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

Percentage Accurate: 88.5% → 99.1%

Time: 12.5s

Alternatives: 19

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: 88.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_0 - \left(x \cdot x\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, x \cdot \left(-x\right), t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (* x (* x x))))
   (if (<= t_1 -2e-305)
     (- t_0 (* (* x x) t_2))
     (if (<= t_1 0.0)
       (* (pow x 4.0) (- (* eps 5.0) (/ (* (* eps eps) -10.0) x)))
       (fma t_2 (* x (- x)) t_0)))))

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 = x * (x * x);
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_0 - ((x * x) * t_2);
	} else if (t_1 <= 0.0) {
		tmp = pow(x, 4.0) * ((eps * 5.0) - (((eps * eps) * -10.0) / x));
	} else {
		tmp = fma(t_2, (x * -x), t_0);
	}
	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(x * Float64(x * x))
	tmp = 0.0
	if (t_1 <= -2e-305)
		tmp = Float64(t_0 - Float64(Float64(x * x) * t_2));
	elseif (t_1 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) - Float64(Float64(Float64(eps * eps) * -10.0) / x)));
	else
		tmp = fma(t_2, Float64(x * Float64(-x)), t_0);
	end
	return 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[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-305], N[(t$95$0 - N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(x * (-x)), $MachinePrecision] + t$95$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))) < -1.99999999999999999e-305

   1. Initial program 97.5%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 97.5

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. cube-mult N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 97.5

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

   4. Applied rewrites 97.5%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

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

   1. Initial program 96.8%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 96.8

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. cube-mult N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 96.8

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

   4. Applied rewrites 96.8%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 96.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 96.9

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

   6. Applied rewrites 96.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), x \cdot \left(-x\right), {\left(x + \varepsilon\right)}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_0 - \left(x \cdot x\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, x \cdot \left(-x\right), t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (* x (* x x))))
   (if (<= t_1 -2e-305)
     (- t_0 (* (* x x) t_2))
     (if (<= t_1 0.0) (* (pow x 4.0) (* eps 5.0)) (fma t_2 (* x (- x)) t_0)))))

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 = x * (x * x);
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_0 - ((x * x) * t_2);
	} else if (t_1 <= 0.0) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = fma(t_2, (x * -x), t_0);
	}
	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(x * Float64(x * x))
	tmp = 0.0
	if (t_1 <= -2e-305)
		tmp = Float64(t_0 - Float64(Float64(x * x) * t_2));
	elseif (t_1 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = fma(t_2, Float64(x * Float64(-x)), t_0);
	end
	return 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[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-305], N[(t$95$0 - N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(x * (-x)), $MachinePrecision] + t$95$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))) < -1.99999999999999999e-305

   1. Initial program 97.5%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 97.5

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. cube-mult N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 97.5

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

   4. Applied rewrites 97.5%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

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

   1. Initial program 96.8%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 96.8

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. cube-mult N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 96.8

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

   4. Applied rewrites 96.8%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 96.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 96.9

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

   6. Applied rewrites 96.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), x \cdot \left(-x\right), {\left(x + \varepsilon\right)}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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-305)
     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-305) {
		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-305)) 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-305) {
		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-305:
		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-305)
		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-305)
		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-305], 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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.2%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 97.2

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. cube-mult N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 97.2

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

   4. Applied rewrites 97.2%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := {\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (pow eps 5.0) (fma 5.0 (/ x eps) 1.0))))
   (if (<= t_0 -2e-305)
     t_1
     (if (<= t_0 0.0) (* (pow x 4.0) (* eps 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 = pow(eps, 5.0) * fma(5.0, (x / eps), 1.0);
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = pow(x, 4.0) * (eps * 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((eps ^ 5.0) * fma(5.0, Float64(x / eps), 1.0))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	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[Power[eps, 5.0], $MachinePrecision] * N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.2%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-305)
     (pow eps 5.0)
     (if (<= t_0 0.0)
       (* (pow x 4.0) (* eps 5.0))
       (* (pow eps 4.0) (fma 5.0 x 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-305) {
		tmp = pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = pow(eps, 4.0) * fma(5.0, x, 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-305)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = Float64((eps ^ 4.0) * fma(5.0, x, 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-305], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(5.0 * x + eps), $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))) < -1.99999999999999999e-305

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

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

   1. Initial program 96.8%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 90.2

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

   8. Applied rewrites 90.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \varepsilon, t\_1 \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* eps (* eps (* eps eps)))))
   (if (<= t_0 -2e-305)
     (pow eps 5.0)
     (if (<= t_0 0.0)
       (* (pow x 4.0) (* eps 5.0))
       (fma t_1 eps (* t_1 (* x 5.0)))))))

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));
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = fma(t_1, eps, (t_1 * (x * 5.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(eps * Float64(eps * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = fma(t_1, eps, Float64(t_1 * Float64(x * 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]}, Block[{t$95$1 = N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * eps + N[(t$95$1 * N[(x * 5.0), $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))) < -1.99999999999999999e-305

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

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

   1. Initial program 96.8%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 90.2

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

   8. Applied rewrites 90.2%

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

      1. lift-pow.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow-prod-up N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lift-*.f64 N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 90.0

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

   10. Applied rewrites 90.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, 10 \cdot \frac{\varepsilon \cdot \varepsilon}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \varepsilon, t\_1 \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* eps (* eps (* eps eps)))))
   (if (<= t_0 -2e-305)
     (pow eps 5.0)
     (if (<= t_0 0.0)
       (* (fma eps 5.0 (* 10.0 (/ (* eps eps) x))) (* x (* x (* x x))))
       (fma t_1 eps (* t_1 (* x 5.0)))))))

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));
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = fma(eps, 5.0, (10.0 * ((eps * eps) / x))) * (x * (x * (x * x)));
	} else {
		tmp = fma(t_1, eps, (t_1 * (x * 5.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(eps * Float64(eps * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(eps, 5.0, Float64(10.0 * Float64(Float64(eps * eps) / x))) * Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = fma(t_1, eps, Float64(t_1 * Float64(x * 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]}, Block[{t$95$1 = N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(eps * 5.0 + N[(10.0 * N[(N[(eps * eps), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * eps + N[(t$95$1 * N[(x * 5.0), $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))) < -1.99999999999999999e-305

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

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

   1. Initial program 96.8%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 90.2

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

   8. Applied rewrites 90.2%

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

      1. lift-pow.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow-prod-up N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lift-*.f64 N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 90.0

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

   10. Applied rewrites 90.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, 10 \cdot \frac{\varepsilon \cdot \varepsilon}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \left(x \cdot 25\right)\right)\\ t_1 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_2 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ t_3 := \varepsilon \cdot t\_2\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\frac{t\_2 \cdot t\_0}{t\_0} \cdot \frac{\varepsilon}{\frac{1}{\mathsf{fma}\left(5, x, \varepsilon\right)}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, 10 \cdot \frac{\varepsilon \cdot \varepsilon}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, \varepsilon, t\_3 \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma eps eps (* x (* x 25.0))))
        (t_1 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_2 (* eps (* eps eps)))
        (t_3 (* eps t_2)))
   (if (<= t_1 -2e-305)
     (* (/ (* t_2 t_0) t_0) (/ eps (/ 1.0 (fma 5.0 x eps))))
     (if (<= t_1 0.0)
       (* (fma eps 5.0 (* 10.0 (/ (* eps eps) x))) (* x (* x (* x x))))
       (fma t_3 eps (* t_3 (* x 5.0)))))))

C

double code(double x, double eps) {
	double t_0 = fma(eps, eps, (x * (x * 25.0)));
	double t_1 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_2 = eps * (eps * eps);
	double t_3 = eps * t_2;
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = ((t_2 * t_0) / t_0) * (eps / (1.0 / fma(5.0, x, eps)));
	} else if (t_1 <= 0.0) {
		tmp = fma(eps, 5.0, (10.0 * ((eps * eps) / x))) * (x * (x * (x * x)));
	} else {
		tmp = fma(t_3, eps, (t_3 * (x * 5.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(eps, eps, Float64(x * Float64(x * 25.0)))
	t_1 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_2 = Float64(eps * Float64(eps * eps))
	t_3 = Float64(eps * t_2)
	tmp = 0.0
	if (t_1 <= -2e-305)
		tmp = Float64(Float64(Float64(t_2 * t_0) / t_0) * Float64(eps / Float64(1.0 / fma(5.0, x, eps))));
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(eps, 5.0, Float64(10.0 * Float64(Float64(eps * eps) / x))) * Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = fma(t_3, eps, Float64(t_3 * Float64(x * 5.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * eps + N[(x * N[(x * 25.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(eps * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-305], N[(N[(N[(t$95$2 * t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(eps / N[(1.0 / N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(eps * 5.0 + N[(10.0 * N[(N[(eps * eps), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * eps + N[(t$95$3 * N[(x * 5.0), $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))) < -1.99999999999999999e-305

   1. Initial program 97.5%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 96.4

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

   8. Applied rewrites 96.4%

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

      1. lift-pow.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow-prod-up N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lift-*.f64 N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   12. Applied rewrites 96.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \left(x \cdot 25\right)\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \left(x \cdot 25\right)\right)} \cdot \frac{\varepsilon}{\frac{1}{\mathsf{fma}\left(5, x, \varepsilon\right)}}} \]

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

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

   1. Initial program 96.8%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 90.2

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

   8. Applied rewrites 90.2%

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

      1. lift-pow.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow-prod-up N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lift-*.f64 N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 90.0

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

   10. Applied rewrites 90.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \left(x \cdot 25\right)\right)}{\mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \left(x \cdot 25\right)\right)} \cdot \frac{\varepsilon}{\frac{1}{\mathsf{fma}\left(5, x, \varepsilon\right)}}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, 10 \cdot \frac{\varepsilon \cdot \varepsilon}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, 10 \cdot \frac{\varepsilon \cdot \varepsilon}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \varepsilon, t\_1 \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* eps (* eps (* eps eps)))))
   (if (<= t_0 -2e-305)
     (* (fma 5.0 x eps) t_1)
     (if (<= t_0 0.0)
       (* (fma eps 5.0 (* 10.0 (/ (* eps eps) x))) (* x (* x (* x x))))
       (fma t_1 eps (* t_1 (* x 5.0)))))))

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));
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = fma(5.0, x, eps) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = fma(eps, 5.0, (10.0 * ((eps * eps) / x))) * (x * (x * (x * x)));
	} else {
		tmp = fma(t_1, eps, (t_1 * (x * 5.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(eps * Float64(eps * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = Float64(fma(5.0, x, eps) * t_1);
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(eps, 5.0, Float64(10.0 * Float64(Float64(eps * eps) / x))) * Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = fma(t_1, eps, Float64(t_1 * Float64(x * 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]}, Block[{t$95$1 = N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], N[(N[(5.0 * x + eps), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(eps * 5.0 + N[(10.0 * N[(N[(eps * eps), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * eps + N[(t$95$1 * N[(x * 5.0), $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))) < -1.99999999999999999e-305

   1. Initial program 97.5%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 96.4

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

   8. Applied rewrites 96.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 96.4

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

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

   1. Initial program 96.8%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 90.2

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

   8. Applied rewrites 90.2%

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

      1. lift-pow.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow-prod-up N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lift-*.f64 N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 90.0

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

   10. Applied rewrites 90.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5, 10 \cdot \frac{\varepsilon \cdot \varepsilon}{x}\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \varepsilon, t\_1 \cdot \left(x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* eps (* eps (* eps eps)))))
   (if (<= t_0 -2e-305)
     (* (fma 5.0 x eps) t_1)
     (if (<= t_0 0.0)
       (* (fma eps 10.0 (* x 5.0)) (* eps (* x (* x x))))
       (fma t_1 eps (* t_1 (* x 5.0)))))))

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));
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = fma(5.0, x, eps) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = fma(eps, 10.0, (x * 5.0)) * (eps * (x * (x * x)));
	} else {
		tmp = fma(t_1, eps, (t_1 * (x * 5.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(eps * Float64(eps * Float64(eps * eps)))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = Float64(fma(5.0, x, eps) * t_1);
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(eps, 10.0, Float64(x * 5.0)) * Float64(eps * Float64(x * Float64(x * x))));
	else
		tmp = fma(t_1, eps, Float64(t_1 * Float64(x * 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]}, Block[{t$95$1 = N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], N[(N[(5.0 * x + eps), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(eps * 10.0 + N[(x * 5.0), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * eps + N[(t$95$1 * N[(x * 5.0), $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))) < -1.99999999999999999e-305

   1. Initial program 97.5%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 96.4

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

   8. Applied rewrites 96.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 96.4

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.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. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

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

   1. Initial program 96.8%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 90.2

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

   8. Applied rewrites 90.2%

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

      1. lift-pow.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow-prod-up N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lift-*.f64 N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 90.0

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

   10. Applied rewrites 90.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (fma 5.0 x eps) (* eps (* eps (* eps eps))))))
   (if (<= t_0 -2e-305)
     t_1
     (if (<= t_0 0.0)
       (* (fma eps 10.0 (* x 5.0)) (* eps (* 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 = fma(5.0, x, eps) * (eps * (eps * (eps * eps)));
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = fma(eps, 10.0, (x * 5.0)) * (eps * (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(fma(5.0, x, eps) * Float64(eps * Float64(eps * Float64(eps * eps))))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(eps, 10.0, Float64(x * 5.0)) * Float64(eps * Float64(x * Float64(x * x))));
	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[(5.0 * x + eps), $MachinePrecision] * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(eps * 10.0 + N[(x * 5.0), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(x * N[(x * x), $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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.2%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 93.8

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 93.5

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

   8. Applied rewrites 93.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 93.5

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 93.2

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

   10. Applied rewrites 93.2%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.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. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 10, x \cdot 5\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (fma 5.0 x eps) (* eps (* eps (* eps eps))))))
   (if (<= t_0 -2e-305)
     t_1
     (if (<= t_0 0.0)
       (* (* x (* x x)) (* eps (fma 5.0 x (* 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 = fma(5.0, x, eps) * (eps * (eps * (eps * eps)));
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (x * (x * x)) * (eps * fma(5.0, x, (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(fma(5.0, x, eps) * Float64(eps * Float64(eps * Float64(eps * eps))))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(eps * fma(5.0, x, Float64(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[(5.0 * x + eps), $MachinePrecision] * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$1, 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], 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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.2%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 93.8

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 93.5

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

   8. Applied rewrites 93.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 93.5

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 93.2

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

   10. Applied rewrites 93.2%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.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. Applied rewrites 99.9%

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

Alternative 13: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (* (fma 5.0 x eps) (* eps (* eps (* eps eps))))))
   (if (<= t_0 -2e-305)
     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 = fma(5.0, x, eps) * (eps * (eps * (eps * eps)));
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = t_1;
	} else if (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(fma(5.0, x, eps) * Float64(eps * Float64(eps * Float64(eps * eps))))
	tmp = 0.0
	if (t_0 <= -2e-305)
		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

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[(5.0 * x + eps), $MachinePrecision] * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], 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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.2%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 93.8

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 93.5

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

   8. Applied rewrites 93.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 93.5

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 93.2

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

   10. Applied rewrites 93.2%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.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. Applied rewrites 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 x around inf

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

       1. lower-*.f64 99.9

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

   11. Applied rewrites 99.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-305)
     (* eps (* eps (* eps (* eps eps))))
     (if (<= t_0 0.0)
       (* (* x (* x x)) (* eps (* x 5.0)))
       (* (* eps eps) (* (* eps eps) (fma 5.0 x 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-305) {
		tmp = eps * (eps * (eps * (eps * eps)));
	} else if (t_0 <= 0.0) {
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	} else {
		tmp = (eps * eps) * ((eps * eps) * fma(5.0, x, 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-305)
		tmp = Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(eps * Float64(x * 5.0)));
	else
		tmp = Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * fma(5.0, x, 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-305], N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(5.0 * x + 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))) < -1.99999999999999999e-305

   1. Initial program 97.5%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 96.4

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

   8. Applied rewrites 96.4%

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

      1. lift-pow.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow-prod-up N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lift-*.f64 N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   11. Taylor expanded in eps around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. metadata-eval N/A

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

       6. pow-plus N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 95.9

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

   13. Applied rewrites 95.9%

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

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.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. Applied rewrites 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 x around inf

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

       1. lower-*.f64 99.9

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

   11. Applied rewrites 99.9%

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

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

   1. Initial program 96.8%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 90.2

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

   8. Applied rewrites 90.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{4}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{4}} \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{\color{blue}{\left(2 \cdot 2\right)}} \]

      6. pow-pow N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \color{blue}{{\left({\varepsilon}^{2}\right)}^{2}} \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\color{blue}{\left(\varepsilon \cdot \varepsilon\right)}}^{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\color{blue}{\left(\varepsilon \cdot \varepsilon\right)}}^{2} \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

      12. lower-*.f64 89.6

          \[\leadsto \color{blue}{\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \cdot \left(\varepsilon \cdot \varepsilon\right) \]

   10. Applied rewrites 89.6%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot t\_1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))) (t_1 (* eps (* eps eps))))
   (if (<= t_0 -2e-305)
     (* eps (* eps t_1))
     (if (<= t_0 0.0)
       (* (* x (* x x)) (* eps (* x 5.0)))
       (* t_1 (* eps (fma 5.0 x eps)))))))

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);
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = eps * (eps * t_1);
	} else if (t_0 <= 0.0) {
		tmp = (x * (x * x)) * (eps * (x * 5.0));
	} else {
		tmp = t_1 * (eps * fma(5.0, x, eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(eps * Float64(eps * eps))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = Float64(eps * Float64(eps * t_1));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(eps * Float64(x * 5.0)));
	else
		tmp = Float64(t_1 * Float64(eps * fma(5.0, x, 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]}, Block[{t$95$1 = N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], N[(eps * N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(eps * N[(5.0 * x + 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))) < -1.99999999999999999e-305

   1. Initial program 97.5%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5}} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \]

      5. metadata-eval N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

      7. lower-/.f64 96.7

         \[\leadsto {\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \color{blue}{\frac{x}{\varepsilon}}, 1\right) \]

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4}} \cdot \left(\varepsilon + 5 \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\left(5 \cdot x + \varepsilon\right)} \]

      4. lower-fma.f64 96.4

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right)} \]

   8. Applied rewrites 96.4%

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \mathsf{fma}\left(5, x, \varepsilon\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4}} \cdot \left(5 \cdot x + \varepsilon\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(5 \cdot x\right) \cdot {\varepsilon}^{4} + \varepsilon \cdot {\varepsilon}^{4}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4} + \left(5 \cdot x\right) \cdot {\varepsilon}^{4}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} + \left(5 \cdot x\right) \cdot {\varepsilon}^{4} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{4}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({\varepsilon}^{\color{blue}{\left(2 + 2\right)}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x\right)}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x\right)}\right) \]

      17. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4}} \cdot \left(5 \cdot x\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, {\varepsilon}^{\color{blue}{\left(2 + 2\right)}} \cdot \left(5 \cdot x\right)\right) \]

      19. pow-prod-up N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \cdot \left(5 \cdot x\right)\right) \]

      20. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \cdot \left(5 \cdot x\right)\right) \]

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      23. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \left(5 \cdot x\right)\right) \]

      24. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \left(5 \cdot x\right)\right) \]

      26. lower-*.f64 95.9

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]

   10. Applied rewrites 95.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(5 \cdot x\right)\right)} \]

   11. Taylor expanded in eps around inf

       \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto {\varepsilon}^{\color{blue}{\left(4 + 1\right)}} \]

       2. pow-plus N/A

          \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

       5. metadata-eval N/A

          \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(3 + 1\right)}} \]

       6. pow-plus N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{3} \cdot \varepsilon\right)} \]

       7. *-commutative N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{3}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{3}\right)} \]

       9. cube-mult N/A

          \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)}\right) \]

       12. unpow2 N/A

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       13. lower-*.f64 95.9

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   13. Applied rewrites 95.9%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]

      2. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot \varepsilon} + 10 \cdot {\varepsilon}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(5 \cdot x\right) \cdot \varepsilon + 10 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(5 \cdot x\right) \cdot \varepsilon + \color{blue}{\left(10 \cdot \varepsilon\right) \cdot \varepsilon}\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

      15. lower-*.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. Applied rewrites 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 x around inf

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 99.9

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5}} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \]

      5. metadata-eval N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

      7. lower-/.f64 90.4

         \[\leadsto {\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \color{blue}{\frac{x}{\varepsilon}}, 1\right) \]

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4}} \cdot \left(\varepsilon + 5 \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\left(5 \cdot x + \varepsilon\right)} \]

      4. lower-fma.f64 90.2

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right)} \]

   8. Applied rewrites 90.2%

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \mathsf{fma}\left(5, x, \varepsilon\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4}} \cdot \left(5 \cdot x + \varepsilon\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(5 \cdot x\right) \cdot {\varepsilon}^{4} + \varepsilon \cdot {\varepsilon}^{4}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4} + \left(5 \cdot x\right) \cdot {\varepsilon}^{4}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} + \left(5 \cdot x\right) \cdot {\varepsilon}^{4} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{4}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({\varepsilon}^{\color{blue}{\left(2 + 2\right)}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x\right)}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x\right)}\right) \]

      17. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4}} \cdot \left(5 \cdot x\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, {\varepsilon}^{\color{blue}{\left(2 + 2\right)}} \cdot \left(5 \cdot x\right)\right) \]

      19. pow-prod-up N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \cdot \left(5 \cdot x\right)\right) \]

      20. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \cdot \left(5 \cdot x\right)\right) \]

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      23. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \left(5 \cdot x\right)\right) \]

      24. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \left(5 \cdot x\right)\right) \]

      26. lower-*.f64 90.0

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]

   10. Applied rewrites 90.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(5 \cdot x\right)\right)} \]

   12. Applied rewrites 89.5%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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-305)
     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-305) {
		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-305)) 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-305) {
		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-305:
		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(eps * Float64(eps * Float64(eps * Float64(eps * eps))))
	tmp = 0.0
	if (t_0 <= -2e-305)
		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-305)
		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[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], 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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5}} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \]

      5. metadata-eval N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

      7. lower-/.f64 93.8

         \[\leadsto {\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \color{blue}{\frac{x}{\varepsilon}}, 1\right) \]

   5. Applied rewrites 93.8%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4}} \cdot \left(\varepsilon + 5 \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\left(5 \cdot x + \varepsilon\right)} \]

      4. lower-fma.f64 93.5

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right)} \]

   8. Applied rewrites 93.5%

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \mathsf{fma}\left(5, x, \varepsilon\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4}} \cdot \left(5 \cdot x + \varepsilon\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(5 \cdot x\right) \cdot {\varepsilon}^{4} + \varepsilon \cdot {\varepsilon}^{4}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4} + \left(5 \cdot x\right) \cdot {\varepsilon}^{4}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} + \left(5 \cdot x\right) \cdot {\varepsilon}^{4} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{4}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({\varepsilon}^{\color{blue}{\left(2 + 2\right)}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x\right)}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x\right)}\right) \]

      17. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4}} \cdot \left(5 \cdot x\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, {\varepsilon}^{\color{blue}{\left(2 + 2\right)}} \cdot \left(5 \cdot x\right)\right) \]

      19. pow-prod-up N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \cdot \left(5 \cdot x\right)\right) \]

      20. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \cdot \left(5 \cdot x\right)\right) \]

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      23. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \left(5 \cdot x\right)\right) \]

      24. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \left(5 \cdot x\right)\right) \]

      26. lower-*.f64 93.2

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]

   10. Applied rewrites 93.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(5 \cdot x\right)\right)} \]

   11. Taylor expanded in eps around inf

       \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto {\varepsilon}^{\color{blue}{\left(4 + 1\right)}} \]

       2. pow-plus N/A

          \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

       5. metadata-eval N/A

          \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(3 + 1\right)}} \]

       6. pow-plus N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{3} \cdot \varepsilon\right)} \]

       7. *-commutative N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{3}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{3}\right)} \]

       9. cube-mult N/A

          \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)}\right) \]

       12. unpow2 N/A

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       13. lower-*.f64 92.3

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   13. Applied rewrites 92.3%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]

      2. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot \varepsilon} + 10 \cdot {\varepsilon}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(5 \cdot x\right) \cdot \varepsilon + 10 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(5 \cdot x\right) \cdot \varepsilon + \color{blue}{\left(10 \cdot \varepsilon\right) \cdot \varepsilon}\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

      15. lower-*.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. Applied rewrites 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 x around inf

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 99.9

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \left(x \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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-305)
     t_1
     (if (<= t_0 0.0) (* (* x (* x x)) (* 5.0 (* x eps))) 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-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (x * (x * x)) * (5.0 * (x * eps));
	} 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-305)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (x * (x * x)) * (5.0d0 * (x * eps))
    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-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (x * (x * x)) * (5.0 * (x * eps));
	} 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-305:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (x * (x * x)) * (5.0 * (x * eps))
	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(eps * Float64(eps * Float64(eps * Float64(eps * eps))))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(5.0 * Float64(x * eps)));
	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-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = (x * (x * x)) * (5.0 * (x * eps));
	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[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(5.0 * N[(x * eps), $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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5}} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \]

      5. metadata-eval N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

      7. lower-/.f64 93.8

         \[\leadsto {\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \color{blue}{\frac{x}{\varepsilon}}, 1\right) \]

   5. Applied rewrites 93.8%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4}} \cdot \left(\varepsilon + 5 \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\left(5 \cdot x + \varepsilon\right)} \]

      4. lower-fma.f64 93.5

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right)} \]

   8. Applied rewrites 93.5%

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \mathsf{fma}\left(5, x, \varepsilon\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4}} \cdot \left(5 \cdot x + \varepsilon\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(5 \cdot x\right) \cdot {\varepsilon}^{4} + \varepsilon \cdot {\varepsilon}^{4}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4} + \left(5 \cdot x\right) \cdot {\varepsilon}^{4}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} + \left(5 \cdot x\right) \cdot {\varepsilon}^{4} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{4}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({\varepsilon}^{\color{blue}{\left(2 + 2\right)}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x\right)}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x\right)}\right) \]

      17. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4}} \cdot \left(5 \cdot x\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, {\varepsilon}^{\color{blue}{\left(2 + 2\right)}} \cdot \left(5 \cdot x\right)\right) \]

      19. pow-prod-up N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \cdot \left(5 \cdot x\right)\right) \]

      20. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \cdot \left(5 \cdot x\right)\right) \]

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      23. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \left(5 \cdot x\right)\right) \]

      24. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \left(5 \cdot x\right)\right) \]

      26. lower-*.f64 93.2

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]

   10. Applied rewrites 93.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(5 \cdot x\right)\right)} \]

   11. Taylor expanded in eps around inf

       \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto {\varepsilon}^{\color{blue}{\left(4 + 1\right)}} \]

       2. pow-plus N/A

          \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

       5. metadata-eval N/A

          \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(3 + 1\right)}} \]

       6. pow-plus N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{3} \cdot \varepsilon\right)} \]

       7. *-commutative N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{3}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{3}\right)} \]

       9. cube-mult N/A

          \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)}\right) \]

       12. unpow2 N/A

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       13. lower-*.f64 92.3

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   13. Applied rewrites 92.3%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]

      2. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot \varepsilon} + 10 \cdot {\varepsilon}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(5 \cdot x\right) \cdot \varepsilon + 10 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(5 \cdot x\right) \cdot \varepsilon + \color{blue}{\left(10 \cdot \varepsilon\right) \cdot \varepsilon}\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

      15. lower-*.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. Applied rewrites 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. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right)\right)} \]

       2. lower-*.f64 99.9

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \left(x \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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-305)
     t_1
     (if (<= t_0 0.0) (* eps (* 5.0 (* x (* 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-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * (x * (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-305)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x * (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-305) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * (x * (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-305:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = eps * (5.0 * (x * (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(eps * Float64(eps * Float64(eps * Float64(eps * eps))))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * Float64(x * 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-305)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = eps * (5.0 * (x * (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[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-305], t$95$1, If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[(x * 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))) < -1.99999999999999999e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{5}} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \]

      5. metadata-eval N/A

         \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

      7. lower-/.f64 93.8

         \[\leadsto {\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \color{blue}{\frac{x}{\varepsilon}}, 1\right) \]

   5. Applied rewrites 93.8%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4}} \cdot \left(\varepsilon + 5 \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\left(5 \cdot x + \varepsilon\right)} \]

      4. lower-fma.f64 93.5

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right)} \]

   8. Applied rewrites 93.5%

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \mathsf{fma}\left(5, x, \varepsilon\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4}} \cdot \left(5 \cdot x + \varepsilon\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(5 \cdot x\right) \cdot {\varepsilon}^{4} + \varepsilon \cdot {\varepsilon}^{4}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4} + \left(5 \cdot x\right) \cdot {\varepsilon}^{4}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} + \left(5 \cdot x\right) \cdot {\varepsilon}^{4} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{4}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({\varepsilon}^{\color{blue}{\left(2 + 2\right)}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x\right)}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x\right)}\right) \]

      17. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4}} \cdot \left(5 \cdot x\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, {\varepsilon}^{\color{blue}{\left(2 + 2\right)}} \cdot \left(5 \cdot x\right)\right) \]

      19. pow-prod-up N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \cdot \left(5 \cdot x\right)\right) \]

      20. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \cdot \left(5 \cdot x\right)\right) \]

      21. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      23. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \left(5 \cdot x\right)\right) \]

      24. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \left(5 \cdot x\right)\right) \]

      26. lower-*.f64 93.2

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]

   10. Applied rewrites 93.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(5 \cdot x\right)\right)} \]

   11. Taylor expanded in eps around inf

       \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto {\varepsilon}^{\color{blue}{\left(4 + 1\right)}} \]

       2. pow-plus N/A

          \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

       5. metadata-eval N/A

          \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(3 + 1\right)}} \]

       6. pow-plus N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{3} \cdot \varepsilon\right)} \]

       7. *-commutative N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{3}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{3}\right)} \]

       9. cube-mult N/A

          \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)}\right) \]

       12. unpow2 N/A

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       13. lower-*.f64 92.3

           \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   13. Applied rewrites 92.3%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]

   if -1.99999999999999999e-305 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 83.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.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. lower-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. lower--.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. lower-*.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. lower-/.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. Applied rewrites 99.9%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10}{x}\right)} \]

   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)} \]

      2. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + 10 \cdot {\varepsilon}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(5 \cdot x\right) \cdot \varepsilon} + 10 \cdot {\varepsilon}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(5 \cdot x\right) \cdot \varepsilon + 10 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(5 \cdot x\right) \cdot \varepsilon + \color{blue}{\left(10 \cdot \varepsilon\right) \cdot \varepsilon}\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(5, x, \color{blue}{\varepsilon \cdot 10}\right)\right) \]

      15. lower-*.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. Applied rewrites 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 x around inf

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot {x}^{4}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left(5 \cdot {x}^{4}\right)} \]

       6. metadata-eval N/A

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \]

       7. pow-plus N/A

          \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \varepsilon \cdot \left(5 \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right) \]

       10. cube-mult N/A

           \[\leadsto \varepsilon \cdot \left(5 \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \varepsilon \cdot \left(5 \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       14. lower-*.f64 99.9

           \[\leadsto \varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 87.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (* eps (* eps (* eps eps)))))

C

double code(double x, double eps) {
	return eps * (eps * (eps * (eps * eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (eps * (eps * (eps * eps)))
end function

Java

public static double code(double x, double eps) {
	return eps * (eps * (eps * (eps * eps)));
}

Python

def code(x, eps):
	return eps * (eps * (eps * (eps * eps)))

Julia

function code(x, eps)
	return Float64(eps * Float64(eps * Float64(eps * Float64(eps * eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (eps * (eps * (eps * eps)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.9%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]

   2. lower-pow.f64 N/A

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \]

   5. metadata-eval N/A

      \[\leadsto {\varepsilon}^{5} \cdot \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto {\varepsilon}^{5} \cdot \color{blue}{\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

   7. lower-/.f64 85.2

      \[\leadsto {\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \color{blue}{\frac{x}{\varepsilon}}, 1\right) \]

5. Applied rewrites 85.2%

   \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right)} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)} \]

   2. lower-pow.f64 N/A

      \[\leadsto \color{blue}{{\varepsilon}^{4}} \cdot \left(\varepsilon + 5 \cdot x\right) \]

   3. +-commutative N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\left(5 \cdot x + \varepsilon\right)} \]

   4. lower-fma.f64 85.2

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right)} \]

8. Applied rewrites 85.2%

   \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \mathsf{fma}\left(5, x, \varepsilon\right)} \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \color{blue}{{\varepsilon}^{4}} \cdot \left(5 \cdot x + \varepsilon\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(5 \cdot x\right) \cdot {\varepsilon}^{4} + \varepsilon \cdot {\varepsilon}^{4}} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4} + \left(5 \cdot x\right) \cdot {\varepsilon}^{4}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} + \left(5 \cdot x\right) \cdot {\varepsilon}^{4} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right)} \]

   6. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{4}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({\varepsilon}^{\color{blue}{\left(2 + 2\right)}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

   8. pow-prod-up N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot {\varepsilon}^{2}}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

   9. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

   10. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon, \left(5 \cdot x\right) \cdot {\varepsilon}^{4}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x\right)}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x\right)}\right) \]

   17. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{{\varepsilon}^{4}} \cdot \left(5 \cdot x\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, {\varepsilon}^{\color{blue}{\left(2 + 2\right)}} \cdot \left(5 \cdot x\right)\right) \]

   19. pow-prod-up N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right)} \cdot \left(5 \cdot x\right)\right) \]

   20. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {\varepsilon}^{2}\right) \cdot \left(5 \cdot x\right)\right) \]

   21. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

   22. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

   23. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \left(5 \cdot x\right)\right) \]

   24. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\right) \cdot \left(5 \cdot x\right)\right) \]

   25. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \left(5 \cdot x\right)\right) \]

   26. lower-*.f64 85.1

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \color{blue}{\left(5 \cdot x\right)}\right) \]

10. Applied rewrites 85.1%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(5 \cdot x\right)\right)} \]

11. Taylor expanded in eps around inf

    \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
12. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto {\varepsilon}^{\color{blue}{\left(4 + 1\right)}} \]

    2. pow-plus N/A

       \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

    3. *-commutative N/A

       \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

    4. lower-*.f64 N/A

       \[\leadsto \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

    5. metadata-eval N/A

       \[\leadsto \varepsilon \cdot {\varepsilon}^{\color{blue}{\left(3 + 1\right)}} \]

    6. pow-plus N/A

       \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{3} \cdot \varepsilon\right)} \]

    7. *-commutative N/A

       \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{3}\right)} \]

    8. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{3}\right)} \]

    9. cube-mult N/A

       \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\right) \]

    10. unpow2 N/A

        \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{{\varepsilon}^{2}}\right)\right) \]

    11. lower-*.f64 N/A

        \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot {\varepsilon}^{2}\right)}\right) \]

    12. unpow2 N/A

        \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

    13. lower-*.f64 85.0

        \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

13. Applied rewrites 85.0%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))