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

Percentage Accurate: 88.8% → 99.3%

Time: 11.6s

Alternatives: 16

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (- t_0 (* (* x x) (* x (* x x))))))
   (if (<= t_1 -2e-304)
     t_2
     (if (<= t_1 0.0) (* (pow x 4.0) (* eps 5.0)) t_2))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double t_2 = t_0 - ((x * x) * (x * (x * x)));
	double tmp;
	if (t_1 <= -2e-304) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    t_2 = t_0 - ((x * x) * (x * (x * x)))
    if (t_1 <= (-2d-304)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double t_2 = t_0 - ((x * x) * (x * (x * x)));
	double tmp;
	if (t_1 <= -2e-304) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	t_2 = t_0 - ((x * x) * (x * (x * x)))
	tmp = 0
	if t_1 <= -2e-304:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = Float64(t_0 - Float64(Float64(x * x) * Float64(x * Float64(x * x))))
	tmp = 0.0
	if (t_1 <= -2e-304)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	t_2 = t_0 - ((x * x) * (x * (x * x)));
	tmp = 0.0;
	if (t_1 <= -2e-304)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = (x ^ 4.0) * (eps * 5.0);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-304], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

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))) < -1.99999999999999994e-304 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. *-lowering-*.f64 96.2

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

   4. Applied egg-rr 96.2%

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

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

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

   3. Taylor expanded in x around -inf

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

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

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

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

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot 10\right) \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\right)}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-304)
     (/
      1.0
      (/
       1.0
       (fma
        (* (* eps eps) (* eps eps))
        (fma 5.0 x eps)
        (* (* eps eps) (* (* x 10.0) (* x (+ x eps)))))))
     (if (<= t_0 0.0) (* (pow x 4.0) (* eps 5.0)) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = 1.0 / (1.0 / fma(((eps * eps) * (eps * eps)), fma(5.0, x, eps), ((eps * eps) * ((x * 10.0) * (x * (x + eps))))));
	} else if (t_0 <= 0.0) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = Float64(1.0 / Float64(1.0 / fma(Float64(Float64(eps * eps) * Float64(eps * eps)), fma(5.0, x, eps), Float64(Float64(eps * eps) * Float64(Float64(x * 10.0) * Float64(x * Float64(x + eps)))))));
	elseif (t_0 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], N[(1.0 / N[(1.0 / N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(x * 10.0), $MachinePrecision] * N[(x * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.99999999999999994e-304

   1. Initial program 97.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. *-lowering-*.f64 97.3

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 95.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   9. Applied egg-rr 95.5%

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

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

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

   3. Taylor expanded in x around -inf

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

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

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

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

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

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

   1. Initial program 94.7%

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

   3. Taylor expanded in x around 0

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

      1. pow-lowering-pow.f64 91.6

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

   5. Simplified 91.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot 10\right) \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\right)}}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot 10\right) \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\right)}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-304)
     (/
      1.0
      (/
       1.0
       (fma
        (* (* eps eps) (* eps eps))
        (fma 5.0 x eps)
        (* (* eps eps) (* (* x 10.0) (* x (+ x eps)))))))
     (if (<= t_0 0.0)
       (* (fma eps 5.0 (/ (* (* eps eps) 10.0) x)) (* x (* x (* x x))))
       (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = 1.0 / (1.0 / fma(((eps * eps) * (eps * eps)), fma(5.0, x, eps), ((eps * eps) * ((x * 10.0) * (x * (x + eps))))));
	} else if (t_0 <= 0.0) {
		tmp = fma(eps, 5.0, (((eps * eps) * 10.0) / x)) * (x * (x * (x * x)));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = Float64(1.0 / Float64(1.0 / fma(Float64(Float64(eps * eps) * Float64(eps * eps)), fma(5.0, x, eps), Float64(Float64(eps * eps) * Float64(Float64(x * 10.0) * Float64(x * Float64(x + eps)))))));
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(eps, 5.0, Float64(Float64(Float64(eps * eps) * 10.0) / x)) * Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], N[(1.0 / N[(1.0 / N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(x * 10.0), $MachinePrecision] * N[(x * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(eps * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.99999999999999994e-304

   1. Initial program 97.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. *-lowering-*.f64 97.3

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 95.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   9. Applied egg-rr 95.5%

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

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

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

   3. Taylor expanded in x around -inf

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

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

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

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

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

      1. *-commutative N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. distribute-neg-frac N/A

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

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

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

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

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

      8. metadata-eval N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. pow-plus N/A

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

      13. cube-unmult N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 94.7%

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

   3. Taylor expanded in x around 0

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

      1. pow-lowering-pow.f64 91.6

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

   5. Simplified 91.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot 10\right) \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\right)}}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot 10\right) \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\right)}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 5, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-304)
     (/
      1.0
      (/
       1.0
       (fma
        (* (* eps eps) (* eps eps))
        (fma 5.0 x eps)
        (* (* eps eps) (* (* x 10.0) (* x (+ x eps)))))))
     (if (<= t_0 0.0)
       (* (fma eps 5.0 (/ (* (* eps eps) 10.0) x)) (* x (* x (* x x))))
       (* (* eps eps) (* eps (fma eps (fma x 5.0 eps) (* (* x x) 10.0))))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = 1.0 / (1.0 / fma(((eps * eps) * (eps * eps)), fma(5.0, x, eps), ((eps * eps) * ((x * 10.0) * (x * (x + eps))))));
	} else if (t_0 <= 0.0) {
		tmp = fma(eps, 5.0, (((eps * eps) * 10.0) / x)) * (x * (x * (x * x)));
	} else {
		tmp = (eps * eps) * (eps * fma(eps, fma(x, 5.0, eps), ((x * x) * 10.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = Float64(1.0 / Float64(1.0 / fma(Float64(Float64(eps * eps) * Float64(eps * eps)), fma(5.0, x, eps), Float64(Float64(eps * eps) * Float64(Float64(x * 10.0) * Float64(x * Float64(x + eps)))))));
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(eps, 5.0, Float64(Float64(Float64(eps * eps) * 10.0) / x)) * Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(eps * eps) * Float64(eps * fma(eps, fma(x, 5.0, eps), Float64(Float64(x * x) * 10.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], N[(1.0 / N[(1.0 / N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(x * 10.0), $MachinePrecision] * N[(x * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(eps * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * N[(x * 5.0 + eps), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.99999999999999994e-304

   1. Initial program 97.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. *-lowering-*.f64 97.3

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 95.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   9. Applied egg-rr 95.5%

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

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

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

   3. Taylor expanded in x around -inf

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

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

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

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

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

      1. *-commutative N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. distribute-neg-frac N/A

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

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

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

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

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

      8. metadata-eval N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. pow-plus N/A

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

      13. cube-unmult N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 94.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. *-lowering-*.f64 95.1

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

   4. Applied egg-rr 95.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 90.7%

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

   8. Taylor expanded in eps around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   10. Simplified 90.1%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-+r+ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. distribute-rgt-in N/A

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

       10. unpow2 N/A

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

       11. associate-*r* N/A

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

       12. associate-*r* N/A

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

       13. associate-*r* N/A

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

       14. unpow2 N/A

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. distribute-lft-in N/A

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

       18. +-commutative N/A

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

   13. Simplified 90.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot 10\right) \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\right)}}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 5, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot 10\right) \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 5, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-304)
     (fma
      (fma 5.0 x eps)
      (* (* eps eps) (* eps eps))
      (* (* eps eps) (* (* x 10.0) (* x (+ x eps)))))
     (if (<= t_0 0.0)
       (* (fma eps 5.0 (/ (* (* eps eps) 10.0) x)) (* x (* x (* x x))))
       (* (* eps eps) (* eps (fma eps (fma x 5.0 eps) (* (* x x) 10.0))))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = fma(fma(5.0, x, eps), ((eps * eps) * (eps * eps)), ((eps * eps) * ((x * 10.0) * (x * (x + eps)))));
	} else if (t_0 <= 0.0) {
		tmp = fma(eps, 5.0, (((eps * eps) * 10.0) / x)) * (x * (x * (x * x)));
	} else {
		tmp = (eps * eps) * (eps * fma(eps, fma(x, 5.0, eps), ((x * x) * 10.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = fma(fma(5.0, x, eps), Float64(Float64(eps * eps) * Float64(eps * eps)), Float64(Float64(eps * eps) * Float64(Float64(x * 10.0) * Float64(x * Float64(x + eps)))));
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(eps, 5.0, Float64(Float64(Float64(eps * eps) * 10.0) / x)) * Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(eps * eps) * Float64(eps * fma(eps, fma(x, 5.0, eps), Float64(Float64(x * x) * 10.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(x * 10.0), $MachinePrecision] * N[(x * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(eps * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * N[(x * 5.0 + eps), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.99999999999999994e-304

   1. Initial program 97.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. *-lowering-*.f64 97.3

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 95.8%

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

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. unpow3 N/A

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

      7. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

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

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

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

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

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

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

      18. +-lowering-+.f64 95.5

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

   9. Applied egg-rr 95.5%

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

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

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

   3. Taylor expanded in x around -inf

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

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

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

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

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

      1. *-commutative N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. distribute-neg-frac N/A

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

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

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

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

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

      8. metadata-eval N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. pow-plus N/A

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

      13. cube-unmult N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 94.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. *-lowering-*.f64 95.1

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

   4. Applied egg-rr 95.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 90.7%

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

   8. Taylor expanded in eps around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   10. Simplified 90.1%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-+r+ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. distribute-rgt-in N/A

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

       10. unpow2 N/A

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

       11. associate-*r* N/A

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

       12. associate-*r* N/A

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

       13. associate-*r* N/A

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

       14. unpow2 N/A

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. distribute-lft-in N/A

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

       18. +-commutative N/A

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

   13. Simplified 90.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot 10\right) \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 5, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(5, x, \varepsilon\right), x \cdot \left(x \cdot \left(10 \cdot \left(x + \varepsilon\right)\right)\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 5, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-304)
     (*
      eps
      (* eps (fma (* eps eps) (fma 5.0 x eps) (* x (* x (* 10.0 (+ x eps)))))))
     (if (<= t_0 0.0)
       (* (fma eps 5.0 (/ (* (* eps eps) 10.0) x)) (* x (* x (* x x))))
       (* (* eps eps) (* eps (fma eps (fma x 5.0 eps) (* (* x x) 10.0))))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = eps * (eps * fma((eps * eps), fma(5.0, x, eps), (x * (x * (10.0 * (x + eps))))));
	} else if (t_0 <= 0.0) {
		tmp = fma(eps, 5.0, (((eps * eps) * 10.0) / x)) * (x * (x * (x * x)));
	} else {
		tmp = (eps * eps) * (eps * fma(eps, fma(x, 5.0, eps), ((x * x) * 10.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = Float64(eps * Float64(eps * fma(Float64(eps * eps), fma(5.0, x, eps), Float64(x * Float64(x * Float64(10.0 * Float64(x + eps)))))));
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(eps, 5.0, Float64(Float64(Float64(eps * eps) * 10.0) / x)) * Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(eps * eps) * Float64(eps * fma(eps, fma(x, 5.0, eps), Float64(Float64(x * x) * 10.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], N[(eps * N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision] + N[(x * N[(x * N[(10.0 * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(eps * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * N[(x * 5.0 + eps), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.99999999999999994e-304

   1. Initial program 97.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. *-lowering-*.f64 97.3

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 95.8%

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

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

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

      4. cube-mult N/A

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

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

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

      6. *-lowering-*.f64 95.4

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

   9. Applied egg-rr 95.4%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. associate-*r* N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. distribute-lft-in N/A

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

   11. Applied egg-rr 95.4%

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

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

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

   3. Taylor expanded in x around -inf

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

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

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

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

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

      1. *-commutative N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. distribute-neg-frac N/A

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

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

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

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

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

      8. metadata-eval N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. pow-plus N/A

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

      13. cube-unmult N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-lowering-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 94.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. *-lowering-*.f64 95.1

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

   4. Applied egg-rr 95.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 90.7%

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

   8. Taylor expanded in eps around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   10. Simplified 90.1%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-+r+ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. distribute-rgt-in N/A

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

       10. unpow2 N/A

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

       11. associate-*r* N/A

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

       12. associate-*r* N/A

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

       13. associate-*r* N/A

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

       14. unpow2 N/A

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. distribute-lft-in N/A

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

       18. +-commutative N/A

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

   13. Simplified 90.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(5, x, \varepsilon\right), x \cdot \left(x \cdot \left(10 \cdot \left(x + \varepsilon\right)\right)\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 5, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(5, x, \varepsilon\right), x \cdot \left(x \cdot \left(10 \cdot \left(x + \varepsilon\right)\right)\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot \varepsilon, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 5, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-304)
     (*
      eps
      (* eps (fma (* eps eps) (fma 5.0 x eps) (* x (* x (* 10.0 (+ x eps)))))))
     (if (<= t_0 0.0)
       (* (* x (* x x)) (fma (* x eps) 5.0 (* (* eps eps) 10.0)))
       (* (* eps eps) (* eps (fma eps (fma x 5.0 eps) (* (* x x) 10.0))))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = eps * (eps * fma((eps * eps), fma(5.0, x, eps), (x * (x * (10.0 * (x + eps))))));
	} else if (t_0 <= 0.0) {
		tmp = (x * (x * x)) * fma((x * eps), 5.0, ((eps * eps) * 10.0));
	} else {
		tmp = (eps * eps) * (eps * fma(eps, fma(x, 5.0, eps), ((x * x) * 10.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = Float64(eps * Float64(eps * fma(Float64(eps * eps), fma(5.0, x, eps), Float64(x * Float64(x * Float64(10.0 * Float64(x + eps)))))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(x * eps), 5.0, Float64(Float64(eps * eps) * 10.0)));
	else
		tmp = Float64(Float64(eps * eps) * Float64(eps * fma(eps, fma(x, 5.0, eps), Float64(Float64(x * x) * 10.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], N[(eps * N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision] + N[(x * N[(x * N[(10.0 * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * eps), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * N[(x * 5.0 + eps), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.99999999999999994e-304

   1. Initial program 97.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. *-lowering-*.f64 97.3

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 95.8%

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

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

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

      4. cube-mult N/A

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

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

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

      6. *-lowering-*.f64 95.4

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

   9. Applied egg-rr 95.4%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. associate-*r* N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. distribute-lft-in N/A

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

   11. Applied egg-rr 95.4%

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

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

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

   3. Taylor expanded in x around -inf

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

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

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

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

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

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

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. distribute-lft-out N/A

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

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

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

      20. accelerator-lowering-fma.f64 N/A

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

      21. *-commutative N/A

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

      22. *-lowering-*.f64 99.9

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

   8. Simplified 99.9%

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

      1. distribute-lft-in N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      8. *-lowering-*.f64 99.9

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

   10. Applied egg-rr 99.9%

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

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

   1. Initial program 94.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. *-lowering-*.f64 95.1

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

   4. Applied egg-rr 95.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 90.7%

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

   8. Taylor expanded in eps around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   10. Simplified 90.1%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-+r+ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. distribute-rgt-in N/A

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

       10. unpow2 N/A

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

       11. associate-*r* N/A

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

       12. associate-*r* N/A

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

       13. associate-*r* N/A

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

       14. unpow2 N/A

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. distribute-lft-in N/A

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

       18. +-commutative N/A

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

   13. Simplified 90.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(5, x, \varepsilon\right), x \cdot \left(x \cdot \left(10 \cdot \left(x + \varepsilon\right)\right)\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot \varepsilon, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 5, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 5, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot \varepsilon, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1
         (* (* eps eps) (* eps (fma eps (fma x 5.0 eps) (* (* x x) 10.0))))))
   (if (<= t_0 -2e-304)
     t_1
     (if (<= t_0 0.0)
       (* (* x (* x x)) (fma (* x eps) 5.0 (* (* eps eps) 10.0)))
       t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * fma(eps, fma(x, 5.0, eps), ((x * x) * 10.0)));
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (x * (x * x)) * fma((x * eps), 5.0, ((eps * eps) * 10.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * Float64(eps * fma(eps, fma(x, 5.0, eps), Float64(Float64(x * x) * 10.0))))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(x * eps), 5.0, Float64(Float64(eps * eps) * 10.0)));
	else
		tmp = t_1;
	end
	return 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]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * N[(x * 5.0 + eps), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * eps), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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))) < -1.99999999999999994e-304 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. *-lowering-*.f64 96.2

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

   4. Applied egg-rr 96.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 93.4%

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

   8. Taylor expanded in eps around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   10. Simplified 92.7%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-+r+ N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. distribute-rgt-in N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\varepsilon + 5 \cdot x\right)} + \left(10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot x\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\varepsilon + 5 \cdot x\right) + \left(10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot x\right) \]

       11. associate-*r* N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)} + \left(10 \cdot \left(\varepsilon \cdot x\right)\right) \cdot x\right) \]

       12. associate-*r* N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) + \color{blue}{\left(\left(10 \cdot \varepsilon\right) \cdot x\right)} \cdot x\right) \]

       13. associate-*r* N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) + \color{blue}{\left(10 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) + \left(10 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{2}}\right) \]

       15. *-commutative N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) + \color{blue}{\left(\varepsilon \cdot 10\right)} \cdot {x}^{2}\right) \]

       16. associate-*r* N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) + \color{blue}{\varepsilon \cdot \left(10 \cdot {x}^{2}\right)}\right) \]

       17. distribute-lft-in N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right) + 10 \cdot {x}^{2}\right)\right)} \]

       18. +-commutative N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

   13. Simplified 93.0%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 5, \varepsilon\right), 10 \cdot \left(x \cdot x\right)\right)\right)} \]

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{x}^{4}} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. associate-+r+ N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      9. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\varepsilon \cdot 5 - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot {\varepsilon}^{2}\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto {x}^{3} \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      4. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot {\varepsilon}^{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot \varepsilon} + 10 \cdot {\varepsilon}^{2}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\varepsilon \cdot \left(5 \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{{\varepsilon}^{2} \cdot 10}\right) \]

      15. unpow2 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 10\right) \]

      16. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot 10\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \varepsilon \cdot \color{blue}{\left(10 \cdot \varepsilon\right)}\right) \]

      18. distribute-lft-out N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      20. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right)}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

      22. *-lowering-*.f64 99.9

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x\right) + \varepsilon \cdot \left(\varepsilon \cdot 10\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(x \cdot 5\right)} + \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(\varepsilon \cdot x\right) \cdot 5} + \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot x}, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\varepsilon \cdot x, 5, \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}\right) \]

      8. *-lowering-*.f64 99.9

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\varepsilon \cdot x, 5, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 10\right) \]

   10. Applied egg-rr 99.9%

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 5, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot \varepsilon, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, 5, \varepsilon\right), \left(x \cdot x\right) \cdot 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot \varepsilon, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-304)
     (* (* eps eps) (* (* eps eps) (fma x 5.0 eps)))
     (if (<= t_0 0.0)
       (* (* x (* x x)) (fma (* x eps) 5.0 (* (* eps eps) 10.0)))
       (* (* eps eps) (* eps (* eps eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = (eps * eps) * ((eps * eps) * fma(x, 5.0, eps));
	} else if (t_0 <= 0.0) {
		tmp = (x * (x * x)) * fma((x * eps), 5.0, ((eps * eps) * 10.0));
	} else {
		tmp = (eps * eps) * (eps * (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * fma(x, 5.0, eps)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(x * eps), 5.0, Float64(Float64(eps * eps) * 10.0)));
	else
		tmp = Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(x * 5.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * eps), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.99999999999999994e-304

   1. Initial program 97.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} - {x}^{5} \]

      4. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      5. pow-prod-up N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      6. pow2 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{3} \cdot \color{blue}{\left(x \cdot x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot \left(x \cdot x\right)} \]

      8. cube-mult N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

      11. *-lowering-*.f64 97.3

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. Applied egg-rr 97.3%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot \left(x \cdot 10\right)\right) \cdot \left(\varepsilon + x\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3} + \color{blue}{\left(\left(10 \cdot {x}^{2}\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(10 \cdot {x}^{3} + \left(10 \cdot {x}^{2}\right) \cdot \varepsilon\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + \color{blue}{10 \cdot \left({x}^{2} \cdot \varepsilon\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + 10 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(10, \varepsilon \cdot {x}^{2} + {x}^{3}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   10. Simplified 95.1%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, \left(x \cdot x\right) \cdot \left(\varepsilon + x\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(5 \cdot \left({\varepsilon}^{2} \cdot x\right) + {\varepsilon}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot {\varepsilon}^{2}\right)} + {\varepsilon}^{3}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot {\varepsilon}^{2}} + {\varepsilon}^{3}\right) \]

       3. cube-mult N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(5 \cdot x\right) \cdot {\varepsilon}^{2} + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(5 \cdot x\right) \cdot {\varepsilon}^{2} + \varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(5 \cdot x + \varepsilon\right)\right)} \]

       6. +-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]

       8. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\varepsilon + 5 \cdot x\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\varepsilon + 5 \cdot x\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(5 \cdot x + \varepsilon\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{x \cdot 5} + \varepsilon\right)\right) \]

       12. accelerator-lowering-fma.f64 93.9

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(x, 5, \varepsilon\right)}\right) \]

   13. Simplified 93.9%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{x}^{4}} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. associate-+r+ N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      9. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\varepsilon \cdot 5 - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot {\varepsilon}^{2}\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto {x}^{3} \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      4. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot {\varepsilon}^{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot \varepsilon} + 10 \cdot {\varepsilon}^{2}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\varepsilon \cdot \left(5 \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{{\varepsilon}^{2} \cdot 10}\right) \]

      15. unpow2 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 10\right) \]

      16. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot 10\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \varepsilon \cdot \color{blue}{\left(10 \cdot \varepsilon\right)}\right) \]

      18. distribute-lft-out N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      20. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right)}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

      22. *-lowering-*.f64 99.9

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x\right) + \varepsilon \cdot \left(\varepsilon \cdot 10\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(x \cdot 5\right)} + \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(\varepsilon \cdot x\right) \cdot 5} + \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot x}, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\varepsilon \cdot x, 5, \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}\right) \]

      8. *-lowering-*.f64 99.9

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\varepsilon \cdot x, 5, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 10\right) \]

   10. Applied egg-rr 99.9%

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)} \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 94.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} - {x}^{5} \]

      4. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      5. pow-prod-up N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      6. pow2 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{3} \cdot \color{blue}{\left(x \cdot x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot \left(x \cdot x\right)} \]

      8. cube-mult N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

      11. *-lowering-*.f64 95.1

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 90.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot \left(x \cdot 10\right)\right) \cdot \left(\varepsilon + x\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3} + \color{blue}{\left(\left(10 \cdot {x}^{2}\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(10 \cdot {x}^{3} + \left(10 \cdot {x}^{2}\right) \cdot \varepsilon\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + \color{blue}{10 \cdot \left({x}^{2} \cdot \varepsilon\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + 10 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(10, \varepsilon \cdot {x}^{2} + {x}^{3}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   10. Simplified 90.1%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, \left(x \cdot x\right) \cdot \left(\varepsilon + x\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

       5. *-lowering-*.f64 90.6

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   13. Simplified 90.6%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot \varepsilon, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-304)
     (* (* eps eps) (* (* eps eps) (fma x 5.0 eps)))
     (if (<= t_0 0.0)
       (* (* x (* x x)) (* eps (fma 5.0 x (* eps 10.0))))
       (* (* eps eps) (* eps (* eps eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = (eps * eps) * ((eps * eps) * fma(x, 5.0, eps));
	} else if (t_0 <= 0.0) {
		tmp = (x * (x * x)) * (eps * fma(5.0, x, (eps * 10.0)));
	} else {
		tmp = (eps * eps) * (eps * (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * fma(x, 5.0, eps)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(eps * fma(5.0, x, Float64(eps * 10.0))));
	else
		tmp = Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(x * 5.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(5.0 * x + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.99999999999999994e-304

   1. Initial program 97.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} - {x}^{5} \]

      4. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      5. pow-prod-up N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      6. pow2 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{3} \cdot \color{blue}{\left(x \cdot x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot \left(x \cdot x\right)} \]

      8. cube-mult N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

      11. *-lowering-*.f64 97.3

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. Applied egg-rr 97.3%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot \left(x \cdot 10\right)\right) \cdot \left(\varepsilon + x\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3} + \color{blue}{\left(\left(10 \cdot {x}^{2}\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(10 \cdot {x}^{3} + \left(10 \cdot {x}^{2}\right) \cdot \varepsilon\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + \color{blue}{10 \cdot \left({x}^{2} \cdot \varepsilon\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + 10 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(10, \varepsilon \cdot {x}^{2} + {x}^{3}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   10. Simplified 95.1%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, \left(x \cdot x\right) \cdot \left(\varepsilon + x\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(5 \cdot \left({\varepsilon}^{2} \cdot x\right) + {\varepsilon}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot {\varepsilon}^{2}\right)} + {\varepsilon}^{3}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot {\varepsilon}^{2}} + {\varepsilon}^{3}\right) \]

       3. cube-mult N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(5 \cdot x\right) \cdot {\varepsilon}^{2} + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(5 \cdot x\right) \cdot {\varepsilon}^{2} + \varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(5 \cdot x + \varepsilon\right)\right)} \]

       6. +-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]

       8. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\varepsilon + 5 \cdot x\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\varepsilon + 5 \cdot x\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(5 \cdot x + \varepsilon\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{x \cdot 5} + \varepsilon\right)\right) \]

       12. accelerator-lowering-fma.f64 93.9

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(x, 5, \varepsilon\right)}\right) \]

   13. Simplified 93.9%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{x}^{4}} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. associate-+r+ N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      9. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\varepsilon \cdot 5 - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot {\varepsilon}^{2}\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto {x}^{3} \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      4. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot {\varepsilon}^{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot \varepsilon} + 10 \cdot {\varepsilon}^{2}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\varepsilon \cdot \left(5 \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{{\varepsilon}^{2} \cdot 10}\right) \]

      15. unpow2 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 10\right) \]

      16. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot 10\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \varepsilon \cdot \color{blue}{\left(10 \cdot \varepsilon\right)}\right) \]

      18. distribute-lft-out N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      20. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right)}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

      22. *-lowering-*.f64 99.9

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)} \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 94.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} - {x}^{5} \]

      4. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      5. pow-prod-up N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      6. pow2 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{3} \cdot \color{blue}{\left(x \cdot x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot \left(x \cdot x\right)} \]

      8. cube-mult N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

      11. *-lowering-*.f64 95.1

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 90.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot \left(x \cdot 10\right)\right) \cdot \left(\varepsilon + x\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3} + \color{blue}{\left(\left(10 \cdot {x}^{2}\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(10 \cdot {x}^{3} + \left(10 \cdot {x}^{2}\right) \cdot \varepsilon\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + \color{blue}{10 \cdot \left({x}^{2} \cdot \varepsilon\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + 10 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(10, \varepsilon \cdot {x}^{2} + {x}^{3}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   10. Simplified 90.1%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, \left(x \cdot x\right) \cdot \left(\varepsilon + x\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

       5. *-lowering-*.f64 90.6

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   13. Simplified 90.6%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-304)
     (* (* eps eps) (* (* eps eps) (fma x 5.0 eps)))
     (if (<= t_0 0.0)
       (* eps (* (* x (* x x)) (* x 5.0)))
       (* (* eps eps) (* eps (* eps eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = (eps * eps) * ((eps * eps) * fma(x, 5.0, eps));
	} else if (t_0 <= 0.0) {
		tmp = eps * ((x * (x * x)) * (x * 5.0));
	} else {
		tmp = (eps * eps) * (eps * (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * fma(x, 5.0, eps)));
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(Float64(x * Float64(x * x)) * Float64(x * 5.0)));
	else
		tmp = Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(x * 5.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.99999999999999994e-304

   1. Initial program 97.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} - {x}^{5} \]

      4. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      5. pow-prod-up N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      6. pow2 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{3} \cdot \color{blue}{\left(x \cdot x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot \left(x \cdot x\right)} \]

      8. cube-mult N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

      11. *-lowering-*.f64 97.3

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. Applied egg-rr 97.3%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot \left(x \cdot 10\right)\right) \cdot \left(\varepsilon + x\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3} + \color{blue}{\left(\left(10 \cdot {x}^{2}\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(10 \cdot {x}^{3} + \left(10 \cdot {x}^{2}\right) \cdot \varepsilon\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + \color{blue}{10 \cdot \left({x}^{2} \cdot \varepsilon\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + 10 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(10, \varepsilon \cdot {x}^{2} + {x}^{3}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   10. Simplified 95.1%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, \left(x \cdot x\right) \cdot \left(\varepsilon + x\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(5 \cdot \left({\varepsilon}^{2} \cdot x\right) + {\varepsilon}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot {\varepsilon}^{2}\right)} + {\varepsilon}^{3}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot {\varepsilon}^{2}} + {\varepsilon}^{3}\right) \]

       3. cube-mult N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(5 \cdot x\right) \cdot {\varepsilon}^{2} + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(5 \cdot x\right) \cdot {\varepsilon}^{2} + \varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(5 \cdot x + \varepsilon\right)\right)} \]

       6. +-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \color{blue}{\left(\varepsilon + 5 \cdot x\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\varepsilon + 5 \cdot x\right)\right)} \]

       8. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\varepsilon + 5 \cdot x\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\varepsilon + 5 \cdot x\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(5 \cdot x + \varepsilon\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{x \cdot 5} + \varepsilon\right)\right) \]

       12. accelerator-lowering-fma.f64 93.9

           \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(x, 5, \varepsilon\right)}\right) \]

   13. Simplified 93.9%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{x}^{4}} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. associate-+r+ N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      9. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\varepsilon \cdot 5 - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot {\varepsilon}^{2}\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto {x}^{3} \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      4. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot {\varepsilon}^{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot \varepsilon} + 10 \cdot {\varepsilon}^{2}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\varepsilon \cdot \left(5 \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{{\varepsilon}^{2} \cdot 10}\right) \]

      15. unpow2 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 10\right) \]

      16. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot 10\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \varepsilon \cdot \color{blue}{\left(10 \cdot \varepsilon\right)}\right) \]

      18. distribute-lft-out N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      20. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right)}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

      22. *-lowering-*.f64 99.9

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(5 \cdot x + \varepsilon \cdot 10\right) \cdot \varepsilon\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right) \cdot \varepsilon} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right) \cdot \varepsilon} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right)} \cdot \varepsilon \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right) \cdot \varepsilon \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right) \cdot \varepsilon \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)}\right) \cdot \varepsilon \]

      8. *-lowering-*.f64 99.9

         \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \cdot \varepsilon \]

   10. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right) \cdot \varepsilon} \]

   11. Taylor expanded in x around inf

       \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot x\right)}\right) \cdot \varepsilon \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot 5\right)}\right) \cdot \varepsilon \]

       2. *-lowering-*.f64 99.9

          \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot 5\right)}\right) \cdot \varepsilon \]

   13. Simplified 99.9%

       \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot 5\right)}\right) \cdot \varepsilon \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 94.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} - {x}^{5} \]

      4. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      5. pow-prod-up N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      6. pow2 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{3} \cdot \color{blue}{\left(x \cdot x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot \left(x \cdot x\right)} \]

      8. cube-mult N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

      11. *-lowering-*.f64 95.1

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 90.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot \left(x \cdot 10\right)\right) \cdot \left(\varepsilon + x\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3} + \color{blue}{\left(\left(10 \cdot {x}^{2}\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(10 \cdot {x}^{3} + \left(10 \cdot {x}^{2}\right) \cdot \varepsilon\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + \color{blue}{10 \cdot \left({x}^{2} \cdot \varepsilon\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + 10 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(10, \varepsilon \cdot {x}^{2} + {x}^{3}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   10. Simplified 90.1%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, \left(x \cdot x\right) \cdot \left(\varepsilon + x\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

       5. *-lowering-*.f64 90.6

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   13. Simplified 90.6%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* eps eps) (* eps (* eps eps)))))
   (if (<= t_0 -2e-304)
     t_1
     (if (<= t_0 0.0) (* eps (* (* x (* x x)) (* x 5.0))) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * (eps * eps));
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * ((x * (x * x)) * (x * 5.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = (eps * eps) * (eps * (eps * eps))
    if (t_0 <= (-2d-304)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = eps * ((x * (x * x)) * (x * 5.0d0))
    else
        tmp = t_1
    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 t_1 = (eps * eps) * (eps * (eps * eps));
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * ((x * (x * x)) * (x * 5.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = (eps * eps) * (eps * (eps * eps))
	tmp = 0
	if t_0 <= -2e-304:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = eps * ((x * (x * x)) * (x * 5.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(Float64(x * Float64(x * x)) * Float64(x * 5.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = (eps * eps) * (eps * (eps * eps));
	tmp = 0.0;
	if (t_0 <= -2e-304)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = eps * ((x * (x * x)) * (x * 5.0));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], t$95$1, If[LessEqual[t$95$0, 0.0], N[(eps * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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))) < -1.99999999999999994e-304 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} - {x}^{5} \]

      4. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      5. pow-prod-up N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      6. pow2 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{3} \cdot \color{blue}{\left(x \cdot x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot \left(x \cdot x\right)} \]

      8. cube-mult N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

      11. *-lowering-*.f64 96.2

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot \left(x \cdot 10\right)\right) \cdot \left(\varepsilon + x\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3} + \color{blue}{\left(\left(10 \cdot {x}^{2}\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(10 \cdot {x}^{3} + \left(10 \cdot {x}^{2}\right) \cdot \varepsilon\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + \color{blue}{10 \cdot \left({x}^{2} \cdot \varepsilon\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + 10 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(10, \varepsilon \cdot {x}^{2} + {x}^{3}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   10. Simplified 92.7%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, \left(x \cdot x\right) \cdot \left(\varepsilon + x\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

       5. *-lowering-*.f64 91.7

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   13. Simplified 91.7%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{x}^{4}} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. associate-+r+ N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      9. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\varepsilon \cdot 5 - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot {\varepsilon}^{2}\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto {x}^{3} \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      4. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot {\varepsilon}^{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot \varepsilon} + 10 \cdot {\varepsilon}^{2}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\varepsilon \cdot \left(5 \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{{\varepsilon}^{2} \cdot 10}\right) \]

      15. unpow2 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 10\right) \]

      16. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot 10\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \varepsilon \cdot \color{blue}{\left(10 \cdot \varepsilon\right)}\right) \]

      18. distribute-lft-out N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      20. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right)}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

      22. *-lowering-*.f64 99.9

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(5 \cdot x + \varepsilon \cdot 10\right) \cdot \varepsilon\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right) \cdot \varepsilon} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right) \cdot \varepsilon} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right)} \cdot \varepsilon \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right) \cdot \varepsilon \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right) \cdot \varepsilon \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)}\right) \cdot \varepsilon \]

      8. *-lowering-*.f64 99.9

         \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \cdot \varepsilon \]

   10. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right) \cdot \varepsilon} \]

   11. Taylor expanded in x around inf

       \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot x\right)}\right) \cdot \varepsilon \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot 5\right)}\right) \cdot \varepsilon \]

       2. *-lowering-*.f64 99.9

          \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot 5\right)}\right) \cdot \varepsilon \]

   13. Simplified 99.9%

       \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot 5\right)}\right) \cdot \varepsilon \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* eps eps) (* eps (* eps eps)))))
   (if (<= t_0 -2e-304)
     t_1
     (if (<= t_0 0.0) (* (* x (* x x)) (* eps (* x 5.0))) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * (eps * eps));
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = (eps * eps) * (eps * (eps * eps))
    if (t_0 <= (-2d-304)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (x * (x * x)) * (eps * (x * 5.0d0))
    else
        tmp = t_1
    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 t_1 = (eps * eps) * (eps * (eps * eps));
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = (eps * eps) * (eps * (eps * eps))
	tmp = 0
	if t_0 <= -2e-304:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (x * (x * x)) * (eps * (x * 5.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(eps * Float64(x * 5.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = (eps * eps) * (eps * (eps * eps));
	tmp = 0.0;
	if (t_0 <= -2e-304)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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))) < -1.99999999999999994e-304 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} - {x}^{5} \]

      4. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      5. pow-prod-up N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      6. pow2 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{3} \cdot \color{blue}{\left(x \cdot x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot \left(x \cdot x\right)} \]

      8. cube-mult N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

      11. *-lowering-*.f64 96.2

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot \left(x \cdot 10\right)\right) \cdot \left(\varepsilon + x\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3} + \color{blue}{\left(\left(10 \cdot {x}^{2}\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(10 \cdot {x}^{3} + \left(10 \cdot {x}^{2}\right) \cdot \varepsilon\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + \color{blue}{10 \cdot \left({x}^{2} \cdot \varepsilon\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + 10 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(10, \varepsilon \cdot {x}^{2} + {x}^{3}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   10. Simplified 92.7%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, \left(x \cdot x\right) \cdot \left(\varepsilon + x\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

       5. *-lowering-*.f64 91.7

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   13. Simplified 91.7%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{x}^{4}} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. associate-+r+ N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      9. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\varepsilon \cdot 5 - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot {\varepsilon}^{2}\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto {x}^{3} \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      4. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot {\varepsilon}^{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot \varepsilon} + 10 \cdot {\varepsilon}^{2}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\varepsilon \cdot \left(5 \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{{\varepsilon}^{2} \cdot 10}\right) \]

      15. unpow2 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 10\right) \]

      16. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot 10\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \varepsilon \cdot \color{blue}{\left(10 \cdot \varepsilon\right)}\right) \]

      18. distribute-lft-out N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      20. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right)}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

      22. *-lowering-*.f64 99.9

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(\varepsilon \cdot x\right) \cdot 5\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(x \cdot 5\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(x \cdot 5\right)}\right) \]

       6. *-lowering-*.f64 99.9

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(x \cdot 5\right)}\right) \]

   11. Simplified 99.9%

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(x \cdot 5\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* eps eps) (* eps (* eps eps)))))
   (if (<= t_0 -2e-304)
     t_1
     (if (<= t_0 0.0) (* (* x x) (* x (* eps (* x 5.0)))) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * (eps * eps));
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (x * x) * (x * (eps * (x * 5.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = (eps * eps) * (eps * (eps * eps))
    if (t_0 <= (-2d-304)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (x * x) * (x * (eps * (x * 5.0d0)))
    else
        tmp = t_1
    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 t_1 = (eps * eps) * (eps * (eps * eps));
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (x * x) * (x * (eps * (x * 5.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = (eps * eps) * (eps * (eps * eps))
	tmp = 0
	if t_0 <= -2e-304:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (x * x) * (x * (eps * (x * 5.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x * x) * Float64(x * Float64(eps * Float64(x * 5.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = (eps * eps) * (eps * (eps * eps));
	tmp = 0.0;
	if (t_0 <= -2e-304)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = (x * x) * (x * (eps * (x * 5.0)));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(x * x), $MachinePrecision] * N[(x * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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))) < -1.99999999999999994e-304 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} - {x}^{5} \]

      4. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      5. pow-prod-up N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      6. pow2 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{3} \cdot \color{blue}{\left(x \cdot x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot \left(x \cdot x\right)} \]

      8. cube-mult N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

      11. *-lowering-*.f64 96.2

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot \left(x \cdot 10\right)\right) \cdot \left(\varepsilon + x\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3} + \color{blue}{\left(\left(10 \cdot {x}^{2}\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(10 \cdot {x}^{3} + \left(10 \cdot {x}^{2}\right) \cdot \varepsilon\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + \color{blue}{10 \cdot \left({x}^{2} \cdot \varepsilon\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + 10 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(10, \varepsilon \cdot {x}^{2} + {x}^{3}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   10. Simplified 92.7%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, \left(x \cdot x\right) \cdot \left(\varepsilon + x\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

       5. *-lowering-*.f64 91.7

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   13. Simplified 91.7%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{x}^{4}} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. associate-+r+ N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      9. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\varepsilon \cdot 5 - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot {x}^{4}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}} \]

      3. pow-sqr N/A

         \[\leadsto \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \]

      4. pow2 N/A

         \[\leadsto \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right) \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      9. sub-neg N/A

         \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot 5 + \left(\mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\varepsilon, 5, \mathsf{neg}\left(\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \left(\mathsf{fma}\left(\varepsilon, 5, \color{blue}{\frac{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)}{x}}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\varepsilon, 5, \color{blue}{\frac{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right)}{x}}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \left(\mathsf{fma}\left(\varepsilon, 5, \frac{\color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\mathsf{neg}\left(-10\right)\right)}}{x}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\varepsilon, 5, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{10}}{x}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\varepsilon, 5, \frac{\color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}}{x}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\varepsilon, 5, \frac{\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 10}{x}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\varepsilon, 5, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

      18. *-lowering-*.f64 99.9

          \[\leadsto \left(\mathsf{fma}\left(\varepsilon, 5, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\varepsilon, 5, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\left(5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)} \cdot \left(x \cdot x\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(5 \cdot \left(\varepsilon \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \left(x \cdot x\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(5 \cdot \color{blue}{\left(\left(\varepsilon \cdot x\right) \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(5 \cdot \left(\varepsilon \cdot x\right)\right) \cdot x\right)} \cdot \left(x \cdot x\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right)\right)\right)} \cdot \left(x \cdot x\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right)\right)\right)} \cdot \left(x \cdot x\right) \]

      6. *-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\left(\varepsilon \cdot x\right) \cdot 5\right)}\right) \cdot \left(x \cdot x\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\varepsilon \cdot \left(x \cdot 5\right)\right)}\right) \cdot \left(x \cdot x\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\varepsilon \cdot \color{blue}{\left(5 \cdot x\right)}\right)\right) \cdot \left(x \cdot x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x\right)\right)}\right) \cdot \left(x \cdot x\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\varepsilon \cdot \color{blue}{\left(x \cdot 5\right)}\right)\right) \cdot \left(x \cdot x\right) \]

      11. *-lowering-*.f64 99.9

          \[\leadsto \left(x \cdot \left(\varepsilon \cdot \color{blue}{\left(x \cdot 5\right)}\right)\right) \cdot \left(x \cdot x\right) \]

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\left(x \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)} \cdot \left(x \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (* eps eps) (* eps (* eps eps)))))
   (if (<= t_0 -2e-304)
     t_1
     (if (<= t_0 0.0) (* eps (* x (* 5.0 (* x (* x x))))) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (eps * eps) * (eps * (eps * eps));
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (x * (5.0 * (x * (x * x))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = (eps * eps) * (eps * (eps * eps))
    if (t_0 <= (-2d-304)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = eps * (x * (5.0d0 * (x * (x * x))))
    else
        tmp = t_1
    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 t_1 = (eps * eps) * (eps * (eps * eps));
	double tmp;
	if (t_0 <= -2e-304) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (x * (5.0 * (x * (x * x))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = (eps * eps) * (eps * (eps * eps))
	tmp = 0
	if t_0 <= -2e-304:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = eps * (x * (5.0 * (x * (x * x))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-304)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(x * Float64(5.0 * Float64(x * Float64(x * x)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = (eps * eps) * (eps * (eps * eps));
	tmp = 0.0;
	if (t_0 <= -2e-304)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = eps * (x * (5.0 * (x * (x * x))));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-304], t$95$1, If[LessEqual[t$95$0, 0.0], N[(eps * N[(x * N[(5.0 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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))) < -1.99999999999999994e-304 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} - {x}^{5} \]

      4. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

      5. pow-prod-up N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

      6. pow2 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{3} \cdot \color{blue}{\left(x \cdot x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot \left(x \cdot x\right)} \]

      8. cube-mult N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

      11. *-lowering-*.f64 96.2

          \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

   7. Simplified 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot \left(x \cdot 10\right)\right) \cdot \left(\varepsilon + x\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3} + \color{blue}{\left(\left(10 \cdot {x}^{2}\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)}\right) \]

      5. associate-+r+ N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(10 \cdot {x}^{3} + \left(10 \cdot {x}^{2}\right) \cdot \varepsilon\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + \color{blue}{10 \cdot \left({x}^{2} \cdot \varepsilon\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + 10 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(10, \varepsilon \cdot {x}^{2} + {x}^{3}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   10. Simplified 92.7%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, \left(x \cdot x\right) \cdot \left(\varepsilon + x\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

       5. *-lowering-*.f64 91.7

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   13. Simplified 91.7%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

   if -1.99999999999999994e-304 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{x}^{4}} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. associate-+r+ N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      9. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      10. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto {x}^{4} \cdot \left(\varepsilon \cdot 5 - \color{blue}{\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot {\varepsilon}^{2}\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto {x}^{3} \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right)} \]

      4. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) - -10 \cdot {\varepsilon}^{2}\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot {\varepsilon}^{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{10} \cdot {\varepsilon}^{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot \varepsilon} + 10 \cdot {\varepsilon}^{2}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\varepsilon \cdot \left(5 \cdot x\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{{\varepsilon}^{2} \cdot 10}\right) \]

      15. unpow2 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 10\right) \]

      16. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot 10\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(5 \cdot x\right) + \varepsilon \cdot \color{blue}{\left(10 \cdot \varepsilon\right)}\right) \]

      18. distribute-lft-out N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      20. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right)}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

      22. *-lowering-*.f64 99.9

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(5 \cdot x + \varepsilon \cdot 10\right) \cdot \varepsilon\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right) \cdot \varepsilon} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right) \cdot \varepsilon} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right)} \cdot \varepsilon \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right) \cdot \varepsilon \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot x + \varepsilon \cdot 10\right)\right) \cdot \varepsilon \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)}\right) \cdot \varepsilon \]

      8. *-lowering-*.f64 99.9

         \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \cdot \varepsilon \]

   10. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right) \cdot \varepsilon} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\left(5 \cdot {x}^{4}\right)} \cdot \varepsilon \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{4} \cdot 5\right)} \cdot \varepsilon \]

       2. metadata-eval N/A

          \[\leadsto \left({x}^{\color{blue}{\left(3 + 1\right)}} \cdot 5\right) \cdot \varepsilon \]

       3. pow-plus N/A

          \[\leadsto \left(\color{blue}{\left({x}^{3} \cdot x\right)} \cdot 5\right) \cdot \varepsilon \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left({x}^{3} \cdot \left(x \cdot 5\right)\right)} \cdot \varepsilon \]

       5. *-commutative N/A

          \[\leadsto \left({x}^{3} \cdot \color{blue}{\left(5 \cdot x\right)}\right) \cdot \varepsilon \]

       6. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(5 \cdot x\right) \cdot {x}^{3}\right)} \cdot \varepsilon \]

       7. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot 5\right)} \cdot {x}^{3}\right) \cdot \varepsilon \]

       8. associate-*l* N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(5 \cdot {x}^{3}\right)\right)} \cdot \varepsilon \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(5 \cdot {x}^{3}\right)\right)} \cdot \varepsilon \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(x \cdot \color{blue}{\left(5 \cdot {x}^{3}\right)}\right) \cdot \varepsilon \]

       11. cube-mult N/A

           \[\leadsto \left(x \cdot \left(5 \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \varepsilon \]

       12. unpow2 N/A

           \[\leadsto \left(x \cdot \left(5 \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right)\right) \cdot \varepsilon \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(x \cdot \left(5 \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \cdot \varepsilon \]

       14. unpow2 N/A

           \[\leadsto \left(x \cdot \left(5 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \cdot \varepsilon \]

       15. *-lowering-*.f64 99.9

           \[\leadsto \left(x \cdot \left(5 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \cdot \varepsilon \]

   13. Simplified 99.9%

       \[\leadsto \color{blue}{\left(x \cdot \left(5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \cdot \varepsilon \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 87.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* eps eps) (* eps (* eps eps))))

C

double code(double x, double eps) {
	return (eps * eps) * (eps * (eps * eps));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * eps) * (eps * (eps * eps))
end function

Java

public static double code(double x, double eps) {
	return (eps * eps) * (eps * (eps * eps));
}

Python

def code(x, eps):
	return (eps * eps) * (eps * (eps * eps))

Julia

function code(x, eps)
	return Float64(Float64(eps * eps) * Float64(eps * Float64(eps * eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * eps) * (eps * (eps * eps));
end

Wolfram

code[x_, eps_] := N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - {x}^{5}} \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} - {x}^{5} \]

   3. +-lowering-+.f64 N/A

      \[\leadsto {\color{blue}{\left(x + \varepsilon\right)}}^{5} - {x}^{5} \]

   4. metadata-eval N/A

      \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{\left(3 + 2\right)}} \]

   5. pow-prod-up N/A

      \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot {x}^{2}} \]

   6. pow2 N/A

      \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{3} \cdot \color{blue}{\left(x \cdot x\right)} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{3} \cdot \left(x \cdot x\right)} \]

   8. cube-mult N/A

      \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right) \]

   11. *-lowering-*.f64 86.8

       \[\leadsto {\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

4. Applied egg-rr 86.8%

   \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]

7. Simplified 88.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \mathsf{fma}\left(5, x, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot \left(x \cdot 10\right)\right) \cdot \left(\varepsilon + x\right)\right)\right)} \]

8. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right) \]

   4. distribute-rgt-in N/A

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3} + \color{blue}{\left(\left(10 \cdot {x}^{2}\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)}\right) \]

   5. associate-+r+ N/A

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(10 \cdot {x}^{3} + \left(10 \cdot {x}^{2}\right) \cdot \varepsilon\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right)} \]

   6. associate-*r* N/A

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + \color{blue}{10 \cdot \left({x}^{2} \cdot \varepsilon\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

   7. *-commutative N/A

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(10 \cdot {x}^{3} + 10 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right) + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

   8. +-commutative N/A

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

   9. distribute-lft-out N/A

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right)} + \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right) \cdot \varepsilon\right) \]

   10. *-commutative N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2} + {x}^{3}\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)}\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(10, \varepsilon \cdot {x}^{2} + {x}^{3}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]

10. Simplified 88.6%

    \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(10, \left(x \cdot x\right) \cdot \left(\varepsilon + x\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon\right)\right)} \]

11. Taylor expanded in x around 0

    \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
12. Step-by-step derivation

    1. cube-mult N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

    2. unpow2 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)} \]

    4. unpow2 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

    5. *-lowering-*.f64 88.4

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

13. Simplified 88.4%

    \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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)))