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

Percentage Accurate: 88.4% → 99.3%

Time: 12.9s

Alternatives: 22

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: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-309)
     t_0
     (if (<= t_0 0.0)
       (* (pow x 4.0) (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-309) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-1d-309)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (x ** 4.0d0) * ((eps * 5.0d0) - (((eps * eps) * (-10.0d0)) / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-309) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = Math.pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -1e-309:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = math.pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-309)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -1e-309)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (x ^ 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-309], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.000000000000002e-309 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.8%

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

   if -1.000000000000002e-309 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.9%

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

Alternative 2: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-40}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(5 \cdot \left(x \cdot \varepsilon\right) + \left(x \cdot x\right) \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-58}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\left(\left(\frac{\varepsilon \cdot \left(\varepsilon \cdot 8\right)}{x} + \frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 4\right)\right)}{x \cdot x}\right) + \varepsilon \cdot \left(4 + \frac{\frac{\varepsilon \cdot \left(\varepsilon \cdot 6\right)}{x}}{x}\right)\right) + \left(\varepsilon + \frac{\varepsilon \cdot \left(\varepsilon \cdot 2\right)}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.4e-40)
   (*
    eps
    (+
     (* 5.0 (pow x 4.0))
     (*
      eps
      (+
       (* (* x (* x x)) 10.0)
       (* eps (+ (* 5.0 (* x eps)) (* (* x x) 10.0)))))))
   (if (<= x 4.9e-58)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (*
      (pow x 4.0)
      (+
       (+
        (+ (/ (* eps (* eps 8.0)) x) (/ (* eps (* eps (* eps 4.0))) (* x x)))
        (* eps (+ 4.0 (/ (/ (* eps (* eps 6.0)) x) x))))
       (+ eps (/ (* eps (* eps 2.0)) x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.4e-40) {
		tmp = eps * ((5.0 * pow(x, 4.0)) + (eps * (((x * (x * x)) * 10.0) + (eps * ((5.0 * (x * eps)) + ((x * x) * 10.0))))));
	} else if (x <= 4.9e-58) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = pow(x, 4.0) * (((((eps * (eps * 8.0)) / x) + ((eps * (eps * (eps * 4.0))) / (x * x))) + (eps * (4.0 + (((eps * (eps * 6.0)) / x) / x)))) + (eps + ((eps * (eps * 2.0)) / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.4d-40)) then
        tmp = eps * ((5.0d0 * (x ** 4.0d0)) + (eps * (((x * (x * x)) * 10.0d0) + (eps * ((5.0d0 * (x * eps)) + ((x * x) * 10.0d0))))))
    else if (x <= 4.9d-58) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = (x ** 4.0d0) * (((((eps * (eps * 8.0d0)) / x) + ((eps * (eps * (eps * 4.0d0))) / (x * x))) + (eps * (4.0d0 + (((eps * (eps * 6.0d0)) / x) / x)))) + (eps + ((eps * (eps * 2.0d0)) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.4e-40) {
		tmp = eps * ((5.0 * Math.pow(x, 4.0)) + (eps * (((x * (x * x)) * 10.0) + (eps * ((5.0 * (x * eps)) + ((x * x) * 10.0))))));
	} else if (x <= 4.9e-58) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = Math.pow(x, 4.0) * (((((eps * (eps * 8.0)) / x) + ((eps * (eps * (eps * 4.0))) / (x * x))) + (eps * (4.0 + (((eps * (eps * 6.0)) / x) / x)))) + (eps + ((eps * (eps * 2.0)) / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.4e-40:
		tmp = eps * ((5.0 * math.pow(x, 4.0)) + (eps * (((x * (x * x)) * 10.0) + (eps * ((5.0 * (x * eps)) + ((x * x) * 10.0))))))
	elif x <= 4.9e-58:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = math.pow(x, 4.0) * (((((eps * (eps * 8.0)) / x) + ((eps * (eps * (eps * 4.0))) / (x * x))) + (eps * (4.0 + (((eps * (eps * 6.0)) / x) / x)))) + (eps + ((eps * (eps * 2.0)) / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.4e-40)
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(eps * Float64(Float64(Float64(x * Float64(x * x)) * 10.0) + Float64(eps * Float64(Float64(5.0 * Float64(x * eps)) + Float64(Float64(x * x) * 10.0)))))));
	elseif (x <= 4.9e-58)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64((x ^ 4.0) * Float64(Float64(Float64(Float64(Float64(eps * Float64(eps * 8.0)) / x) + Float64(Float64(eps * Float64(eps * Float64(eps * 4.0))) / Float64(x * x))) + Float64(eps * Float64(4.0 + Float64(Float64(Float64(eps * Float64(eps * 6.0)) / x) / x)))) + Float64(eps + Float64(Float64(eps * Float64(eps * 2.0)) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.4e-40)
		tmp = eps * ((5.0 * (x ^ 4.0)) + (eps * (((x * (x * x)) * 10.0) + (eps * ((5.0 * (x * eps)) + ((x * x) * 10.0))))));
	elseif (x <= 4.9e-58)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = (x ^ 4.0) * (((((eps * (eps * 8.0)) / x) + ((eps * (eps * (eps * 4.0))) / (x * x))) + (eps * (4.0 + (((eps * (eps * 6.0)) / x) / x)))) + (eps + ((eps * (eps * 2.0)) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.4e-40], N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision] + N[(eps * N[(N[(5.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9e-58], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(N[(N[(N[(eps * N[(eps * 8.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(eps * N[(eps * N[(eps * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(4.0 + N[(N[(N[(eps * N[(eps * 6.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps + N[(N[(eps * N[(eps * 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e-40

   1. Initial program 26.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 93.0%

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

   if -1.4e-40 < x < 4.9000000000000003e-58

   1. Initial program 98.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.3%

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

   5. Simplified 98.3%

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

   if 4.9000000000000003e-58 < x

   1. Initial program 32.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\left(\left(\frac{\varepsilon \cdot \left(\varepsilon \cdot 8\right)}{x} + \frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 4\right)\right)}{x \cdot x}\right) + \varepsilon \cdot \left(\frac{\frac{\varepsilon \cdot \left(\varepsilon \cdot 6\right)}{x}}{x} + 4\right)\right) + \left(\varepsilon + \frac{\left(2 \cdot \varepsilon\right) \cdot \varepsilon}{x}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-40}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(5 \cdot \left(x \cdot \varepsilon\right) + \left(x \cdot x\right) \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-58}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\left(\left(\frac{\varepsilon \cdot \left(\varepsilon \cdot 8\right)}{x} + \frac{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 4\right)\right)}{x \cdot x}\right) + \varepsilon \cdot \left(4 + \frac{\frac{\varepsilon \cdot \left(\varepsilon \cdot 6\right)}{x}}{x}\right)\right) + \left(\varepsilon + \frac{\varepsilon \cdot \left(\varepsilon \cdot 2\right)}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(5 \cdot \left(x \cdot \varepsilon\right) + \left(x \cdot x\right) \cdot 10\right)\right)\right)\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-55}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          eps
          (+
           (* 5.0 (pow x 4.0))
           (*
            eps
            (+
             (* (* x (* x x)) 10.0)
             (* eps (+ (* 5.0 (* x eps)) (* (* x x) 10.0)))))))))
   (if (<= x -1.3e-40)
     t_0
     (if (<= x 9e-55) (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps)))) t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * ((5.0 * pow(x, 4.0)) + (eps * (((x * (x * x)) * 10.0) + (eps * ((5.0 * (x * eps)) + ((x * x) * 10.0))))));
	double tmp;
	if (x <= -1.3e-40) {
		tmp = t_0;
	} else if (x <= 9e-55) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * ((5.0d0 * (x ** 4.0d0)) + (eps * (((x * (x * x)) * 10.0d0) + (eps * ((5.0d0 * (x * eps)) + ((x * x) * 10.0d0))))))
    if (x <= (-1.3d-40)) then
        tmp = t_0
    else if (x <= 9d-55) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * ((5.0 * Math.pow(x, 4.0)) + (eps * (((x * (x * x)) * 10.0) + (eps * ((5.0 * (x * eps)) + ((x * x) * 10.0))))));
	double tmp;
	if (x <= -1.3e-40) {
		tmp = t_0;
	} else if (x <= 9e-55) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * ((5.0 * math.pow(x, 4.0)) + (eps * (((x * (x * x)) * 10.0) + (eps * ((5.0 * (x * eps)) + ((x * x) * 10.0))))))
	tmp = 0
	if x <= -1.3e-40:
		tmp = t_0
	elif x <= 9e-55:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(eps * Float64(Float64(Float64(x * Float64(x * x)) * 10.0) + Float64(eps * Float64(Float64(5.0 * Float64(x * eps)) + Float64(Float64(x * x) * 10.0)))))))
	tmp = 0.0
	if (x <= -1.3e-40)
		tmp = t_0;
	elseif (x <= 9e-55)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * ((5.0 * (x ^ 4.0)) + (eps * (((x * (x * x)) * 10.0) + (eps * ((5.0 * (x * eps)) + ((x * x) * 10.0))))));
	tmp = 0.0;
	if (x <= -1.3e-40)
		tmp = t_0;
	elseif (x <= 9e-55)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision] + N[(eps * N[(N[(5.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e-40], t$95$0, If[LessEqual[x, 9e-55], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3000000000000001e-40 or 8.99999999999999941e-55 < x

   1. Initial program 29.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 96.3%

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

   if -1.3000000000000001e-40 < x < 8.99999999999999941e-55

   1. Initial program 98.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.3%

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

   5. Simplified 98.3%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-40}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(5 \cdot \left(x \cdot \varepsilon\right) + \left(x \cdot x\right) \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-55}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(5 \cdot \left(x \cdot \varepsilon\right) + \left(x \cdot x\right) \cdot 10\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \left(5 \cdot \left(\varepsilon \cdot t\_0\right) + x \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 + \frac{\frac{10 \cdot t\_0}{x} - \left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* eps eps))))
   (if (<= x -1.35e-40)
     (*
      x
      (+
       (* 5.0 (* eps t_0))
       (* x (+ (* 10.0 (* eps (* eps (+ x eps)))) (* x (* eps (* x 5.0)))))))
     (if (<= x 1.1e-56)
       (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
       (*
        (pow x 4.0)
        (+ (* eps 5.0) (/ (- (/ (* 10.0 t_0) x) (* (* eps eps) -10.0)) x)))))))

C

double code(double x, double eps) {
	double t_0 = eps * (eps * eps);
	double tmp;
	if (x <= -1.35e-40) {
		tmp = x * ((5.0 * (eps * t_0)) + (x * ((10.0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0))))));
	} else if (x <= 1.1e-56) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = pow(x, 4.0) * ((eps * 5.0) + ((((10.0 * t_0) / x) - ((eps * eps) * -10.0)) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * (eps * eps)
    if (x <= (-1.35d-40)) then
        tmp = x * ((5.0d0 * (eps * t_0)) + (x * ((10.0d0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0d0))))))
    else if (x <= 1.1d-56) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = (x ** 4.0d0) * ((eps * 5.0d0) + ((((10.0d0 * t_0) / x) - ((eps * eps) * (-10.0d0))) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (eps * eps);
	double tmp;
	if (x <= -1.35e-40) {
		tmp = x * ((5.0 * (eps * t_0)) + (x * ((10.0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0))))));
	} else if (x <= 1.1e-56) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = Math.pow(x, 4.0) * ((eps * 5.0) + ((((10.0 * t_0) / x) - ((eps * eps) * -10.0)) / x));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (eps * eps)
	tmp = 0
	if x <= -1.35e-40:
		tmp = x * ((5.0 * (eps * t_0)) + (x * ((10.0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0))))))
	elif x <= 1.1e-56:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = math.pow(x, 4.0) * ((eps * 5.0) + ((((10.0 * t_0) / x) - ((eps * eps) * -10.0)) / x))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(eps * eps))
	tmp = 0.0
	if (x <= -1.35e-40)
		tmp = Float64(x * Float64(Float64(5.0 * Float64(eps * t_0)) + Float64(x * Float64(Float64(10.0 * Float64(eps * Float64(eps * Float64(x + eps)))) + Float64(x * Float64(eps * Float64(x * 5.0)))))));
	elseif (x <= 1.1e-56)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) + Float64(Float64(Float64(Float64(10.0 * t_0) / x) - Float64(Float64(eps * eps) * -10.0)) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * (eps * eps);
	tmp = 0.0;
	if (x <= -1.35e-40)
		tmp = x * ((5.0 * (eps * t_0)) + (x * ((10.0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0))))));
	elseif (x <= 1.1e-56)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = (x ^ 4.0) * ((eps * 5.0) + ((((10.0 * t_0) / x) - ((eps * eps) * -10.0)) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e-40], N[(x * N[(N[(5.0 * N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(10.0 * N[(eps * N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e-56], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] + N[(N[(N[(N[(10.0 * t$95$0), $MachinePrecision] / x), $MachinePrecision] - N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35e-40

   1. Initial program 26.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 93.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 92.7%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 92.8%

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

   if -1.35e-40 < x < 1.10000000000000002e-56

   1. Initial program 98.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.3%

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

   5. Simplified 98.3%

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

   if 1.10000000000000002e-56 < x

   1. Initial program 32.9%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \left(5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + x \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 + \frac{\frac{10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{x} - \left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \left(5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + x \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right) + x \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-40)
   (*
    x
    (+
     (* 5.0 (* eps (* eps (* eps eps))))
     (* x (+ (* 10.0 (* eps (* eps (+ x eps)))) (* x (* eps (* x 5.0)))))))
   (if (<= x 5.2e-52)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (*
      eps
      (*
       x
       (+
        (* (* eps eps) (+ (* eps 5.0) (* x 10.0)))
        (* x (* x (+ (* x 5.0) (* eps 10.0))))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-40) {
		tmp = x * ((5.0 * (eps * (eps * (eps * eps)))) + (x * ((10.0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0))))));
	} else if (x <= 5.2e-52) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.45d-40)) then
        tmp = x * ((5.0d0 * (eps * (eps * (eps * eps)))) + (x * ((10.0d0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0d0))))))
    else if (x <= 5.2d-52) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = eps * (x * (((eps * eps) * ((eps * 5.0d0) + (x * 10.0d0))) + (x * (x * ((x * 5.0d0) + (eps * 10.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-40) {
		tmp = x * ((5.0 * (eps * (eps * (eps * eps)))) + (x * ((10.0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0))))));
	} else if (x <= 5.2e-52) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.45e-40:
		tmp = x * ((5.0 * (eps * (eps * (eps * eps)))) + (x * ((10.0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0))))))
	elif x <= 5.2e-52:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-40)
		tmp = Float64(x * Float64(Float64(5.0 * Float64(eps * Float64(eps * Float64(eps * eps)))) + Float64(x * Float64(Float64(10.0 * Float64(eps * Float64(eps * Float64(x + eps)))) + Float64(x * Float64(eps * Float64(x * 5.0)))))));
	elseif (x <= 5.2e-52)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64(eps * Float64(x * Float64(Float64(Float64(eps * eps) * Float64(Float64(eps * 5.0) + Float64(x * 10.0))) + Float64(x * Float64(x * Float64(Float64(x * 5.0) + Float64(eps * 10.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.45e-40)
		tmp = x * ((5.0 * (eps * (eps * (eps * eps)))) + (x * ((10.0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0))))));
	elseif (x <= 5.2e-52)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-40], N[(x * N[(N[(5.0 * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(10.0 * N[(eps * N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-52], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(x * N[(N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4499999999999999e-40

   1. Initial program 26.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 93.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 92.7%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 92.8%

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

   if -1.4499999999999999e-40 < x < 5.1999999999999997e-52

   1. Initial program 98.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.3%

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

   5. Simplified 98.3%

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

   if 5.1999999999999997e-52 < x

   1. Initial program 32.9%

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

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 99.3%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

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

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

   10. Simplified 99.5%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \left(5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + x \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right) + x \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ t_1 := \varepsilon \cdot t\_0\\ \mathbf{if}\;x \leq -7 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \left(5 \cdot t\_1 + x \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-46}:\\ \;\;\;\;t\_1 \cdot \left(\varepsilon + x \cdot 5\right) + \left(x \cdot x\right) \cdot \left(10 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right) + x \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* eps eps))) (t_1 (* eps t_0)))
   (if (<= x -7e-40)
     (*
      x
      (+
       (* 5.0 t_1)
       (* x (+ (* 10.0 (* eps (* eps (+ x eps)))) (* x (* eps (* x 5.0)))))))
     (if (<= x 1.15e-46)
       (+ (* t_1 (+ eps (* x 5.0))) (* (* x x) (* 10.0 t_0)))
       (*
        eps
        (*
         x
         (+
          (* (* eps eps) (+ (* eps 5.0) (* x 10.0)))
          (* x (* x (+ (* x 5.0) (* eps 10.0)))))))))))

C

double code(double x, double eps) {
	double t_0 = eps * (eps * eps);
	double t_1 = eps * t_0;
	double tmp;
	if (x <= -7e-40) {
		tmp = x * ((5.0 * t_1) + (x * ((10.0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0))))));
	} else if (x <= 1.15e-46) {
		tmp = (t_1 * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_0));
	} else {
		tmp = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = eps * (eps * eps)
    t_1 = eps * t_0
    if (x <= (-7d-40)) then
        tmp = x * ((5.0d0 * t_1) + (x * ((10.0d0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0d0))))))
    else if (x <= 1.15d-46) then
        tmp = (t_1 * (eps + (x * 5.0d0))) + ((x * x) * (10.0d0 * t_0))
    else
        tmp = eps * (x * (((eps * eps) * ((eps * 5.0d0) + (x * 10.0d0))) + (x * (x * ((x * 5.0d0) + (eps * 10.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (eps * eps);
	double t_1 = eps * t_0;
	double tmp;
	if (x <= -7e-40) {
		tmp = x * ((5.0 * t_1) + (x * ((10.0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0))))));
	} else if (x <= 1.15e-46) {
		tmp = (t_1 * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_0));
	} else {
		tmp = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (eps * eps)
	t_1 = eps * t_0
	tmp = 0
	if x <= -7e-40:
		tmp = x * ((5.0 * t_1) + (x * ((10.0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0))))))
	elif x <= 1.15e-46:
		tmp = (t_1 * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_0))
	else:
		tmp = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(eps * eps))
	t_1 = Float64(eps * t_0)
	tmp = 0.0
	if (x <= -7e-40)
		tmp = Float64(x * Float64(Float64(5.0 * t_1) + Float64(x * Float64(Float64(10.0 * Float64(eps * Float64(eps * Float64(x + eps)))) + Float64(x * Float64(eps * Float64(x * 5.0)))))));
	elseif (x <= 1.15e-46)
		tmp = Float64(Float64(t_1 * Float64(eps + Float64(x * 5.0))) + Float64(Float64(x * x) * Float64(10.0 * t_0)));
	else
		tmp = Float64(eps * Float64(x * Float64(Float64(Float64(eps * eps) * Float64(Float64(eps * 5.0) + Float64(x * 10.0))) + Float64(x * Float64(x * Float64(Float64(x * 5.0) + Float64(eps * 10.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * (eps * eps);
	t_1 = eps * t_0;
	tmp = 0.0;
	if (x <= -7e-40)
		tmp = x * ((5.0 * t_1) + (x * ((10.0 * (eps * (eps * (x + eps)))) + (x * (eps * (x * 5.0))))));
	elseif (x <= 1.15e-46)
		tmp = (t_1 * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_0));
	else
		tmp = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps * t$95$0), $MachinePrecision]}, If[LessEqual[x, -7e-40], N[(x * N[(N[(5.0 * t$95$1), $MachinePrecision] + N[(x * N[(N[(10.0 * N[(eps * N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e-46], N[(N[(t$95$1 * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(10.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(x * N[(N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.0000000000000003e-40

   1. Initial program 26.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 93.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 92.7%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 92.8%

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

   if -7.0000000000000003e-40 < x < 1.15e-46

   1. Initial program 98.5%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 89.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. distribute-rgt1-in N/A

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

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

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

      15. distribute-rgt1-in N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 98.2%

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

   8. Simplified 98.2%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. associate-*r* N/A

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

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

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

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

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

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

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

   10. Applied egg-rr 98.3%

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

   if 1.15e-46 < x

   1. Initial program 32.9%

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

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 99.3%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

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

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

   10. Simplified 99.5%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \left(5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + x \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-46}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(\varepsilon + x \cdot 5\right) + \left(x \cdot x\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right) + x \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-40}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \varepsilon \cdot \left(x \cdot \left(5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 10 \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-55}:\\ \;\;\;\;\left(\varepsilon \cdot t\_0\right) \cdot \left(\varepsilon + x \cdot 5\right) + \left(x \cdot x\right) \cdot \left(10 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right) + x \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* eps eps))))
   (if (<= x -2.2e-40)
     (*
      eps
      (+
       (* 5.0 (* x (* x (* x x))))
       (* eps (* x (+ (* 5.0 (* eps eps)) (* 10.0 (* x (+ x eps))))))))
     (if (<= x 2.65e-55)
       (+ (* (* eps t_0) (+ eps (* x 5.0))) (* (* x x) (* 10.0 t_0)))
       (*
        eps
        (*
         x
         (+
          (* (* eps eps) (+ (* eps 5.0) (* x 10.0)))
          (* x (* x (+ (* x 5.0) (* eps 10.0)))))))))))

C

double code(double x, double eps) {
	double t_0 = eps * (eps * eps);
	double tmp;
	if (x <= -2.2e-40) {
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (x * ((5.0 * (eps * eps)) + (10.0 * (x * (x + eps)))))));
	} else if (x <= 2.65e-55) {
		tmp = ((eps * t_0) * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_0));
	} else {
		tmp = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * (eps * eps)
    if (x <= (-2.2d-40)) then
        tmp = eps * ((5.0d0 * (x * (x * (x * x)))) + (eps * (x * ((5.0d0 * (eps * eps)) + (10.0d0 * (x * (x + eps)))))))
    else if (x <= 2.65d-55) then
        tmp = ((eps * t_0) * (eps + (x * 5.0d0))) + ((x * x) * (10.0d0 * t_0))
    else
        tmp = eps * (x * (((eps * eps) * ((eps * 5.0d0) + (x * 10.0d0))) + (x * (x * ((x * 5.0d0) + (eps * 10.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (eps * eps);
	double tmp;
	if (x <= -2.2e-40) {
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (x * ((5.0 * (eps * eps)) + (10.0 * (x * (x + eps)))))));
	} else if (x <= 2.65e-55) {
		tmp = ((eps * t_0) * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_0));
	} else {
		tmp = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (eps * eps)
	tmp = 0
	if x <= -2.2e-40:
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (x * ((5.0 * (eps * eps)) + (10.0 * (x * (x + eps)))))))
	elif x <= 2.65e-55:
		tmp = ((eps * t_0) * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_0))
	else:
		tmp = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(eps * eps))
	tmp = 0.0
	if (x <= -2.2e-40)
		tmp = Float64(eps * Float64(Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))) + Float64(eps * Float64(x * Float64(Float64(5.0 * Float64(eps * eps)) + Float64(10.0 * Float64(x * Float64(x + eps))))))));
	elseif (x <= 2.65e-55)
		tmp = Float64(Float64(Float64(eps * t_0) * Float64(eps + Float64(x * 5.0))) + Float64(Float64(x * x) * Float64(10.0 * t_0)));
	else
		tmp = Float64(eps * Float64(x * Float64(Float64(Float64(eps * eps) * Float64(Float64(eps * 5.0) + Float64(x * 10.0))) + Float64(x * Float64(x * Float64(Float64(x * 5.0) + Float64(eps * 10.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * (eps * eps);
	tmp = 0.0;
	if (x <= -2.2e-40)
		tmp = eps * ((5.0 * (x * (x * (x * x)))) + (eps * (x * ((5.0 * (eps * eps)) + (10.0 * (x * (x + eps)))))));
	elseif (x <= 2.65e-55)
		tmp = ((eps * t_0) * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_0));
	else
		tmp = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.2e-40], N[(eps * N[(N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(x * N[(N[(5.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[(10.0 * N[(x * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.65e-55], N[(N[(N[(eps * t$95$0), $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(10.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(x * N[(N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.20000000000000009e-40

   1. Initial program 26.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 93.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 92.7%

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

   8. Taylor expanded in eps around 0

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. distribute-rgt-in N/A

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

   10. Simplified 92.7%

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

   if -2.20000000000000009e-40 < x < 2.6500000000000001e-55

   1. Initial program 98.5%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 89.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. distribute-rgt1-in N/A

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

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

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

      15. distribute-rgt1-in N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 98.2%

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

   8. Simplified 98.2%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. associate-*r* N/A

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

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

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

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

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

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

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

   10. Applied egg-rr 98.3%

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

   if 2.6500000000000001e-55 < x

   1. Initial program 32.9%

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

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 99.3%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

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

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

   10. Simplified 99.5%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-40}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \varepsilon \cdot \left(x \cdot \left(5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 10 \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-55}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(\varepsilon + x \cdot 5\right) + \left(x \cdot x\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right) + x \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right) + x \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\right)\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-54}:\\ \;\;\;\;\left(\varepsilon \cdot t\_1\right) \cdot \left(\varepsilon + x \cdot 5\right) + \left(x \cdot x\right) \cdot \left(10 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          eps
          (*
           x
           (+
            (* (* eps eps) (+ (* eps 5.0) (* x 10.0)))
            (* x (* x (+ (* x 5.0) (* eps 10.0))))))))
        (t_1 (* eps (* eps eps))))
   (if (<= x -1.3e-40)
     t_0
     (if (<= x 1.05e-54)
       (+ (* (* eps t_1) (+ eps (* x 5.0))) (* (* x x) (* 10.0 t_1)))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))));
	double t_1 = eps * (eps * eps);
	double tmp;
	if (x <= -1.3e-40) {
		tmp = t_0;
	} else if (x <= 1.05e-54) {
		tmp = ((eps * t_1) * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_1));
	} 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) :: t_1
    real(8) :: tmp
    t_0 = eps * (x * (((eps * eps) * ((eps * 5.0d0) + (x * 10.0d0))) + (x * (x * ((x * 5.0d0) + (eps * 10.0d0))))))
    t_1 = eps * (eps * eps)
    if (x <= (-1.3d-40)) then
        tmp = t_0
    else if (x <= 1.05d-54) then
        tmp = ((eps * t_1) * (eps + (x * 5.0d0))) + ((x * x) * (10.0d0 * t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))));
	double t_1 = eps * (eps * eps);
	double tmp;
	if (x <= -1.3e-40) {
		tmp = t_0;
	} else if (x <= 1.05e-54) {
		tmp = ((eps * t_1) * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))))
	t_1 = eps * (eps * eps)
	tmp = 0
	if x <= -1.3e-40:
		tmp = t_0
	elif x <= 1.05e-54:
		tmp = ((eps * t_1) * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(x * Float64(Float64(Float64(eps * eps) * Float64(Float64(eps * 5.0) + Float64(x * 10.0))) + Float64(x * Float64(x * Float64(Float64(x * 5.0) + Float64(eps * 10.0)))))))
	t_1 = Float64(eps * Float64(eps * eps))
	tmp = 0.0
	if (x <= -1.3e-40)
		tmp = t_0;
	elseif (x <= 1.05e-54)
		tmp = Float64(Float64(Float64(eps * t_1) * Float64(eps + Float64(x * 5.0))) + Float64(Float64(x * x) * Float64(10.0 * t_1)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * (x * (((eps * eps) * ((eps * 5.0) + (x * 10.0))) + (x * (x * ((x * 5.0) + (eps * 10.0))))));
	t_1 = eps * (eps * eps);
	tmp = 0.0;
	if (x <= -1.3e-40)
		tmp = t_0;
	elseif (x <= 1.05e-54)
		tmp = ((eps * t_1) * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_1));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(x * N[(N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e-40], t$95$0, If[LessEqual[x, 1.05e-54], N[(N[(N[(eps * t$95$1), $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(10.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3000000000000001e-40 or 1.05e-54 < x

   1. Initial program 29.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 96.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 96.0%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

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

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

   10. Simplified 96.1%

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

   if -1.3000000000000001e-40 < x < 1.05e-54

   1. Initial program 98.5%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 89.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. distribute-rgt1-in N/A

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

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

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

      15. distribute-rgt1-in N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 98.2%

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

   8. Simplified 98.2%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. associate-*r* N/A

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

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

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

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

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

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

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

   10. Applied egg-rr 98.3%

       \[\leadsto \color{blue}{\left(\varepsilon + x \cdot 5\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + \left(x \cdot x\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-40}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right) + x \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-54}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(\varepsilon + x \cdot 5\right) + \left(x \cdot x\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right) + x \cdot \left(x \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(x + \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\right)\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-56}:\\ \;\;\;\;\left(\varepsilon \cdot t\_1\right) \cdot \left(\varepsilon + x \cdot 5\right) + \left(x \cdot x\right) \cdot \left(10 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          eps
          (+
           (* 5.0 (* x (* x (* x x))))
           (* (+ x eps) (* 10.0 (* eps (* x x)))))))
        (t_1 (* eps (* eps eps))))
   (if (<= x -1.3e-40)
     t_0
     (if (<= x 1.65e-56)
       (+ (* (* eps t_1) (+ eps (* x 5.0))) (* (* x x) (* 10.0 t_1)))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * ((5.0 * (x * (x * (x * x)))) + ((x + eps) * (10.0 * (eps * (x * x)))));
	double t_1 = eps * (eps * eps);
	double tmp;
	if (x <= -1.3e-40) {
		tmp = t_0;
	} else if (x <= 1.65e-56) {
		tmp = ((eps * t_1) * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_1));
	} 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) :: t_1
    real(8) :: tmp
    t_0 = eps * ((5.0d0 * (x * (x * (x * x)))) + ((x + eps) * (10.0d0 * (eps * (x * x)))))
    t_1 = eps * (eps * eps)
    if (x <= (-1.3d-40)) then
        tmp = t_0
    else if (x <= 1.65d-56) then
        tmp = ((eps * t_1) * (eps + (x * 5.0d0))) + ((x * x) * (10.0d0 * t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * ((5.0 * (x * (x * (x * x)))) + ((x + eps) * (10.0 * (eps * (x * x)))));
	double t_1 = eps * (eps * eps);
	double tmp;
	if (x <= -1.3e-40) {
		tmp = t_0;
	} else if (x <= 1.65e-56) {
		tmp = ((eps * t_1) * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * ((5.0 * (x * (x * (x * x)))) + ((x + eps) * (10.0 * (eps * (x * x)))))
	t_1 = eps * (eps * eps)
	tmp = 0
	if x <= -1.3e-40:
		tmp = t_0
	elif x <= 1.65e-56:
		tmp = ((eps * t_1) * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))) + Float64(Float64(x + eps) * Float64(10.0 * Float64(eps * Float64(x * x))))))
	t_1 = Float64(eps * Float64(eps * eps))
	tmp = 0.0
	if (x <= -1.3e-40)
		tmp = t_0;
	elseif (x <= 1.65e-56)
		tmp = Float64(Float64(Float64(eps * t_1) * Float64(eps + Float64(x * 5.0))) + Float64(Float64(x * x) * Float64(10.0 * t_1)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * ((5.0 * (x * (x * (x * x)))) + ((x + eps) * (10.0 * (eps * (x * x)))));
	t_1 = eps * (eps * eps);
	tmp = 0.0;
	if (x <= -1.3e-40)
		tmp = t_0;
	elseif (x <= 1.65e-56)
		tmp = ((eps * t_1) * (eps + (x * 5.0))) + ((x * x) * (10.0 * t_1));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + eps), $MachinePrecision] * N[(10.0 * N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e-40], t$95$0, If[LessEqual[x, 1.65e-56], N[(N[(N[(eps * t$95$1), $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(10.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3000000000000001e-40 or 1.64999999999999992e-56 < x

   1. Initial program 29.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 96.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 96.0%

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

   8. Taylor expanded in eps around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

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

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

      15. +-lowering-+.f64 95.4%

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

   10. Simplified 95.4%

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

   if -1.3000000000000001e-40 < x < 1.64999999999999992e-56

   1. Initial program 98.5%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 89.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. distribute-rgt1-in N/A

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

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

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

      15. distribute-rgt1-in N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 98.2%

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

   8. Simplified 98.2%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. associate-*r* N/A

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

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

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

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

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

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

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

   10. Applied egg-rr 98.3%

       \[\leadsto \color{blue}{\left(\varepsilon + x \cdot 5\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + \left(x \cdot x\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-40}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(x + \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-56}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(\varepsilon + x \cdot 5\right) + \left(x \cdot x\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(x + \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(x + \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 10 + x \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          eps
          (+
           (* 5.0 (* x (* x (* x x))))
           (* (+ x eps) (* 10.0 (* eps (* x x))))))))
   (if (<= x -1.7e-40)
     t_0
     (if (<= x 9.8e-54)
       (*
        eps
        (* (* eps eps) (+ (* (* x x) 10.0) (* x (* eps (+ 5.0 (/ eps x)))))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * ((5.0 * (x * (x * (x * x)))) + ((x + eps) * (10.0 * (eps * (x * x)))));
	double tmp;
	if (x <= -1.7e-40) {
		tmp = t_0;
	} else if (x <= 9.8e-54) {
		tmp = eps * ((eps * eps) * (((x * x) * 10.0) + (x * (eps * (5.0 + (eps / x))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * ((5.0d0 * (x * (x * (x * x)))) + ((x + eps) * (10.0d0 * (eps * (x * x)))))
    if (x <= (-1.7d-40)) then
        tmp = t_0
    else if (x <= 9.8d-54) then
        tmp = eps * ((eps * eps) * (((x * x) * 10.0d0) + (x * (eps * (5.0d0 + (eps / x))))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * ((5.0 * (x * (x * (x * x)))) + ((x + eps) * (10.0 * (eps * (x * x)))));
	double tmp;
	if (x <= -1.7e-40) {
		tmp = t_0;
	} else if (x <= 9.8e-54) {
		tmp = eps * ((eps * eps) * (((x * x) * 10.0) + (x * (eps * (5.0 + (eps / x))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * ((5.0 * (x * (x * (x * x)))) + ((x + eps) * (10.0 * (eps * (x * x)))))
	tmp = 0
	if x <= -1.7e-40:
		tmp = t_0
	elif x <= 9.8e-54:
		tmp = eps * ((eps * eps) * (((x * x) * 10.0) + (x * (eps * (5.0 + (eps / x))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(Float64(5.0 * Float64(x * Float64(x * Float64(x * x)))) + Float64(Float64(x + eps) * Float64(10.0 * Float64(eps * Float64(x * x))))))
	tmp = 0.0
	if (x <= -1.7e-40)
		tmp = t_0;
	elseif (x <= 9.8e-54)
		tmp = Float64(eps * Float64(Float64(eps * eps) * Float64(Float64(Float64(x * x) * 10.0) + Float64(x * Float64(eps * Float64(5.0 + Float64(eps / x)))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * ((5.0 * (x * (x * (x * x)))) + ((x + eps) * (10.0 * (eps * (x * x)))));
	tmp = 0.0;
	if (x <= -1.7e-40)
		tmp = t_0;
	elseif (x <= 9.8e-54)
		tmp = eps * ((eps * eps) * (((x * x) * 10.0) + (x * (eps * (5.0 + (eps / x))))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(N[(5.0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + eps), $MachinePrecision] * N[(10.0 * N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e-40], t$95$0, If[LessEqual[x, 9.8e-54], N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision] + N[(x * N[(eps * N[(5.0 + N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999992e-40 or 9.80000000000000042e-54 < x

   1. Initial program 29.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 96.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 96.0%

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

   8. Taylor expanded in eps around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-out N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

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

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

      15. +-lowering-+.f64 95.4%

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

   10. Simplified 95.4%

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

   if -1.69999999999999992e-40 < x < 9.80000000000000042e-54

   1. Initial program 98.5%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 89.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. distribute-rgt1-in N/A

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

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

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

      15. distribute-rgt1-in N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 98.2%

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

   8. Simplified 98.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

   10. Applied egg-rr 98.2%

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

   11. Taylor expanded in x around -inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \color{blue}{\left({x}^{2} \cdot \left(10 + -1 \cdot \frac{-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right)\right)}\right), \varepsilon\right) \]
   12. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(10 \cdot {x}^{2} + \left(-1 \cdot \frac{-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{2}\right)\right), \varepsilon\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \left(\left(-1 \cdot \frac{-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left({x}^{2}\right)\right), \left(\left(-1 \cdot \frac{-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left(x \cdot x\right)\right), \left(\left(-1 \cdot \frac{-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \frac{-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)}{x} \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot {x}^{2}}{x}\right)\right)\right), \varepsilon\right) \]

       8. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \frac{{x}^{2}}{x}\right)\right)\right), \varepsilon\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \frac{x \cdot x}{x}\right)\right)\right), \varepsilon\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \left(x \cdot \frac{x}{x}\right)\right)\right)\right), \varepsilon\right) \]

       11. *-inverses N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \left(x \cdot 1\right)\right)\right)\right), \varepsilon\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right)\right)\right)\right)\right), \varepsilon\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot -1\right)\right)\right)\right)\right), \varepsilon\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \left(\mathsf{neg}\left(-1 \cdot x\right)\right)\right)\right)\right), \varepsilon\right) \]

       15. mul-1-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right)\right), \varepsilon\right) \]

       16. remove-double-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot x\right)\right)\right), \varepsilon\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right), x\right)\right)\right), \varepsilon\right) \]

   13. Simplified 98.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-40}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(x + \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-54}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 10 + x \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(x + \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-40}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-53}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 10 + x \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.4e-40)
   (* (* x (* x x)) (* eps (+ (* x 5.0) (* eps 10.0))))
   (if (<= x 2.5e-53)
     (*
      eps
      (* (* eps eps) (+ (* (* x x) 10.0) (* x (* eps (+ 5.0 (/ eps x)))))))
     (* (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)) (* (* x x) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.4e-40) {
		tmp = (x * (x * x)) * (eps * ((x * 5.0) + (eps * 10.0)));
	} else if (x <= 2.5e-53) {
		tmp = eps * ((eps * eps) * (((x * x) * 10.0) + (x * (eps * (5.0 + (eps / x))))));
	} else {
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.4d-40)) then
        tmp = (x * (x * x)) * (eps * ((x * 5.0d0) + (eps * 10.0d0)))
    else if (x <= 2.5d-53) then
        tmp = eps * ((eps * eps) * (((x * x) * 10.0d0) + (x * (eps * (5.0d0 + (eps / x))))))
    else
        tmp = ((eps * 5.0d0) - (((eps * eps) * (-10.0d0)) / x)) * ((x * x) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.4e-40) {
		tmp = (x * (x * x)) * (eps * ((x * 5.0) + (eps * 10.0)));
	} else if (x <= 2.5e-53) {
		tmp = eps * ((eps * eps) * (((x * x) * 10.0) + (x * (eps * (5.0 + (eps / x))))));
	} else {
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.4e-40:
		tmp = (x * (x * x)) * (eps * ((x * 5.0) + (eps * 10.0)))
	elif x <= 2.5e-53:
		tmp = eps * ((eps * eps) * (((x * x) * 10.0) + (x * (eps * (5.0 + (eps / x))))))
	else:
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.4e-40)
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(eps * Float64(Float64(x * 5.0) + Float64(eps * 10.0))));
	elseif (x <= 2.5e-53)
		tmp = Float64(eps * Float64(Float64(eps * eps) * Float64(Float64(Float64(x * x) * 10.0) + Float64(x * Float64(eps * Float64(5.0 + Float64(eps / x)))))));
	else
		tmp = Float64(Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)) * Float64(Float64(x * x) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.4e-40)
		tmp = (x * (x * x)) * (eps * ((x * 5.0) + (eps * 10.0)));
	elseif (x <= 2.5e-53)
		tmp = eps * ((eps * eps) * (((x * x) * 10.0) + (x * (eps * (5.0 + (eps / x))))));
	else
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.4e-40], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-53], N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision] + N[(x * N[(eps * N[(5.0 + N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e-40

   1. Initial program 26.8%

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

   3. Taylor expanded in x around -inf

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

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

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

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

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

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

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

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

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

   5. Simplified 90.3%

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

   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. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 90.2%

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

   8. Simplified 90.2%

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

   if -1.4e-40 < x < 2.5e-53

   1. Initial program 98.5%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 89.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. distribute-rgt1-in N/A

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

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

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

      15. distribute-rgt1-in N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 98.2%

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

   8. Simplified 98.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

   10. Applied egg-rr 98.2%

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

   11. Taylor expanded in x around -inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \color{blue}{\left({x}^{2} \cdot \left(10 + -1 \cdot \frac{-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right)\right)}\right), \varepsilon\right) \]
   12. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(10 \cdot {x}^{2} + \left(-1 \cdot \frac{-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{2}\right)\right), \varepsilon\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \left(\left(-1 \cdot \frac{-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left({x}^{2}\right)\right), \left(\left(-1 \cdot \frac{-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left(x \cdot x\right)\right), \left(\left(-1 \cdot \frac{-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \frac{-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)}{x} \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot {x}^{2}}{x}\right)\right)\right), \varepsilon\right) \]

       8. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \frac{{x}^{2}}{x}\right)\right)\right), \varepsilon\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \frac{x \cdot x}{x}\right)\right)\right), \varepsilon\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \left(x \cdot \frac{x}{x}\right)\right)\right)\right), \varepsilon\right) \]

       11. *-inverses N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \left(x \cdot 1\right)\right)\right)\right), \varepsilon\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right)\right)\right)\right)\right), \varepsilon\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot -1\right)\right)\right)\right)\right), \varepsilon\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \left(\mathsf{neg}\left(-1 \cdot x\right)\right)\right)\right)\right), \varepsilon\right) \]

       15. mul-1-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right)\right), \varepsilon\right) \]

       16. remove-double-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \cdot x\right)\right)\right), \varepsilon\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\left(-1 \cdot \left(-5 \cdot \varepsilon + -1 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right), x\right)\right)\right), \varepsilon\right) \]

   13. Simplified 98.3%

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

   if 2.5e-53 < x

   1. Initial program 32.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.4%

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

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

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

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

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

      7. *-lowering-*.f64 98.2%

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

   7. Applied egg-rr 98.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-40}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-53}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot 10 + x \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon}{x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-50}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon + x \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e-40)
   (* (* x (* x x)) (* eps (+ (* x 5.0) (* eps 10.0))))
   (if (<= x 4.1e-50)
     (* eps (* (* eps eps) (+ (* eps eps) (* x (+ (* eps 5.0) (* x 10.0))))))
     (* (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)) (* (* x x) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e-40) {
		tmp = (x * (x * x)) * (eps * ((x * 5.0) + (eps * 10.0)));
	} else if (x <= 4.1e-50) {
		tmp = eps * ((eps * eps) * ((eps * eps) + (x * ((eps * 5.0) + (x * 10.0)))));
	} else {
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.35d-40)) then
        tmp = (x * (x * x)) * (eps * ((x * 5.0d0) + (eps * 10.0d0)))
    else if (x <= 4.1d-50) then
        tmp = eps * ((eps * eps) * ((eps * eps) + (x * ((eps * 5.0d0) + (x * 10.0d0)))))
    else
        tmp = ((eps * 5.0d0) - (((eps * eps) * (-10.0d0)) / x)) * ((x * x) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e-40) {
		tmp = (x * (x * x)) * (eps * ((x * 5.0) + (eps * 10.0)));
	} else if (x <= 4.1e-50) {
		tmp = eps * ((eps * eps) * ((eps * eps) + (x * ((eps * 5.0) + (x * 10.0)))));
	} else {
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.35e-40:
		tmp = (x * (x * x)) * (eps * ((x * 5.0) + (eps * 10.0)))
	elif x <= 4.1e-50:
		tmp = eps * ((eps * eps) * ((eps * eps) + (x * ((eps * 5.0) + (x * 10.0)))))
	else:
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e-40)
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(eps * Float64(Float64(x * 5.0) + Float64(eps * 10.0))));
	elseif (x <= 4.1e-50)
		tmp = Float64(eps * Float64(Float64(eps * eps) * Float64(Float64(eps * eps) + Float64(x * Float64(Float64(eps * 5.0) + Float64(x * 10.0))))));
	else
		tmp = Float64(Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)) * Float64(Float64(x * x) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.35e-40)
		tmp = (x * (x * x)) * (eps * ((x * 5.0) + (eps * 10.0)));
	elseif (x <= 4.1e-50)
		tmp = eps * ((eps * eps) * ((eps * eps) + (x * ((eps * 5.0) + (x * 10.0)))));
	else
		tmp = ((eps * 5.0) - (((eps * eps) * -10.0) / x)) * ((x * x) * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e-40], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.1e-50], N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] + N[(x * N[(N[(eps * 5.0), $MachinePrecision] + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35e-40

   1. Initial program 26.8%

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

   3. Taylor expanded in x around -inf

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

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

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

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

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

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

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

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

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

   5. Simplified 90.3%

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

   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. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 90.2%

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

   8. Simplified 90.2%

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

   if -1.35e-40 < x < 4.09999999999999985e-50

   1. Initial program 98.5%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 89.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. distribute-rgt1-in N/A

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

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

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

      15. distribute-rgt1-in N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 98.2%

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

   8. Simplified 98.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

   10. Applied egg-rr 98.2%

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

   if 4.09999999999999985e-50 < x

   1. Initial program 32.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.4%

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

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

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

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

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

      7. *-lowering-*.f64 98.2%

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

   7. Applied egg-rr 98.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-50}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon + x \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot 5 + \varepsilon \cdot 10\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-40}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot t\_1\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-53}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon + x \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\varepsilon \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (+ (* x 5.0) (* eps 10.0))))
   (if (<= x -1.65e-40)
     (* t_0 (* eps t_1))
     (if (<= x 4.1e-53)
       (* eps (* (* eps eps) (+ (* eps eps) (* x (+ (* eps 5.0) (* x 10.0))))))
       (* t_1 (* eps t_0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double t_1 = (x * 5.0) + (eps * 10.0);
	double tmp;
	if (x <= -1.65e-40) {
		tmp = t_0 * (eps * t_1);
	} else if (x <= 4.1e-53) {
		tmp = eps * ((eps * eps) * ((eps * eps) + (x * ((eps * 5.0) + (x * 10.0)))));
	} else {
		tmp = t_1 * (eps * 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) :: t_1
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = (x * 5.0d0) + (eps * 10.0d0)
    if (x <= (-1.65d-40)) then
        tmp = t_0 * (eps * t_1)
    else if (x <= 4.1d-53) then
        tmp = eps * ((eps * eps) * ((eps * eps) + (x * ((eps * 5.0d0) + (x * 10.0d0)))))
    else
        tmp = t_1 * (eps * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double t_1 = (x * 5.0) + (eps * 10.0);
	double tmp;
	if (x <= -1.65e-40) {
		tmp = t_0 * (eps * t_1);
	} else if (x <= 4.1e-53) {
		tmp = eps * ((eps * eps) * ((eps * eps) + (x * ((eps * 5.0) + (x * 10.0)))));
	} else {
		tmp = t_1 * (eps * t_0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	t_1 = (x * 5.0) + (eps * 10.0)
	tmp = 0
	if x <= -1.65e-40:
		tmp = t_0 * (eps * t_1)
	elif x <= 4.1e-53:
		tmp = eps * ((eps * eps) * ((eps * eps) + (x * ((eps * 5.0) + (x * 10.0)))))
	else:
		tmp = t_1 * (eps * t_0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(x * 5.0) + Float64(eps * 10.0))
	tmp = 0.0
	if (x <= -1.65e-40)
		tmp = Float64(t_0 * Float64(eps * t_1));
	elseif (x <= 4.1e-53)
		tmp = Float64(eps * Float64(Float64(eps * eps) * Float64(Float64(eps * eps) + Float64(x * Float64(Float64(eps * 5.0) + Float64(x * 10.0))))));
	else
		tmp = Float64(t_1 * Float64(eps * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	t_1 = (x * 5.0) + (eps * 10.0);
	tmp = 0.0;
	if (x <= -1.65e-40)
		tmp = t_0 * (eps * t_1);
	elseif (x <= 4.1e-53)
		tmp = eps * ((eps * eps) * ((eps * eps) + (x * ((eps * 5.0) + (x * 10.0)))));
	else
		tmp = t_1 * (eps * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e-40], N[(t$95$0 * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.1e-53], N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] + N[(x * N[(N[(eps * 5.0), $MachinePrecision] + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.64999999999999996e-40

   1. Initial program 26.8%

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

   3. Taylor expanded in x around -inf

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

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

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

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

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

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

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

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

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

   5. Simplified 90.3%

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

   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. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 90.2%

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

   8. Simplified 90.2%

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

   if -1.64999999999999996e-40 < x < 4.1000000000000001e-53

   1. Initial program 98.5%

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

   3. Taylor expanded in eps around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*l* N/A

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

   5. Simplified 89.3%

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

   6. Taylor expanded in eps around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. distribute-rgt1-in N/A

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

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

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

      15. distribute-rgt1-in N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 98.2%

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

   8. Simplified 98.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

   10. Applied egg-rr 98.2%

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

   if 4.1000000000000001e-53 < x

   1. Initial program 32.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.4%

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. cube-unmult N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. flip-- N/A

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

      7. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 67.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 25 - \frac{\varepsilon \cdot \frac{\varepsilon \cdot -10}{x}}{\frac{\frac{x}{-10}}{\varepsilon \cdot \varepsilon}}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\varepsilon \cdot 5 + \varepsilon \cdot \frac{\varepsilon \cdot -10}{x}}} \]

   8. Taylor expanded in eps around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

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

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

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

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

      22. *-commutative N/A

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

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

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

      24. *-lowering-*.f64 98.1%

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

   10. Simplified 98.1%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-40}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-53}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon + x \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 5 + \varepsilon \cdot 10\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot 5 + \varepsilon \cdot 10\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{-40}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot t\_1\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-58}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 10 + \varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\varepsilon \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (+ (* x 5.0) (* eps 10.0))))
   (if (<= x -1.3e-40)
     (* t_0 (* eps t_1))
     (if (<= x 5.8e-58)
       (* (* eps (* eps eps)) (+ (* (* x x) 10.0) (* eps (+ eps (* x 5.0)))))
       (* t_1 (* eps t_0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double t_1 = (x * 5.0) + (eps * 10.0);
	double tmp;
	if (x <= -1.3e-40) {
		tmp = t_0 * (eps * t_1);
	} else if (x <= 5.8e-58) {
		tmp = (eps * (eps * eps)) * (((x * x) * 10.0) + (eps * (eps + (x * 5.0))));
	} else {
		tmp = t_1 * (eps * 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) :: t_1
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = (x * 5.0d0) + (eps * 10.0d0)
    if (x <= (-1.3d-40)) then
        tmp = t_0 * (eps * t_1)
    else if (x <= 5.8d-58) then
        tmp = (eps * (eps * eps)) * (((x * x) * 10.0d0) + (eps * (eps + (x * 5.0d0))))
    else
        tmp = t_1 * (eps * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double t_1 = (x * 5.0) + (eps * 10.0);
	double tmp;
	if (x <= -1.3e-40) {
		tmp = t_0 * (eps * t_1);
	} else if (x <= 5.8e-58) {
		tmp = (eps * (eps * eps)) * (((x * x) * 10.0) + (eps * (eps + (x * 5.0))));
	} else {
		tmp = t_1 * (eps * t_0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	t_1 = (x * 5.0) + (eps * 10.0)
	tmp = 0
	if x <= -1.3e-40:
		tmp = t_0 * (eps * t_1)
	elif x <= 5.8e-58:
		tmp = (eps * (eps * eps)) * (((x * x) * 10.0) + (eps * (eps + (x * 5.0))))
	else:
		tmp = t_1 * (eps * t_0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(x * 5.0) + Float64(eps * 10.0))
	tmp = 0.0
	if (x <= -1.3e-40)
		tmp = Float64(t_0 * Float64(eps * t_1));
	elseif (x <= 5.8e-58)
		tmp = Float64(Float64(eps * Float64(eps * eps)) * Float64(Float64(Float64(x * x) * 10.0) + Float64(eps * Float64(eps + Float64(x * 5.0)))));
	else
		tmp = Float64(t_1 * Float64(eps * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	t_1 = (x * 5.0) + (eps * 10.0);
	tmp = 0.0;
	if (x <= -1.3e-40)
		tmp = t_0 * (eps * t_1);
	elseif (x <= 5.8e-58)
		tmp = (eps * (eps * eps)) * (((x * x) * 10.0) + (eps * (eps + (x * 5.0))));
	else
		tmp = t_1 * (eps * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e-40], N[(t$95$0 * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-58], N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision] + N[(eps * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3000000000000001e-40

   1. Initial program 26.8%

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

   3. Taylor expanded in x around -inf

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

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

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

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

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

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

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

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

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

   5. Simplified 90.3%

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

   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. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

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

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

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

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

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(x \cdot 5\right), \left(10 \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \left(10 \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      13. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \varepsilon\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \varepsilon\right)\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      17. *-lowering-*.f64 90.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   8. Simplified 90.2%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(x \cdot 5 + 10 \cdot \varepsilon\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if -1.3000000000000001e-40 < x < 5.7999999999999998e-58

   1. Initial program 98.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot {\varepsilon}^{5}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + -1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\left(\frac{5 \cdot x - \frac{\left(x \cdot x\right) \cdot -10}{\varepsilon}}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left({x}^{2}\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left(x \cdot x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(4 + 1\right) \cdot x\right)\right)\right)\right) \]

      13. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(x + \color{blue}{4 \cdot x}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(x + 4 \cdot x\right)}\right)\right)\right)\right) \]

      15. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(\left(4 + 1\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(5 \cdot x\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right)\right)\right) \]

   8. Simplified 98.2%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]

   if 5.7999999999999998e-58 < x

   1. Initial program 32.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.4%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {x}^{\left(3 + 1\right)} \cdot \left(\varepsilon \cdot \color{blue}{5} - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \]

      2. pow-plus N/A

         \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \]

      3. cube-unmult N/A

         \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\color{blue}{\varepsilon} \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]

      6. flip-- N/A

         \[\leadsto \frac{\left(\varepsilon \cdot 5\right) \cdot \left(\varepsilon \cdot 5\right) - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x} \cdot \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}}{\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}} \cdot \left(\color{blue}{x} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\left(\left(\varepsilon \cdot 5\right) \cdot \left(\varepsilon \cdot 5\right) - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x} \cdot \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\varepsilon \cdot 5\right) \cdot \left(\varepsilon \cdot 5\right) - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x} \cdot \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \color{blue}{\left(\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)}\right) \]

   7. Applied egg-rr 67.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 25 - \frac{\varepsilon \cdot \frac{\varepsilon \cdot -10}{x}}{\frac{\frac{x}{-10}}{\varepsilon \cdot \varepsilon}}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\varepsilon \cdot 5 + \varepsilon \cdot \frac{\varepsilon \cdot -10}{x}}} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) + \color{blue}{\varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot 5\right) \cdot {x}^{4} + \color{blue}{\varepsilon} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{\left(3 + 1\right)} + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      5. pow-plus N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{3} \cdot x\right) + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{3}\right) \cdot x + \color{blue}{\varepsilon} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(5 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \cdot x + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(5 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) + \color{blue}{\varepsilon} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(x \cdot 5\right) \cdot \left(\varepsilon \cdot {x}^{3}\right) + \color{blue}{\varepsilon} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \left(5 \cdot x\right) \cdot \left(\varepsilon \cdot {x}^{3}\right) + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(5 \cdot x\right) \cdot \left(\varepsilon \cdot {x}^{3}\right) + \left(\varepsilon \cdot 10\right) \cdot \color{blue}{\left(\varepsilon \cdot {x}^{3}\right)} \]

      12. *-commutative N/A

          \[\leadsto \left(5 \cdot x\right) \cdot \left(\varepsilon \cdot {x}^{3}\right) + \left(10 \cdot \varepsilon\right) \cdot \left(\color{blue}{\varepsilon} \cdot {x}^{3}\right) \]

      13. distribute-rgt-out N/A

          \[\leadsto \left(\varepsilon \cdot {x}^{3}\right) \cdot \color{blue}{\left(5 \cdot x + 10 \cdot \varepsilon\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {x}^{3}\right), \color{blue}{\left(5 \cdot x + 10 \cdot \varepsilon\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({x}^{3}\right)\right), \left(\color{blue}{5 \cdot x} + 10 \cdot \varepsilon\right)\right) \]

      16. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(5 \cdot \color{blue}{x} + 10 \cdot \varepsilon\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(x \cdot {x}^{2}\right)\right), \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(5 \cdot \color{blue}{x} + 10 \cdot \varepsilon\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \]

      21. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(5 \cdot x\right), \color{blue}{\left(10 \cdot \varepsilon\right)}\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(x \cdot 5\right), \left(\color{blue}{10} \cdot \varepsilon\right)\right)\right) \]

      23. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \left(\color{blue}{10} \cdot \varepsilon\right)\right)\right) \]

      24. *-lowering-*.f64 98.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \color{blue}{\varepsilon}\right)\right)\right) \]

   10. Simplified 98.1%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot 5 + 10 \cdot \varepsilon\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-40}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-58}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 10 + \varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 5 + \varepsilon \cdot 10\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot 5 + \varepsilon \cdot 10\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot t\_1\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-57}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\varepsilon \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (+ (* x 5.0) (* eps 10.0))))
   (if (<= x -1.35e-40)
     (* t_0 (* eps t_1))
     (if (<= x 4.8e-57)
       (* eps (* (* eps eps) (* eps (+ eps (* x 5.0)))))
       (* t_1 (* eps t_0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x * x);
	double t_1 = (x * 5.0) + (eps * 10.0);
	double tmp;
	if (x <= -1.35e-40) {
		tmp = t_0 * (eps * t_1);
	} else if (x <= 4.8e-57) {
		tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0))));
	} else {
		tmp = t_1 * (eps * 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) :: t_1
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = (x * 5.0d0) + (eps * 10.0d0)
    if (x <= (-1.35d-40)) then
        tmp = t_0 * (eps * t_1)
    else if (x <= 4.8d-57) then
        tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0d0))))
    else
        tmp = t_1 * (eps * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x * x);
	double t_1 = (x * 5.0) + (eps * 10.0);
	double tmp;
	if (x <= -1.35e-40) {
		tmp = t_0 * (eps * t_1);
	} else if (x <= 4.8e-57) {
		tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0))));
	} else {
		tmp = t_1 * (eps * t_0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x * x)
	t_1 = (x * 5.0) + (eps * 10.0)
	tmp = 0
	if x <= -1.35e-40:
		tmp = t_0 * (eps * t_1)
	elif x <= 4.8e-57:
		tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0))))
	else:
		tmp = t_1 * (eps * t_0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(x * 5.0) + Float64(eps * 10.0))
	tmp = 0.0
	if (x <= -1.35e-40)
		tmp = Float64(t_0 * Float64(eps * t_1));
	elseif (x <= 4.8e-57)
		tmp = Float64(eps * Float64(Float64(eps * eps) * Float64(eps * Float64(eps + Float64(x * 5.0)))));
	else
		tmp = Float64(t_1 * Float64(eps * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x * x);
	t_1 = (x * 5.0) + (eps * 10.0);
	tmp = 0.0;
	if (x <= -1.35e-40)
		tmp = t_0 * (eps * t_1);
	elseif (x <= 4.8e-57)
		tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0))));
	else
		tmp = t_1 * (eps * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e-40], N[(t$95$0 * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e-57], N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35e-40

   1. Initial program 26.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\color{blue}{\varepsilon} + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\varepsilon + \left(4 \cdot \varepsilon + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon + 4 \cdot \varepsilon\right), \color{blue}{\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right)\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\left(4 + 1\right) \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(5 \cdot \varepsilon\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2}} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\left(\varepsilon \cdot 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \left(\frac{\color{blue}{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}}{x}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, 5\right), \mathsf{/.f64}\left(\left(-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

   5. Simplified 90.3%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   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. *-commutative N/A

         \[\leadsto \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{{x}^{3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(5 \cdot \left(x \cdot \varepsilon\right) + 10 \cdot {\varepsilon}^{2}\right), \left({x}^{3}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(5 \cdot x\right) \cdot \varepsilon + 10 \cdot {\varepsilon}^{2}\right), \left({x}^{3}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(5 \cdot x\right) \cdot \varepsilon + 10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{3}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(5 \cdot x\right) \cdot \varepsilon + \left(10 \cdot \varepsilon\right) \cdot \varepsilon\right), \left({x}^{3}\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(5 \cdot x + 10 \cdot \varepsilon\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(5 \cdot x\right), \left(10 \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(x \cdot 5\right), \left(10 \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \left(10 \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      13. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \varepsilon\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \varepsilon\right)\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      17. *-lowering-*.f64 90.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   8. Simplified 90.2%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(x \cdot 5 + 10 \cdot \varepsilon\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if -1.35e-40 < x < 4.80000000000000012e-57

   1. Initial program 98.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot {\varepsilon}^{5}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + -1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\left(\frac{5 \cdot x - \frac{\left(x \cdot x\right) \cdot -10}{\varepsilon}}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left({x}^{2}\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left(x \cdot x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(4 + 1\right) \cdot x\right)\right)\right)\right) \]

      13. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(x + \color{blue}{4 \cdot x}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(x + 4 \cdot x\right)}\right)\right)\right)\right) \]

      15. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(\left(4 + 1\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(5 \cdot x\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right)\right)\right) \]

   8. Simplified 98.2%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right), \color{blue}{\varepsilon}\right) \]

   10. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon + x \cdot \left(10 \cdot x + \varepsilon \cdot 5\right)\right)\right) \cdot \varepsilon} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + {\varepsilon}^{2}\right)}\right), \varepsilon\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right), \varepsilon\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} + \left(\varepsilon \cdot x\right) \cdot 5\right)\right), \varepsilon\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot 5\right)\right)\right), \varepsilon\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} + \varepsilon \cdot \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\varepsilon \cdot \varepsilon + \varepsilon \cdot \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

       6. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right), \varepsilon\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + 5 \cdot x\right)\right)\right), \varepsilon\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(5 \cdot x\right)\right)\right)\right), \varepsilon\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(x \cdot 5\right)\right)\right)\right), \varepsilon\right) \]

       10. *-lowering-*.f64 98.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 5\right)\right)\right)\right), \varepsilon\right) \]

   13. Simplified 98.1%

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)}\right) \cdot \varepsilon \]

   if 4.80000000000000012e-57 < x

   1. Initial program 32.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.4%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {x}^{\left(3 + 1\right)} \cdot \left(\varepsilon \cdot \color{blue}{5} - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \]

      2. pow-plus N/A

         \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \]

      3. cube-unmult N/A

         \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\color{blue}{\varepsilon} \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]

      6. flip-- N/A

         \[\leadsto \frac{\left(\varepsilon \cdot 5\right) \cdot \left(\varepsilon \cdot 5\right) - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x} \cdot \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}}{\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}} \cdot \left(\color{blue}{x} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\left(\left(\varepsilon \cdot 5\right) \cdot \left(\varepsilon \cdot 5\right) - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x} \cdot \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\varepsilon \cdot 5\right) \cdot \left(\varepsilon \cdot 5\right) - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x} \cdot \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \color{blue}{\left(\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)}\right) \]

   7. Applied egg-rr 67.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 25 - \frac{\varepsilon \cdot \frac{\varepsilon \cdot -10}{x}}{\frac{\frac{x}{-10}}{\varepsilon \cdot \varepsilon}}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\varepsilon \cdot 5 + \varepsilon \cdot \frac{\varepsilon \cdot -10}{x}}} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) + \color{blue}{\varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot 5\right) \cdot {x}^{4} + \color{blue}{\varepsilon} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{\left(3 + 1\right)} + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      5. pow-plus N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{3} \cdot x\right) + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{3}\right) \cdot x + \color{blue}{\varepsilon} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(5 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \cdot x + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(5 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) + \color{blue}{\varepsilon} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(x \cdot 5\right) \cdot \left(\varepsilon \cdot {x}^{3}\right) + \color{blue}{\varepsilon} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \left(5 \cdot x\right) \cdot \left(\varepsilon \cdot {x}^{3}\right) + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(5 \cdot x\right) \cdot \left(\varepsilon \cdot {x}^{3}\right) + \left(\varepsilon \cdot 10\right) \cdot \color{blue}{\left(\varepsilon \cdot {x}^{3}\right)} \]

      12. *-commutative N/A

          \[\leadsto \left(5 \cdot x\right) \cdot \left(\varepsilon \cdot {x}^{3}\right) + \left(10 \cdot \varepsilon\right) \cdot \left(\color{blue}{\varepsilon} \cdot {x}^{3}\right) \]

      13. distribute-rgt-out N/A

          \[\leadsto \left(\varepsilon \cdot {x}^{3}\right) \cdot \color{blue}{\left(5 \cdot x + 10 \cdot \varepsilon\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {x}^{3}\right), \color{blue}{\left(5 \cdot x + 10 \cdot \varepsilon\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({x}^{3}\right)\right), \left(\color{blue}{5 \cdot x} + 10 \cdot \varepsilon\right)\right) \]

      16. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(5 \cdot \color{blue}{x} + 10 \cdot \varepsilon\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(x \cdot {x}^{2}\right)\right), \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(5 \cdot \color{blue}{x} + 10 \cdot \varepsilon\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \]

      21. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(5 \cdot x\right), \color{blue}{\left(10 \cdot \varepsilon\right)}\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(x \cdot 5\right), \left(\color{blue}{10} \cdot \varepsilon\right)\right)\right) \]

      23. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \left(\color{blue}{10} \cdot \varepsilon\right)\right)\right) \]

      24. *-lowering-*.f64 98.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \color{blue}{\varepsilon}\right)\right)\right) \]

   10. Simplified 98.1%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot 5 + 10 \cdot \varepsilon\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-57}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 5 + \varepsilon \cdot 10\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot 5 + \varepsilon \cdot 10\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ (* x 5.0) (* eps 10.0)) (* eps (* x (* x x))))))
   (if (<= x -3.7e-40)
     t_0
     (if (<= x 3.4e-55)
       (* eps (* (* eps eps) (* eps (+ eps (* x 5.0)))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = ((x * 5.0) + (eps * 10.0)) * (eps * (x * (x * x)));
	double tmp;
	if (x <= -3.7e-40) {
		tmp = t_0;
	} else if (x <= 3.4e-55) {
		tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * 5.0d0) + (eps * 10.0d0)) * (eps * (x * (x * x)))
    if (x <= (-3.7d-40)) then
        tmp = t_0
    else if (x <= 3.4d-55) then
        tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((x * 5.0) + (eps * 10.0)) * (eps * (x * (x * x)));
	double tmp;
	if (x <= -3.7e-40) {
		tmp = t_0;
	} else if (x <= 3.4e-55) {
		tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((x * 5.0) + (eps * 10.0)) * (eps * (x * (x * x)))
	tmp = 0
	if x <= -3.7e-40:
		tmp = t_0
	elif x <= 3.4e-55:
		tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(x * 5.0) + Float64(eps * 10.0)) * Float64(eps * Float64(x * Float64(x * x))))
	tmp = 0.0
	if (x <= -3.7e-40)
		tmp = t_0;
	elseif (x <= 3.4e-55)
		tmp = Float64(eps * Float64(Float64(eps * eps) * Float64(eps * Float64(eps + Float64(x * 5.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x * 5.0) + (eps * 10.0)) * (eps * (x * (x * x)));
	tmp = 0.0;
	if (x <= -3.7e-40)
		tmp = t_0;
	elseif (x <= 3.4e-55)
		tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e-40], t$95$0, If[LessEqual[x, 3.4e-55], N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.69999999999999998e-40 or 3.39999999999999973e-55 < x

   1. Initial program 29.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 94.3%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {x}^{\left(3 + 1\right)} \cdot \left(\varepsilon \cdot \color{blue}{5} - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \]

      2. pow-plus N/A

         \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \]

      3. cube-unmult N/A

         \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\color{blue}{\varepsilon} \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]

      6. flip-- N/A

         \[\leadsto \frac{\left(\varepsilon \cdot 5\right) \cdot \left(\varepsilon \cdot 5\right) - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x} \cdot \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}}{\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}} \cdot \left(\color{blue}{x} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\left(\left(\varepsilon \cdot 5\right) \cdot \left(\varepsilon \cdot 5\right) - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x} \cdot \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\varepsilon \cdot 5\right) \cdot \left(\varepsilon \cdot 5\right) - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x} \cdot \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \color{blue}{\left(\varepsilon \cdot 5 + \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)}\right) \]

   7. Applied egg-rr 56.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 25 - \frac{\varepsilon \cdot \frac{\varepsilon \cdot -10}{x}}{\frac{\frac{x}{-10}}{\varepsilon \cdot \varepsilon}}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\varepsilon \cdot 5 + \varepsilon \cdot \frac{\varepsilon \cdot -10}{x}}} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4}\right) + \color{blue}{\varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot 5\right) \cdot {x}^{4} + \color{blue}{\varepsilon} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{\left(3 + 1\right)} + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      5. pow-plus N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{3} \cdot x\right) + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{3}\right) \cdot x + \color{blue}{\varepsilon} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(5 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \cdot x + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(5 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) + \color{blue}{\varepsilon} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(x \cdot 5\right) \cdot \left(\varepsilon \cdot {x}^{3}\right) + \color{blue}{\varepsilon} \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \left(5 \cdot x\right) \cdot \left(\varepsilon \cdot {x}^{3}\right) + \varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(5 \cdot x\right) \cdot \left(\varepsilon \cdot {x}^{3}\right) + \left(\varepsilon \cdot 10\right) \cdot \color{blue}{\left(\varepsilon \cdot {x}^{3}\right)} \]

      12. *-commutative N/A

          \[\leadsto \left(5 \cdot x\right) \cdot \left(\varepsilon \cdot {x}^{3}\right) + \left(10 \cdot \varepsilon\right) \cdot \left(\color{blue}{\varepsilon} \cdot {x}^{3}\right) \]

      13. distribute-rgt-out N/A

          \[\leadsto \left(\varepsilon \cdot {x}^{3}\right) \cdot \color{blue}{\left(5 \cdot x + 10 \cdot \varepsilon\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {x}^{3}\right), \color{blue}{\left(5 \cdot x + 10 \cdot \varepsilon\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({x}^{3}\right)\right), \left(\color{blue}{5 \cdot x} + 10 \cdot \varepsilon\right)\right) \]

      16. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(5 \cdot \color{blue}{x} + 10 \cdot \varepsilon\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(x \cdot {x}^{2}\right)\right), \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(5 \cdot \color{blue}{x} + 10 \cdot \varepsilon\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(5 \cdot x + 10 \cdot \varepsilon\right)\right) \]

      21. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(5 \cdot x\right), \color{blue}{\left(10 \cdot \varepsilon\right)}\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(x \cdot 5\right), \left(\color{blue}{10} \cdot \varepsilon\right)\right)\right) \]

      23. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \left(\color{blue}{10} \cdot \varepsilon\right)\right)\right) \]

      24. *-lowering-*.f64 94.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 5\right), \mathsf{*.f64}\left(10, \color{blue}{\varepsilon}\right)\right)\right) \]

   10. Simplified 94.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot 5 + 10 \cdot \varepsilon\right)} \]

   if -3.69999999999999998e-40 < x < 3.39999999999999973e-55

   1. Initial program 98.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot {\varepsilon}^{5}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + -1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\left(\frac{5 \cdot x - \frac{\left(x \cdot x\right) \cdot -10}{\varepsilon}}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left({x}^{2}\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left(x \cdot x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(4 + 1\right) \cdot x\right)\right)\right)\right) \]

      13. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(x + \color{blue}{4 \cdot x}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(x + 4 \cdot x\right)}\right)\right)\right)\right) \]

      15. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(\left(4 + 1\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(5 \cdot x\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right)\right)\right) \]

   8. Simplified 98.2%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right), \color{blue}{\varepsilon}\right) \]

   10. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon + x \cdot \left(10 \cdot x + \varepsilon \cdot 5\right)\right)\right) \cdot \varepsilon} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + {\varepsilon}^{2}\right)}\right), \varepsilon\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right), \varepsilon\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} + \left(\varepsilon \cdot x\right) \cdot 5\right)\right), \varepsilon\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot 5\right)\right)\right), \varepsilon\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} + \varepsilon \cdot \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\varepsilon \cdot \varepsilon + \varepsilon \cdot \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

       6. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right), \varepsilon\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + 5 \cdot x\right)\right)\right), \varepsilon\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(5 \cdot x\right)\right)\right)\right), \varepsilon\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(x \cdot 5\right)\right)\right)\right), \varepsilon\right) \]

       10. *-lowering-*.f64 98.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 5\right)\right)\right)\right), \varepsilon\right) \]

   13. Simplified 98.1%

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)}\right) \cdot \varepsilon \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-40}:\\ \;\;\;\;\left(x \cdot 5 + \varepsilon \cdot 10\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 5 + \varepsilon \cdot 10\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 97.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* 5.0 (* (* x x) (* x x))))))
   (if (<= x -1.35e-40)
     t_0
     (if (<= x 4.2e-56)
       (* eps (* (* eps eps) (* eps (+ eps (* x 5.0)))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (5.0 * ((x * x) * (x * x)));
	double tmp;
	if (x <= -1.35e-40) {
		tmp = t_0;
	} else if (x <= 4.2e-56) {
		tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * (5.0d0 * ((x * x) * (x * x)))
    if (x <= (-1.35d-40)) then
        tmp = t_0
    else if (x <= 4.2d-56) then
        tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (5.0 * ((x * x) * (x * x)));
	double tmp;
	if (x <= -1.35e-40) {
		tmp = t_0;
	} else if (x <= 4.2e-56) {
		tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (5.0 * ((x * x) * (x * x)))
	tmp = 0
	if x <= -1.35e-40:
		tmp = t_0
	elif x <= 4.2e-56:
		tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))))
	tmp = 0.0
	if (x <= -1.35e-40)
		tmp = t_0;
	elseif (x <= 4.2e-56)
		tmp = Float64(eps * Float64(Float64(eps * eps) * Float64(eps * Float64(eps + Float64(x * 5.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * (5.0 * ((x * x) * (x * x)));
	tmp = 0.0;
	if (x <= -1.35e-40)
		tmp = t_0;
	elseif (x <= 4.2e-56)
		tmp = eps * ((eps * eps) * (eps * (eps + (x * 5.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e-40], t$95$0, If[LessEqual[x, 4.2e-56], N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35e-40 or 4.20000000000000012e-56 < x

   1. Initial program 29.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 96.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 96.0%

      \[\leadsto \color{blue}{\left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 10\right) + \varepsilon \cdot \left(x \cdot \left(x \cdot 10\right) + \varepsilon \cdot \left(5 \cdot x\right)\right)\right)\right) \cdot \varepsilon} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \color{blue}{\left(5 \cdot \left({\varepsilon}^{3} \cdot x\right)\right)}\right), \varepsilon\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\left({\varepsilon}^{3} \cdot x\right) \cdot 5\right)\right), \varepsilon\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left({\varepsilon}^{3} \cdot \left(x \cdot 5\right)\right)\right), \varepsilon\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left({\varepsilon}^{3} \cdot \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(x \cdot 5\right)\right)\right), \varepsilon\right) \]

      11. *-lowering-*.f64 91.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(x, 5\right)\right)\right), \varepsilon\right) \]

   10. Simplified 91.9%

       \[\leadsto \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot 5\right)}\right) \cdot \varepsilon \]

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(5 \cdot {x}^{4}\right)}, \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \left({x}^{4}\right)\right), \varepsilon\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \left({x}^{\left(2 \cdot 2\right)}\right)\right), \varepsilon\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \left({x}^{2} \cdot {x}^{2}\right)\right), \varepsilon\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right)\right), \varepsilon\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right)\right), \varepsilon\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right)\right), \varepsilon\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right)\right), \varepsilon\right) \]

       8. *-lowering-*.f64 92.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \varepsilon\right) \]

   13. Simplified 92.0%

       \[\leadsto \color{blue}{\left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)} \cdot \varepsilon \]

   if -1.35e-40 < x < 4.20000000000000012e-56

   1. Initial program 98.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot {\varepsilon}^{5}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + -1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\left(\frac{5 \cdot x - \frac{\left(x \cdot x\right) \cdot -10}{\varepsilon}}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left({x}^{2}\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left(x \cdot x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(4 + 1\right) \cdot x\right)\right)\right)\right) \]

      13. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(x + \color{blue}{4 \cdot x}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(x + 4 \cdot x\right)}\right)\right)\right)\right) \]

      15. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(\left(4 + 1\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(5 \cdot x\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right)\right)\right) \]

   8. Simplified 98.2%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right), \color{blue}{\varepsilon}\right) \]

   10. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon + x \cdot \left(10 \cdot x + \varepsilon \cdot 5\right)\right)\right) \cdot \varepsilon} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + {\varepsilon}^{2}\right)}\right), \varepsilon\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right), \varepsilon\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} + \left(\varepsilon \cdot x\right) \cdot 5\right)\right), \varepsilon\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot 5\right)\right)\right), \varepsilon\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} + \varepsilon \cdot \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\varepsilon \cdot \varepsilon + \varepsilon \cdot \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

       6. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right), \varepsilon\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + 5 \cdot x\right)\right)\right), \varepsilon\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(5 \cdot x\right)\right)\right)\right), \varepsilon\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(x \cdot 5\right)\right)\right)\right), \varepsilon\right) \]

       10. *-lowering-*.f64 98.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 5\right)\right)\right)\right), \varepsilon\right) \]

   13. Simplified 98.1%

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)}\right) \cdot \varepsilon \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 97.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* 5.0 (* (* x x) (* x x))))))
   (if (<= x -1.3e-40)
     t_0
     (if (<= x 1.32e-52)
       (* (* eps (* eps eps)) (* eps (+ eps (* x 5.0))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (5.0 * ((x * x) * (x * x)));
	double tmp;
	if (x <= -1.3e-40) {
		tmp = t_0;
	} else if (x <= 1.32e-52) {
		tmp = (eps * (eps * eps)) * (eps * (eps + (x * 5.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * (5.0d0 * ((x * x) * (x * x)))
    if (x <= (-1.3d-40)) then
        tmp = t_0
    else if (x <= 1.32d-52) then
        tmp = (eps * (eps * eps)) * (eps * (eps + (x * 5.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (5.0 * ((x * x) * (x * x)));
	double tmp;
	if (x <= -1.3e-40) {
		tmp = t_0;
	} else if (x <= 1.32e-52) {
		tmp = (eps * (eps * eps)) * (eps * (eps + (x * 5.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (5.0 * ((x * x) * (x * x)))
	tmp = 0
	if x <= -1.3e-40:
		tmp = t_0
	elif x <= 1.32e-52:
		tmp = (eps * (eps * eps)) * (eps * (eps + (x * 5.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))))
	tmp = 0.0
	if (x <= -1.3e-40)
		tmp = t_0;
	elseif (x <= 1.32e-52)
		tmp = Float64(Float64(eps * Float64(eps * eps)) * Float64(eps * Float64(eps + Float64(x * 5.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * (5.0 * ((x * x) * (x * x)));
	tmp = 0.0;
	if (x <= -1.3e-40)
		tmp = t_0;
	elseif (x <= 1.32e-52)
		tmp = (eps * (eps * eps)) * (eps * (eps + (x * 5.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e-40], t$95$0, If[LessEqual[x, 1.32e-52], N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3000000000000001e-40 or 1.32000000000000002e-52 < x

   1. Initial program 29.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 96.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 96.0%

      \[\leadsto \color{blue}{\left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 10\right) + \varepsilon \cdot \left(x \cdot \left(x \cdot 10\right) + \varepsilon \cdot \left(5 \cdot x\right)\right)\right)\right) \cdot \varepsilon} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \color{blue}{\left(5 \cdot \left({\varepsilon}^{3} \cdot x\right)\right)}\right), \varepsilon\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\left({\varepsilon}^{3} \cdot x\right) \cdot 5\right)\right), \varepsilon\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left({\varepsilon}^{3} \cdot \left(x \cdot 5\right)\right)\right), \varepsilon\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left({\varepsilon}^{3} \cdot \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(x \cdot 5\right)\right)\right), \varepsilon\right) \]

      11. *-lowering-*.f64 91.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(x, 5\right)\right)\right), \varepsilon\right) \]

   10. Simplified 91.9%

       \[\leadsto \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot 5\right)}\right) \cdot \varepsilon \]

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(5 \cdot {x}^{4}\right)}, \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \left({x}^{4}\right)\right), \varepsilon\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \left({x}^{\left(2 \cdot 2\right)}\right)\right), \varepsilon\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \left({x}^{2} \cdot {x}^{2}\right)\right), \varepsilon\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right)\right), \varepsilon\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right)\right), \varepsilon\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right)\right), \varepsilon\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right)\right), \varepsilon\right) \]

       8. *-lowering-*.f64 92.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \varepsilon\right) \]

   13. Simplified 92.0%

       \[\leadsto \color{blue}{\left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)} \cdot \varepsilon \]

   if -1.3000000000000001e-40 < x < 1.32000000000000002e-52

   1. Initial program 98.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot {\varepsilon}^{5}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + -1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\left(\frac{5 \cdot x - \frac{\left(x \cdot x\right) \cdot -10}{\varepsilon}}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left({x}^{2}\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left(x \cdot x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(4 + 1\right) \cdot x\right)\right)\right)\right) \]

      13. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(x + \color{blue}{4 \cdot x}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(x + 4 \cdot x\right)}\right)\right)\right)\right) \]

      15. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(\left(4 + 1\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(5 \cdot x\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right)\right)\right) \]

   8. Simplified 98.2%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + {\varepsilon}^{2}\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(5 \cdot \left(x \cdot \varepsilon\right) + {\varepsilon}^{2}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\left(5 \cdot x\right) \cdot \varepsilon + {\color{blue}{\varepsilon}}^{2}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\left(5 \cdot x\right) \cdot \varepsilon + \varepsilon \cdot \color{blue}{\varepsilon}\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\varepsilon \cdot \color{blue}{\left(5 \cdot x + \varepsilon\right)}\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\varepsilon \cdot \left(\varepsilon + \color{blue}{5 \cdot x}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(5 \cdot x\right)}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(x \cdot \color{blue}{5}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 98.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{5}\right)\right)\right)\right) \]

   11. Simplified 98.1%

       \[\leadsto \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-40}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-52}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \left(\varepsilon + x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 97.2% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.96 \cdot 10^{-56}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* 5.0 (* (* x x) (* x x))))))
   (if (<= x -1.5e-40)
     t_0
     (if (<= x 1.96e-56) (* eps (* eps (* eps (* eps eps)))) t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (5.0 * ((x * x) * (x * x)));
	double tmp;
	if (x <= -1.5e-40) {
		tmp = t_0;
	} else if (x <= 1.96e-56) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * (5.0d0 * ((x * x) * (x * x)))
    if (x <= (-1.5d-40)) then
        tmp = t_0
    else if (x <= 1.96d-56) then
        tmp = eps * (eps * (eps * (eps * eps)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (5.0 * ((x * x) * (x * x)));
	double tmp;
	if (x <= -1.5e-40) {
		tmp = t_0;
	} else if (x <= 1.96e-56) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (5.0 * ((x * x) * (x * x)))
	tmp = 0
	if x <= -1.5e-40:
		tmp = t_0
	elif x <= 1.96e-56:
		tmp = eps * (eps * (eps * (eps * eps)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))))
	tmp = 0.0
	if (x <= -1.5e-40)
		tmp = t_0;
	elseif (x <= 1.96e-56)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * (5.0 * ((x * x) * (x * x)));
	tmp = 0.0;
	if (x <= -1.5e-40)
		tmp = t_0;
	elseif (x <= 1.96e-56)
		tmp = eps * (eps * (eps * (eps * eps)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5e-40], t$95$0, If[LessEqual[x, 1.96e-56], 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 < -1.5000000000000001e-40 or 1.95999999999999995e-56 < x

   1. Initial program 29.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 96.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 10 + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 10 + 5 \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
   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 96.0%

      \[\leadsto \color{blue}{\left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 10\right) + \varepsilon \cdot \left(x \cdot \left(x \cdot 10\right) + \varepsilon \cdot \left(5 \cdot x\right)\right)\right)\right) \cdot \varepsilon} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \color{blue}{\left(5 \cdot \left({\varepsilon}^{3} \cdot x\right)\right)}\right), \varepsilon\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\left({\varepsilon}^{3} \cdot x\right) \cdot 5\right)\right), \varepsilon\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left({\varepsilon}^{3} \cdot \left(x \cdot 5\right)\right)\right), \varepsilon\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left({\varepsilon}^{3} \cdot \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(5 \cdot x\right)\right)\right), \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(x \cdot 5\right)\right)\right), \varepsilon\right) \]

      11. *-lowering-*.f64 91.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(x, 5\right)\right)\right), \varepsilon\right) \]

   10. Simplified 91.9%

       \[\leadsto \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot 5\right)}\right) \cdot \varepsilon \]

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(5 \cdot {x}^{4}\right)}, \varepsilon\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \left({x}^{4}\right)\right), \varepsilon\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \left({x}^{\left(2 \cdot 2\right)}\right)\right), \varepsilon\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \left({x}^{2} \cdot {x}^{2}\right)\right), \varepsilon\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right)\right), \varepsilon\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right)\right), \varepsilon\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right)\right), \varepsilon\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right)\right), \varepsilon\right) \]

       8. *-lowering-*.f64 92.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(5, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \varepsilon\right) \]

   13. Simplified 92.0%

       \[\leadsto \color{blue}{\left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)} \cdot \varepsilon \]

   if -1.5000000000000001e-40 < x < 1.95999999999999995e-56

   1. Initial program 98.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot {\varepsilon}^{5}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + -1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\left(\frac{5 \cdot x - \frac{\left(x \cdot x\right) \cdot -10}{\varepsilon}}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left({x}^{2}\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left(x \cdot x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(4 + 1\right) \cdot x\right)\right)\right)\right) \]

      13. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(x + \color{blue}{4 \cdot x}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(x + 4 \cdot x\right)}\right)\right)\right)\right) \]

      15. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(\left(4 + 1\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(5 \cdot x\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right)\right)\right) \]

   8. Simplified 98.2%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right), \color{blue}{\varepsilon}\right) \]

   10. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon + x \cdot \left(10 \cdot x + \varepsilon \cdot 5\right)\right)\right) \cdot \varepsilon} \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\varepsilon}^{4}\right)}, \varepsilon\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \varepsilon\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \varepsilon\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \varepsilon\right), \varepsilon\right) \]

       4. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \varepsilon\right), \varepsilon\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \varepsilon\right), \varepsilon\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

       8. *-lowering-*.f64 97.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

   13. Simplified 97.7%

       \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)} \cdot \varepsilon \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-40}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.96 \cdot 10^{-56}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 87.1% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (* eps (* eps (* eps eps)))))

C

double code(double x, double eps) {
	return eps * (eps * (eps * (eps * eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (eps * (eps * (eps * eps)))
end function

Java

public static double code(double x, double eps) {
	return eps * (eps * (eps * (eps * eps)));
}

Python

def code(x, eps):
	return eps * (eps * (eps * (eps * eps)))

Julia

function code(x, eps)
	return Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (eps * (eps * (eps * eps)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.2%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in eps around -inf

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot {\varepsilon}^{5}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right)} \]

   3. sub-neg N/A

      \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

   5. metadata-eval N/A

      \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + -1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

   6. distribute-lft1-in N/A

      \[\leadsto \left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

   7. associate-*l* N/A

      \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

5. Simplified 74.3%

   \[\leadsto \color{blue}{\left(\frac{5 \cdot x - \frac{\left(x \cdot x\right) \cdot -10}{\varepsilon}}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

   2. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left({x}^{2}\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left(x \cdot x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(4 + 1\right) \cdot x\right)\right)\right)\right) \]

   13. distribute-rgt1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(x + \color{blue}{4 \cdot x}\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(x + 4 \cdot x\right)}\right)\right)\right)\right) \]

   15. distribute-rgt1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(\left(4 + 1\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(5 \cdot x\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 84.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right)\right)\right) \]

8. Simplified 84.8%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
9. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right), \color{blue}{\varepsilon}\right) \]

10. Applied egg-rr 84.8%

    \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon + x \cdot \left(10 \cdot x + \varepsilon \cdot 5\right)\right)\right) \cdot \varepsilon} \]

11. Taylor expanded in eps around inf

    \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\varepsilon}^{4}\right)}, \varepsilon\right) \]
12. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{\left(3 + 1\right)}\right), \varepsilon\right) \]

    2. pow-plus N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3} \cdot \varepsilon\right), \varepsilon\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \varepsilon\right), \varepsilon\right) \]

    4. cube-mult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \varepsilon\right), \varepsilon\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \varepsilon\right), \varepsilon\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

    8. *-lowering-*.f64 84.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

13. Simplified 84.3%

    \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)} \cdot \varepsilon \]

14. Final simplification 84.3%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]
15. Add Preprocessing

Alternative 21: 87.1% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (* (* eps eps) (* eps eps))))

C

double code(double x, double eps) {
	return eps * ((eps * eps) * (eps * eps));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * eps) * (eps * eps))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * eps) * (eps * eps));
}

Python

def code(x, eps):
	return eps * ((eps * eps) * (eps * eps))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * 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[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.2%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in eps around -inf

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot {\varepsilon}^{5}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right)} \]

   3. sub-neg N/A

      \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

   5. metadata-eval N/A

      \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + -1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

   6. distribute-lft1-in N/A

      \[\leadsto \left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

   7. associate-*l* N/A

      \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

5. Simplified 74.3%

   \[\leadsto \color{blue}{\left(\frac{5 \cdot x - \frac{\left(x \cdot x\right) \cdot -10}{\varepsilon}}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

   2. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left({x}^{2}\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left(x \cdot x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(4 + 1\right) \cdot x\right)\right)\right)\right) \]

   13. distribute-rgt1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(x + \color{blue}{4 \cdot x}\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(x + 4 \cdot x\right)}\right)\right)\right)\right) \]

   15. distribute-rgt1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(\left(4 + 1\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(5 \cdot x\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 84.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right)\right)\right) \]

8. Simplified 84.8%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
9. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right), \color{blue}{\varepsilon}\right) \]

10. Applied egg-rr 84.8%

    \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon + x \cdot \left(10 \cdot x + \varepsilon \cdot 5\right)\right)\right) \cdot \varepsilon} \]

11. Taylor expanded in eps around inf

    \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \color{blue}{\left({\varepsilon}^{2}\right)}\right), \varepsilon\right) \]
12. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right) \]

    2. *-lowering-*.f64 84.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \varepsilon\right) \]

13. Simplified 84.2%

    \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \varepsilon \]

14. Final simplification 84.2%

    \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]
15. Add Preprocessing

Alternative 22: 87.1% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* eps eps) (* eps (* eps eps))))

C

double code(double x, double eps) {
	return (eps * eps) * (eps * (eps * eps));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * eps) * (eps * (eps * eps))
end function

Java

public static double code(double x, double eps) {
	return (eps * eps) * (eps * (eps * eps));
}

Python

def code(x, eps):
	return (eps * eps) * (eps * (eps * eps))

Julia

function code(x, eps)
	return Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * eps) * (eps * (eps * eps));
end

Wolfram

code[x_, eps_] := N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.2%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in eps around -inf

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot {\varepsilon}^{5}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot \color{blue}{\left(-1 \cdot {\varepsilon}^{5}\right)} \]

   3. sub-neg N/A

      \[\leadsto \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

   5. metadata-eval N/A

      \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} \cdot -1 + -1\right) \cdot \left(-1 \cdot {\varepsilon}^{5}\right) \]

   6. distribute-lft1-in N/A

      \[\leadsto \left(\left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot -1\right) \cdot \left(\color{blue}{-1} \cdot {\varepsilon}^{5}\right) \]

   7. associate-*l* N/A

      \[\leadsto \left(\frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} + 1\right) \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot {\varepsilon}^{5}\right)\right)} \]

5. Simplified 74.3%

   \[\leadsto \color{blue}{\left(\frac{5 \cdot x - \frac{\left(x \cdot x\right) \cdot -10}{\varepsilon}}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5}} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

   2. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{10 \cdot {x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(10 \cdot \color{blue}{{x}^{2}} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(10 \cdot {x}^{2}\right), \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left({x}^{2}\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \left(x \cdot x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(4 + 1\right) \cdot x\right)\right)\right)\right) \]

   13. distribute-rgt1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \left(x + \color{blue}{4 \cdot x}\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(x + 4 \cdot x\right)}\right)\right)\right)\right) \]

   15. distribute-rgt1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(\left(4 + 1\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \left(5 \cdot x\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 84.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(10, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(5, \color{blue}{x}\right)\right)\right)\right)\right) \]

8. Simplified 84.8%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \color{blue}{\left({\varepsilon}^{2}\right)}\right) \]
10. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right) \]

    2. *-lowering-*.f64 84.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right)\right) \]

11. Simplified 84.2%

    \[\leadsto \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \]

12. Final simplification 84.2%

    \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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)))