 4.0))
       (*
        eps
        (+ (* t_0 10.0) (* eps (+ (* (* x x) 10.0) (* 5.0 (* x eps))))))))
     (if (<= x 1.32e-52)
       (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
       (*
        eps
        (+
         (* 5.0 (* x t_0))
         (*
          eps
          (+ (* (+ x eps) (* x (* x 10.0))) (* eps (* eps (* x 5.0)))))))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -5.9e-39) {
		tmp = eps * ((5.0 * pow(x, 4.0)) + (eps * ((t_0 * 10.0) + (eps * (((x * x) * 10.0) + (5.0 * (x * eps)))))));
	} else if (x <= 1.32e-52) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * ((5.0 * (x * t_0)) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= (-5.9d-39)) then
        tmp = eps * ((5.0d0 * (x ** 4.0d0)) + (eps * ((t_0 * 10.0d0) + (eps * (((x * x) * 10.0d0) + (5.0d0 * (x * eps)))))))
    else if (x <= 1.32d-52) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = eps * ((5.0d0 * (x * t_0)) + (eps * (((x + eps) * (x * (x * 10.0d0))) + (eps * (eps * (x * 5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -5.9e-39) {
		tmp = eps * ((5.0 * Math.pow(x, 4.0)) + (eps * ((t_0 * 10.0) + (eps * (((x * x) * 10.0) + (5.0 * (x * eps)))))));
	} else if (x <= 1.32e-52) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * ((5.0 * (x * t_0)) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	tmp = 0
	if x <= -5.9e-39:
		tmp = eps * ((5.0 * math.pow(x, 4.0)) + (eps * ((t_0 * 10.0) + (eps * (((x * x) * 10.0) + (5.0 * (x * eps)))))))
	elif x <= 1.32e-52:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = eps * ((5.0 * (x * t_0)) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -5.9e-39)
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(eps * Float64(Float64(t_0 * 10.0) + Float64(eps * Float64(Float64(Float64(x * x) * 10.0) + Float64(5.0 * Float64(x * eps))))))));
	elseif (x <= 1.32e-52)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64(eps * Float64(Float64(5.0 * Float64(x * t_0)) + Float64(eps * Float64(Float64(Float64(x + eps) * Float64(x * Float64(x * 10.0))) + Float64(eps * Float64(eps * Float64(x * 5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= -5.9e-39)
		tmp = eps * ((5.0 * (x ^ 4.0)) + (eps * ((t_0 * 10.0) + (eps * (((x * x) * 10.0) + (5.0 * (x * eps)))))));
	elseif (x <= 1.32e-52)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = eps * ((5.0 * (x * t_0)) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.9e-39], N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(t$95$0 * 10.0), $MachinePrecision] + N[(eps * N[(N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision] + N[(5.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e-52], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(5.0 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(N[(x + eps), $MachinePrecision] * N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8999999999999998e-39

   1. Initial program 37.0%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 96.5%

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

   if -5.8999999999999998e-39 < x < 1.32000000000000002e-52

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf

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

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

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

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

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

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

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. /-lowering-/.f64 99.5%

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

   5. Simplified 99.5%

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

   if 1.32000000000000002e-52 < x

   1. Initial program 39.9%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 95.0%

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

      1. *-commutative N/A

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

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

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

   7. Applied egg-rr 95.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(x \cdot \varepsilon\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \varepsilon \cdot \left(\left(x + \varepsilon\right) \cdot \left(x \cdot \left(x \cdot 10\right)\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \varepsilon \cdot \left(\left(x + \varepsilon\right) \cdot \left(x \cdot \left(x \cdot 10\right)\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.9e-39)
   (* (* x x) (* x (* eps (+ (* x 5.0) (* eps 10.0)))))
   (if (<= x 1.35e-52)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (*
      eps
      (+
       (* 5.0 (* x (* x (* x x))))
       (*
        eps
        (+ (* (+ x eps) (* x (* x 10.0))) (* eps (* eps (* x 5.0))))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.9e-39) {
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))));
	} else if (x <= 1.35e-52) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.9d-39)) then
        tmp = (x * x) * (x * (eps * ((x * 5.0d0) + (eps * 10.0d0))))
    else if (x <= 1.35d-52) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = eps * ((5.0d0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0d0))) + (eps * (eps * (x * 5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.9e-39) {
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))));
	} else if (x <= 1.35e-52) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.9e-39:
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))))
	elif x <= 1.35e-52:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.9e-39)
		tmp = Float64(Float64(x * x) * Float64(x * Float64(eps * Float64(Float64(x * 5.0) + Float64(eps * 10.0)))));
	elseif (x <= 1.35e-52)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64(eps * Float64(Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))) + Float64(eps * Float64(Float64(Float64(x + eps) * Float64(x * Float64(x * 10.0))) + Float64(eps * Float64(eps * Float64(x * 5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.9e-39)
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))));
	elseif (x <= 1.35e-52)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.9e-39], N[(N[(x * x), $MachinePrecision] * N[(x * N[(eps * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e-52], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(N[(x + eps), $MachinePrecision] * N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8999999999999998e-39

   1. Initial program 37.0%

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

   3. Taylor expanded in x around -inf

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

      1. *-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 96.4%

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

   6. Taylor expanded in x around 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. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 96.2%

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

   8. Simplified 96.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 96.4%

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

   10. Applied egg-rr 96.4%

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

   if -5.8999999999999998e-39 < x < 1.35000000000000005e-52

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf

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

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

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

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

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

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

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. /-lowering-/.f64 99.5%

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

   5. Simplified 99.5%

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

   if 1.35000000000000005e-52 < x

   1. Initial program 39.9%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 95.0%

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

      1. *-commutative N/A

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

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

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

   7. Applied egg-rr 95.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \varepsilon \cdot \left(\left(x + \varepsilon\right) \cdot \left(x \cdot \left(x \cdot 10\right)\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \varepsilon \cdot \left(\left(x + \varepsilon\right) \cdot \left(x \cdot \left(x \cdot 10\right)\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.6e-39)
   (* (* x x) (* x (* eps (+ (* x 5.0) (* eps 10.0)))))
   (if (<= x 3e-52)
     (pow eps 5.0)
     (*
      eps
      (+
       (* 5.0 (* x (* x (* x x))))
       (*
        eps
        (+ (* (+ x eps) (* x (* x 10.0))) (* eps (* eps (* x 5.0))))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.6e-39) {
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))));
	} else if (x <= 3e-52) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6.6d-39)) then
        tmp = (x * x) * (x * (eps * ((x * 5.0d0) + (eps * 10.0d0))))
    else if (x <= 3d-52) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * ((5.0d0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0d0))) + (eps * (eps * (x * 5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.6e-39) {
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))));
	} else if (x <= 3e-52) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.6e-39:
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))))
	elif x <= 3e-52:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.6e-39)
		tmp = Float64(Float64(x * x) * Float64(x * Float64(eps * Float64(Float64(x * 5.0) + Float64(eps * 10.0)))));
	elseif (x <= 3e-52)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))) + Float64(eps * Float64(Float64(Float64(x + eps) * Float64(x * Float64(x * 10.0))) + Float64(eps * Float64(eps * Float64(x * 5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.6e-39)
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))));
	elseif (x <= 3e-52)
		tmp = eps ^ 5.0;
	else
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.6e-39], N[(N[(x * x), $MachinePrecision] * N[(x * N[(eps * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e-52], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(N[(x + eps), $MachinePrecision] * N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5999999999999997e-39

   1. Initial program 37.0%

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

   3. Taylor expanded in x around -inf

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

      1. *-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 96.4%

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

   6. Taylor expanded in x around 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. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 96.2%

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

   8. Simplified 96.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 96.4%

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

   10. Applied egg-rr 96.4%

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

   if -6.5999999999999997e-39 < x < 3e-52

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. pow-lowering-pow.f64 99.5%

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

   5. Simplified 99.5%

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

   if 3e-52 < x

   1. Initial program 39.9%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 95.0%

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

      1. *-commutative N/A

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

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

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

   7. Applied egg-rr 95.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \varepsilon \cdot \left(\left(x + \varepsilon\right) \cdot \left(x \cdot \left(x \cdot 10\right)\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \varepsilon \cdot \left(\left(x + \varepsilon\right) \cdot \left(x \cdot \left(x \cdot 10\right)\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.9e-39)
   (* (* x x) (* x (* eps (+ (* x 5.0) (* eps 10.0)))))
   (if (<= x 2.5e-52)
     (* (* eps (* eps (* eps eps))) (* x (+ 5.0 (/ eps x))))
     (*
      eps
      (+
       (* 5.0 (* x (* x (* x x))))
       (*
        eps
        (+ (* (+ x eps) (* x (* x 10.0))) (* eps (* eps (* x 5.0))))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.9e-39) {
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))));
	} else if (x <= 2.5e-52) {
		tmp = (eps * (eps * (eps * eps))) * (x * (5.0 + (eps / x)));
	} else {
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.9d-39)) then
        tmp = (x * x) * (x * (eps * ((x * 5.0d0) + (eps * 10.0d0))))
    else if (x <= 2.5d-52) then
        tmp = (eps * (eps * (eps * eps))) * (x * (5.0d0 + (eps / x)))
    else
        tmp = eps * ((5.0d0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0d0))) + (eps * (eps * (x * 5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.9e-39) {
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))));
	} else if (x <= 2.5e-52) {
		tmp = (eps * (eps * (eps * eps))) * (x * (5.0 + (eps / x)));
	} else {
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.9e-39:
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))))
	elif x <= 2.5e-52:
		tmp = (eps * (eps * (eps * eps))) * (x * (5.0 + (eps / x)))
	else:
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.9e-39)
		tmp = Float64(Float64(x * x) * Float64(x * Float64(eps * Float64(Float64(x * 5.0) + Float64(eps * 10.0)))));
	elseif (x <= 2.5e-52)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) * Float64(x * Float64(5.0 + Float64(eps / x))));
	else
		tmp = Float64(eps * Float64(Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))) + Float64(eps * Float64(Float64(Float64(x + eps) * Float64(x * Float64(x * 10.0))) + Float64(eps * Float64(eps * Float64(x * 5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.9e-39)
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))));
	elseif (x <= 2.5e-52)
		tmp = (eps * (eps * (eps * eps))) * (x * (5.0 + (eps / x)));
	else
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (((x + eps) * (x * (x * 10.0))) + (eps * (eps * (x * 5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.9e-39], N[(N[(x * x), $MachinePrecision] * N[(x * N[(eps * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-52], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(5.0 + N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(N[(x + eps), $MachinePrecision] * N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8999999999999998e-39

   1. Initial program 37.0%

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

   3. Taylor expanded in x around -inf

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

      1. *-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 96.4%

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

   6. Taylor expanded in x around 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. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 96.2%

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

   8. Simplified 96.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 96.4%

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

   10. Applied egg-rr 96.4%

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

   if -5.8999999999999998e-39 < x < 2.5e-52

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf

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

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

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

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

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

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

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. /-lowering-/.f64 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in eps around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 99.4%

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

   8. Simplified 99.4%

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. cube-unmult N/A

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

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

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

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

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

      6. *-lowering-*.f64 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in x around inf

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

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

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

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

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

       3. /-lowering-/.f64 99.4%

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

   13. Simplified 99.4%

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

   if 2.5e-52 < x

   1. Initial program 39.9%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 95.0%

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

      1. *-commutative N/A

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

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

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

   7. Applied egg-rr 95.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \varepsilon \cdot \left(\left(x + \varepsilon\right) \cdot \left(x \cdot \left(x \cdot 10\right)\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-42}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.9e-39)
   (* (* x x) (* x (* eps (+ (* x 5.0) (* eps 10.0)))))
   (if (<= x 2.7e-42)
     (* (* eps (* eps (* eps eps))) (* x (+ 5.0 (/ eps x))))
     (* (* x (* x x)) (* eps (* x 5.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.9e-39) {
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))));
	} else if (x <= 2.7e-42) {
		tmp = (eps * (eps * (eps * eps))) * (x * (5.0 + (eps / x)));
	} else {
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.9d-39)) then
        tmp = (x * x) * (x * (eps * ((x * 5.0d0) + (eps * 10.0d0))))
    else if (x <= 2.7d-42) then
        tmp = (eps * (eps * (eps * eps))) * (x * (5.0d0 + (eps / x)))
    else
        tmp = (x * (x * x)) * (eps * (x * 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.9e-39) {
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))));
	} else if (x <= 2.7e-42) {
		tmp = (eps * (eps * (eps * eps))) * (x * (5.0 + (eps / x)));
	} else {
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.9e-39:
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))))
	elif x <= 2.7e-42:
		tmp = (eps * (eps * (eps * eps))) * (x * (5.0 + (eps / x)))
	else:
		tmp = (x * (x * x)) * (eps * (x * 5.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.9e-39)
		tmp = Float64(Float64(x * x) * Float64(x * Float64(eps * Float64(Float64(x * 5.0) + Float64(eps * 10.0)))));
	elseif (x <= 2.7e-42)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) * Float64(x * Float64(5.0 + Float64(eps / x))));
	else
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(eps * Float64(x * 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.9e-39)
		tmp = (x * x) * (x * (eps * ((x * 5.0) + (eps * 10.0))));
	elseif (x <= 2.7e-42)
		tmp = (eps * (eps * (eps * eps))) * (x * (5.0 + (eps / x)));
	else
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.9e-39], N[(N[(x * x), $MachinePrecision] * N[(x * N[(eps * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-42], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(5.0 + N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8999999999999998e-39

   1. Initial program 37.0%

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

   3. Taylor expanded in x around -inf

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

      1. *-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 96.4%

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

   6. Taylor expanded in x around 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. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 96.2%

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

   8. Simplified 96.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 96.4%

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

   10. Applied egg-rr 96.4%

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

   if -5.8999999999999998e-39 < x < 2.69999999999999999e-42

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

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

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

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

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

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

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

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. /-lowering-/.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in eps around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 99.0%

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

   8. Simplified 99.0%

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. cube-unmult N/A

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

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

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

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

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

      6. *-lowering-*.f64 99.0%

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

   10. Applied egg-rr 99.0%

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

   11. Taylor expanded in x around inf

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

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

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

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

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

       3. /-lowering-/.f64 99.0%

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

   13. Simplified 99.0%

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

   if 2.69999999999999999e-42 < x

   1. Initial program 30.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.7%

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

   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. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in eps around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. *-lowering-*.f64 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-42}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-42}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.2e-39)
   (* (* x x) (* eps (* x (+ (* x 5.0) (* eps 10.0)))))
   (if (<= x 2.9e-42)
     (* (* eps (* eps (* eps eps))) (* x (+ 5.0 (/ eps x))))
     (* (* x (* x x)) (* eps (* x 5.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.2e-39) {
		tmp = (x * x) * (eps * (x * ((x * 5.0) + (eps * 10.0))));
	} else if (x <= 2.9e-42) {
		tmp = (eps * (eps * (eps * eps))) * (x * (5.0 + (eps / x)));
	} else {
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6.2d-39)) then
        tmp = (x * x) * (eps * (x * ((x * 5.0d0) + (eps * 10.0d0))))
    else if (x <= 2.9d-42) then
        tmp = (eps * (eps * (eps * eps))) * (x * (5.0d0 + (eps / x)))
    else
        tmp = (x * (x * x)) * (eps * (x * 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.2e-39) {
		tmp = (x * x) * (eps * (x * ((x * 5.0) + (eps * 10.0))));
	} else if (x <= 2.9e-42) {
		tmp = (eps * (eps * (eps * eps))) * (x * (5.0 + (eps / x)));
	} else {
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.2e-39:
		tmp = (x * x) * (eps * (x * ((x * 5.0) + (eps * 10.0))))
	elif x <= 2.9e-42:
		tmp = (eps * (eps * (eps * eps))) * (x * (5.0 + (eps / x)))
	else:
		tmp = (x * (x * x)) * (eps * (x * 5.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.2e-39)
		tmp = Float64(Float64(x * x) * Float64(eps * Float64(x * Float64(Float64(x * 5.0) + Float64(eps * 10.0)))));
	elseif (x <= 2.9e-42)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) * Float64(x * Float64(5.0 + Float64(eps / x))));
	else
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(eps * Float64(x * 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.2e-39)
		tmp = (x * x) * (eps * (x * ((x * 5.0) + (eps * 10.0))));
	elseif (x <= 2.9e-42)
		tmp = (eps * (eps * (eps * eps))) * (x * (5.0 + (eps / x)));
	else
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.2e-39], N[(N[(x * x), $MachinePrecision] * N[(eps * N[(x * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e-42], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(5.0 + N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999994e-39

   1. Initial program 37.0%

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

   3. Taylor expanded in x around -inf

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

      1. *-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 96.4%

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

   6. Taylor expanded in x around 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. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 96.2%

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

   8. Simplified 96.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 96.2%

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

   10. Applied egg-rr 96.2%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

       10. *-lowering-*.f64 96.3%

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

   12. Applied egg-rr 96.3%

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

   if -6.1999999999999994e-39 < x < 2.9000000000000003e-42

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

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

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

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

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

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

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

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. /-lowering-/.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in eps around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 99.0%

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

   8. Simplified 99.0%

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. cube-unmult N/A

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

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

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

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

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

      6. *-lowering-*.f64 99.0%

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

   10. Applied egg-rr 99.0%

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

   11. Taylor expanded in x around inf

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

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

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

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

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

       3. /-lowering-/.f64 99.0%

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

   13. Simplified 99.0%

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

   if 2.9000000000000003e-42 < x

   1. Initial program 30.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.7%

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

   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. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in eps around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. *-lowering-*.f64 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-42}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.9e-39)
   (* (* x x) (* eps (* x (+ (* x 5.0) (* eps 10.0)))))
   (if (<= x 1.5e-38)
     (* (* eps (* eps (* eps eps))) (+ eps (* x 5.0)))
     (* (* x (* x x)) (* eps (* x 5.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.9e-39) {
		tmp = (x * x) * (eps * (x * ((x * 5.0) + (eps * 10.0))));
	} else if (x <= 1.5e-38) {
		tmp = (eps * (eps * (eps * eps))) * (eps + (x * 5.0));
	} else {
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.9d-39)) then
        tmp = (x * x) * (eps * (x * ((x * 5.0d0) + (eps * 10.0d0))))
    else if (x <= 1.5d-38) then
        tmp = (eps * (eps * (eps * eps))) * (eps + (x * 5.0d0))
    else
        tmp = (x * (x * x)) * (eps * (x * 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.9e-39) {
		tmp = (x * x) * (eps * (x * ((x * 5.0) + (eps * 10.0))));
	} else if (x <= 1.5e-38) {
		tmp = (eps * (eps * (eps * eps))) * (eps + (x * 5.0));
	} else {
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.9e-39:
		tmp = (x * x) * (eps * (x * ((x * 5.0) + (eps * 10.0))))
	elif x <= 1.5e-38:
		tmp = (eps * (eps * (eps * eps))) * (eps + (x * 5.0))
	else:
		tmp = (x * (x * x)) * (eps * (x * 5.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.9e-39)
		tmp = Float64(Float64(x * x) * Float64(eps * Float64(x * Float64(Float64(x * 5.0) + Float64(eps * 10.0)))));
	elseif (x <= 1.5e-38)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) * Float64(eps + Float64(x * 5.0)));
	else
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(eps * Float64(x * 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.9e-39)
		tmp = (x * x) * (eps * (x * ((x * 5.0) + (eps * 10.0))));
	elseif (x <= 1.5e-38)
		tmp = (eps * (eps * (eps * eps))) * (eps + (x * 5.0));
	else
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.9e-39], N[(N[(x * x), $MachinePrecision] * N[(eps * N[(x * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.5e-38], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8999999999999998e-39

   1. Initial program 37.0%

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

   3. Taylor expanded in x around -inf

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

      1. *-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 96.4%

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

   6. Taylor expanded in x around 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. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 96.2%

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

   8. Simplified 96.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 96.2%

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

   10. Applied egg-rr 96.2%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

       10. *-lowering-*.f64 96.3%

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

   12. Applied egg-rr 96.3%

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

   if -5.8999999999999998e-39 < x < 1.49999999999999994e-38

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

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

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

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

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

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

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

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. /-lowering-/.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in eps around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 99.0%

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

   8. Simplified 99.0%

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. cube-unmult N/A

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

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

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

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

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

      6. *-lowering-*.f64 99.0%

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

   10. Applied egg-rr 99.0%

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

   if 1.49999999999999994e-38 < x

   1. Initial program 30.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.7%

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

   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. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in eps around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. *-lowering-*.f64 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(5 \cdot t\_0\right)\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-42}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -5.9e-39)
     (* eps (* x (* 5.0 t_0)))
     (if (<= x 8.2e-42)
       (* (* eps (* eps (* eps eps))) (+ eps (* x 5.0)))
       (* t_0 (* eps (* x 5.0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -5.9e-39) {
		tmp = eps * (x * (5.0 * t_0));
	} else if (x <= 8.2e-42) {
		tmp = (eps * (eps * (eps * eps))) * (eps + (x * 5.0));
	} else {
		tmp = t_0 * (eps * (x * 5.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= (-5.9d-39)) then
        tmp = eps * (x * (5.0d0 * t_0))
    else if (x <= 8.2d-42) then
        tmp = (eps * (eps * (eps * eps))) * (eps + (x * 5.0d0))
    else
        tmp = t_0 * (eps * (x * 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -5.9e-39) {
		tmp = eps * (x * (5.0 * t_0));
	} else if (x <= 8.2e-42) {
		tmp = (eps * (eps * (eps * eps))) * (eps + (x * 5.0));
	} else {
		tmp = t_0 * (eps * (x * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	tmp = 0
	if x <= -5.9e-39:
		tmp = eps * (x * (5.0 * t_0))
	elif x <= 8.2e-42:
		tmp = (eps * (eps * (eps * eps))) * (eps + (x * 5.0))
	else:
		tmp = t_0 * (eps * (x * 5.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -5.9e-39)
		tmp = Float64(eps * Float64(x * Float64(5.0 * t_0)));
	elseif (x <= 8.2e-42)
		tmp = Float64(Float64(eps * Float64(eps * Float64(eps * eps))) * Float64(eps + Float64(x * 5.0)));
	else
		tmp = Float64(t_0 * Float64(eps * Float64(x * 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= -5.9e-39)
		tmp = eps * (x * (5.0 * t_0));
	elseif (x <= 8.2e-42)
		tmp = (eps * (eps * (eps * eps))) * (eps + (x * 5.0));
	else
		tmp = t_0 * (eps * (x * 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.9e-39], N[(eps * N[(x * N[(5.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e-42], N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8999999999999998e-39

   1. Initial program 37.0%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 96.5%

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

      1. *-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. *-lowering-*.f64 96.3%

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

   7. Applied egg-rr 96.3%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

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

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

   10. Simplified 96.3%

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

   11. Taylor expanded in eps around 0

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(5 \cdot {x}^{3}\right)}\right)\right) \]
   12. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 96.1%

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

   13. Simplified 96.1%

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

   if -5.8999999999999998e-39 < x < 8.2000000000000003e-42

   1. Initial program 99.5%

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

   3. Taylor expanded in eps around inf

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

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

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

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

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

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

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. /-lowering-/.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in eps around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 99.0%

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

   8. Simplified 99.0%

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. cube-unmult N/A

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

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

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

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

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

      6. *-lowering-*.f64 99.0%

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

   10. Applied egg-rr 99.0%

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

   if 8.2000000000000003e-42 < x

   1. Initial program 30.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.7%

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

   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. *-lowering-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in eps around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. *-lowering-*.f64 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-42}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(5 \cdot t\_0\right)\right)\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-51}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -5.9e-39)
     (* eps (* x (* 5.0 t_0)))
     (if (<= x 5.1e-51)
       (* eps (* eps (* eps (* eps eps))))
       (* t_0 (* eps (* x 5.0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -5.9e-39) {
		tmp = eps * (x * (5.0 * t_0));
	} else if (x <= 5.1e-51) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (eps * (x * 5.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= (-5.9d-39)) then
        tmp = eps * (x * (5.0d0 * t_0))
    else if (x <= 5.1d-51) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0 * (eps * (x * 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -5.9e-39) {
		tmp = eps * (x * (5.0 * t_0));
	} else if (x <= 5.1e-51) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0 * (eps * (x * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	tmp = 0
	if x <= -5.9e-39:
		tmp = eps * (x * (5.0 * t_0))
	elif x <= 5.1e-51:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0 * (eps * (x * 5.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -5.9e-39)
		tmp = Float64(eps * Float64(x * Float64(5.0 * t_0)));
	elseif (x <= 5.1e-51)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = Float64(t_0 * Float64(eps * Float64(x * 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= -5.9e-39)
		tmp = eps * (x * (5.0 * t_0));
	elseif (x <= 5.1e-51)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0 * (eps * (x * 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.9e-39], N[(eps * N[(x * N[(5.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.1e-51], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8999999999999998e-39

   1. Initial program 37.0%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 96.5%

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

      1. *-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. *-lowering-*.f64 96.3%

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

   7. Applied egg-rr 96.3%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

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

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

   10. Simplified 96.3%

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

   11. Taylor expanded in eps around 0

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(5 \cdot {x}^{3}\right)}\right)\right) \]
   12. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 96.1%

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

   13. Simplified 96.1%

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

   if -5.8999999999999998e-39 < x < 5.0999999999999997e-51

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{5}\right), \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(\color{blue}{1} + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \left(4 + 1\right) \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + 5 \cdot \frac{\color{blue}{x}}{\varepsilon}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{5 \cdot x}{\color{blue}{\varepsilon}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{x + 4 \cdot x}{\varepsilon}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x + 4 \cdot x}{\varepsilon}\right)}\right)\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \left(\frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \left(\frac{5 \cdot x}{\varepsilon}\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \left(5 \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(5, \color{blue}{\left(\frac{x}{\varepsilon}\right)}\right)\right)\right) \]

      13. /-lowering-/.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{/.f64}\left(x, \color{blue}{\varepsilon}\right)\right)\right)\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{4}\right), \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 4\right), \left(\color{blue}{\varepsilon} + 5 \cdot x\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 4\right), \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(5 \cdot x\right)}\right)\right) \]

      4. *-lowering-*.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 4\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right) \]

   8. Simplified 99.4%

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \mathsf{+.f64}\left(\color{blue}{\varepsilon}, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

      3. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \mathsf{+.f64}\left(\color{blue}{\varepsilon}, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

      6. *-lowering-*.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

   10. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)} \cdot \left(\varepsilon + 5 \cdot x\right) \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \color{blue}{\varepsilon}\right) \]
   12. Step-by-step derivation

   13. Simplified 99.4%

       \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon} \]

   if 5.0999999999999997e-51 < x

   1. Initial program 39.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 94.8%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   6. Taylor expanded in x around 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\color{blue}{5 \cdot \left(\varepsilon \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), \left(\color{blue}{5 \cdot \left(\varepsilon \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(5 \cdot \left(x \cdot \varepsilon\right) + 10 \cdot {\varepsilon}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(5 \cdot x\right) \cdot \varepsilon + \color{blue}{10} \cdot {\varepsilon}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(5 \cdot x\right) \cdot \varepsilon + 10 \cdot \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(5 \cdot x\right) \cdot \varepsilon + \left(10 \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon}\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \color{blue}{\left(5 \cdot x + 10 \cdot \varepsilon\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(5 \cdot x + 10 \cdot \varepsilon\right)}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(5 \cdot x\right), \color{blue}{\left(10 \cdot \varepsilon\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(5, x\right), \left(\color{blue}{10} \cdot \varepsilon\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(5, x\right), \left(\varepsilon \cdot \color{blue}{10}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 95.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(5, x\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{10}\right)\right)\right)\right) \]

   8. Simplified 95.0%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(5 \cdot \varepsilon\right) \cdot \color{blue}{x}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(\varepsilon \cdot 5\right) \cdot x\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \color{blue}{\left(5 \cdot x\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(5 \cdot x\right)}\right)\right) \]

       5. *-lowering-*.f64 94.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right) \]

   11. Simplified 94.9%

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-51}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(x \cdot \left(5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* x (* 5.0 (* x (* x x)))))))
   (if (<= x -5.9e-39)
     t_0
     (if (<= x 3.1e-52) (* eps (* eps (* eps (* eps eps)))) t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (x * (5.0 * (x * (x * x))));
	double tmp;
	if (x <= -5.9e-39) {
		tmp = t_0;
	} else if (x <= 3.1e-52) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * (x * (5.0d0 * (x * (x * x))))
    if (x <= (-5.9d-39)) then
        tmp = t_0
    else if (x <= 3.1d-52) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (x * (5.0 * (x * (x * x))));
	double tmp;
	if (x <= -5.9e-39) {
		tmp = t_0;
	} else if (x <= 3.1e-52) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (x * (5.0 * (x * (x * x))))
	tmp = 0
	if x <= -5.9e-39:
		tmp = t_0
	elif x <= 3.1e-52:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(x * Float64(5.0 * Float64(x * Float64(x * x)))))
	tmp = 0.0
	if (x <= -5.9e-39)
		tmp = t_0;
	elseif (x <= 3.1e-52)
		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 * (5.0 * (x * (x * x))));
	tmp = 0.0;
	if (x <= -5.9e-39)
		tmp = t_0;
	elseif (x <= 3.1e-52)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(x * N[(5.0 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.9e-39], t$95$0, If[LessEqual[x, 3.1e-52], 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 < -5.8999999999999998e-39 or 3.0999999999999999e-52 < x

   1. Initial program 38.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 95.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left({x}^{4} \cdot 5\right), \mathsf{*.f64}\left(\color{blue}{\varepsilon}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), 10\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 10\right), \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, x\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({x}^{4}\right), 5\right), \mathsf{*.f64}\left(\color{blue}{\varepsilon}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), 10\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 10\right), \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, x\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), 5\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), 10\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 10\right), \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, x\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), 5\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), 10\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 10\right), \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, x\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right), 5\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), 10\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 10\right), \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, x\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 5\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), 10\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 10\right), \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, x\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 5\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), 10\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 10\right), \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, x\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), 5\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), 10\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 10\right), \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, x\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), 5\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), 10\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 10\right), \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, x\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 95.8%

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), 5\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), 10\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 10\right), \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\varepsilon, x\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 95.8%

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 5} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(x \cdot \left(5 \cdot {\varepsilon}^{3} + x \cdot \left(10 \cdot {\varepsilon}^{2} + x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(5 \cdot {\varepsilon}^{3} + x \cdot \left(10 \cdot {\varepsilon}^{2} + x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)\right)}\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(5 \cdot {\varepsilon}^{3} + \left(\left(10 \cdot {\varepsilon}^{2}\right) \cdot x + \color{blue}{\left(x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \cdot x}\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(5 \cdot {\varepsilon}^{3} + \left(10 \cdot \left({\varepsilon}^{2} \cdot x\right) + \color{blue}{\left(x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \cdot x\right)\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(\left(5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right) + \color{blue}{\left(x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \cdot x}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(5 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right), \color{blue}{\left(\left(x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \cdot x\right)}\right)\right)\right) \]

   10. Simplified 95.6%

       \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot 10 + 5 \cdot \varepsilon\right) + x \cdot \left(x \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right)\right)\right)} \]

   11. Taylor expanded in eps around 0

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(5 \cdot {x}^{3}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(5, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(5, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(5, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 95.5%

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   13. Simplified 95.5%

       \[\leadsto \varepsilon \cdot \left(x \cdot \color{blue}{\left(5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right) \]

   if -5.8999999999999998e-39 < x < 3.0999999999999999e-52

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{5}\right), \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(\color{blue}{1} + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \left(4 + 1\right) \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + 5 \cdot \frac{\color{blue}{x}}{\varepsilon}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{5 \cdot x}{\color{blue}{\varepsilon}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{x + 4 \cdot x}{\varepsilon}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x + 4 \cdot x}{\varepsilon}\right)}\right)\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \left(\frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \left(\frac{5 \cdot x}{\varepsilon}\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \left(5 \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(5, \color{blue}{\left(\frac{x}{\varepsilon}\right)}\right)\right)\right) \]

      13. /-lowering-/.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{/.f64}\left(x, \color{blue}{\varepsilon}\right)\right)\right)\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{4}\right), \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 4\right), \left(\color{blue}{\varepsilon} + 5 \cdot x\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 4\right), \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(5 \cdot x\right)}\right)\right) \]

      4. *-lowering-*.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 4\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right) \]

   8. Simplified 99.4%

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \mathsf{+.f64}\left(\color{blue}{\varepsilon}, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

      3. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \mathsf{+.f64}\left(\color{blue}{\varepsilon}, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

      6. *-lowering-*.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

   10. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)} \cdot \left(\varepsilon + 5 \cdot x\right) \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \color{blue}{\varepsilon}\right) \]
   12. Step-by-step derivation

   13. Simplified 99.4%

       \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-39}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.3% 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 88.6%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{5}\right), \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)}\right) \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(\color{blue}{1} + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)\right) \]

   3. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \left(4 + 1\right) \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + 5 \cdot \frac{\color{blue}{x}}{\varepsilon}\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{5 \cdot x}{\color{blue}{\varepsilon}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right) \]

   7. distribute-rgt1-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \left(1 + \frac{x + 4 \cdot x}{\varepsilon}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{x + 4 \cdot x}{\varepsilon}\right)}\right)\right) \]

   9. distribute-rgt1-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \left(\frac{\left(4 + 1\right) \cdot x}{\varepsilon}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \left(\frac{5 \cdot x}{\varepsilon}\right)\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \left(5 \cdot \color{blue}{\frac{x}{\varepsilon}}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(5, \color{blue}{\left(\frac{x}{\varepsilon}\right)}\right)\right)\right) \]

   13. /-lowering-/.f64 87.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 5\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{/.f64}\left(x, \color{blue}{\varepsilon}\right)\right)\right)\right) \]

5. Simplified 87.9%

   \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{4}\right), \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right) \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 4\right), \left(\color{blue}{\varepsilon} + 5 \cdot x\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 4\right), \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(5 \cdot x\right)}\right)\right) \]

   4. *-lowering-*.f64 87.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\varepsilon, 4\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right) \]

8. Simplified 87.8%

   \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]
9. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

   2. pow-plus N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \mathsf{+.f64}\left(\color{blue}{\varepsilon}, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

   3. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \mathsf{+.f64}\left(\color{blue}{\varepsilon}, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

   6. *-lowering-*.f64 87.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, x\right)\right)\right) \]

10. Applied egg-rr 87.8%

    \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)} \cdot \left(\varepsilon + 5 \cdot x\right) \]

11. Taylor expanded in eps around inf

    \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \color{blue}{\varepsilon}\right) \]
12. Step-by-step derivation

13. Simplified 87.8%

    \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon} \]

14. Final simplification 87.8%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(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)))