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

Percentage Accurate: 88.6% → 99.5%

Time: 9.4s

Alternatives: 14

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: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot {x}^{3}, x, t\_0\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(10, \frac{\varepsilon \cdot \varepsilon}{x}, 4 \cdot \varepsilon\right), {x}^{4}, {x}^{4} \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(x \cdot x\right) \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ eps x) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -1e-321)
     (fma (* (- x) (pow x 3.0)) x t_0)
     (if (<= t_1 0.0)
       (fma
        (fma 10.0 (/ (* eps eps) x) (* 4.0 eps))
        (pow x 4.0)
        (* (pow x 4.0) eps))
       (- t_0 (* (* x x) (pow x 3.0)))))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = fma((-x * pow(x, 3.0)), x, t_0);
	} else if (t_1 <= 0.0) {
		tmp = fma(fma(10.0, ((eps * eps) / x), (4.0 * eps)), pow(x, 4.0), (pow(x, 4.0) * eps));
	} else {
		tmp = t_0 - ((x * x) * pow(x, 3.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps + x) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = fma(Float64(Float64(-x) * (x ^ 3.0)), x, t_0);
	elseif (t_1 <= 0.0)
		tmp = fma(fma(10.0, Float64(Float64(eps * eps) / x), Float64(4.0 * eps)), (x ^ 4.0), Float64((x ^ 4.0) * eps));
	else
		tmp = Float64(t_0 - Float64(Float64(x * x) * (x ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-321], N[(N[((-x) * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] * x + t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(10.0 * N[(N[(eps * eps), $MachinePrecision] / x), $MachinePrecision] + N[(4.0 * eps), $MachinePrecision]), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(x * x), $MachinePrecision] * N[Power[x, 3.0], $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))) < -9.98013e-322

   1. Initial program 96.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 96.3

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 96.3

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

   6. Applied rewrites 96.3%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \frac{{\varepsilon}^{2}}{x} + \left(4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 94.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 94.5

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

   4. Applied rewrites 94.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot {x}^{3}, x, {\left(\varepsilon + x\right)}^{5}\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(10, \frac{\varepsilon \cdot \varepsilon}{x}, 4 \cdot \varepsilon\right), {x}^{4}, {x}^{4} \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - \left(x \cdot x\right) \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot {x}^{3}, x, t\_0\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(x \cdot x\right) \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ eps x) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -1e-321)
     (fma (* (- x) (pow x 3.0)) x t_0)
     (if (<= t_1 0.0)
       (* (* (pow x 4.0) eps) 5.0)
       (- t_0 (* (* x x) (pow x 3.0)))))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = fma((-x * pow(x, 3.0)), x, t_0);
	} else if (t_1 <= 0.0) {
		tmp = (pow(x, 4.0) * eps) * 5.0;
	} else {
		tmp = t_0 - ((x * x) * pow(x, 3.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps + x) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = fma(Float64(Float64(-x) * (x ^ 3.0)), x, t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * eps) * 5.0);
	else
		tmp = Float64(t_0 - Float64(Float64(x * x) * (x ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-321], N[(N[((-x) * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] * x + t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision], N[(t$95$0 - N[(N[(x * x), $MachinePrecision] * N[Power[x, 3.0], $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))) < -9.98013e-322

   1. Initial program 96.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 96.3

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 96.3

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

   6. Applied rewrites 96.3%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 94.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 94.5

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

   4. Applied rewrites 94.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot {x}^{3}, x, {\left(\varepsilon + x\right)}^{5}\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - \left(x \cdot x\right) \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(x \cdot x\right) \cdot {x}^{3}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ eps x) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (- t_0 (* (* x x) (pow x 3.0)))))
   (if (<= t_1 -1e-321)
     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((eps + x), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double t_2 = t_0 - ((x * x) * pow(x, 3.0));
	double tmp;
	if (t_1 <= -1e-321) {
		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 = (eps + x) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    t_2 = t_0 - ((x * x) * (x ** 3.0d0))
    if (t_1 <= (-1d-321)) 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((eps + x), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double t_2 = t_0 - ((x * x) * Math.pow(x, 3.0));
	double tmp;
	if (t_1 <= -1e-321) {
		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((eps + x), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	t_2 = t_0 - ((x * x) * math.pow(x, 3.0))
	tmp = 0
	if t_1 <= -1e-321:
		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(eps + x) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = Float64(t_0 - Float64(Float64(x * x) * (x ^ 3.0)))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * eps) * 5.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (eps + x) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	t_2 = t_0 - ((x * x) * (x ^ 3.0));
	tmp = 0.0;
	if (t_1 <= -1e-321)
		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[(eps + x), $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[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-321], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * 5.0), $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))) < -9.98013e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 95.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 95.4

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

   4. Applied rewrites 95.4%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - \left(x \cdot x\right) \cdot {x}^{3}\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - \left(x \cdot x\right) \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, -x, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ eps x) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -1e-321)
     t_1
     (if (<= t_1 0.0)
       (* (* (pow x 4.0) eps) 5.0)
       (fma (pow x 4.0) (- x) t_0)))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (pow(x, 4.0) * eps) * 5.0;
	} else {
		tmp = fma(pow(x, 4.0), -x, t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps + x) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * eps) * 5.0);
	else
		tmp = fma((x ^ 4.0), Float64(-x), t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-321], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * (-x) + 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))) < -9.98013e-322

   1. Initial program 96.2%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 94.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 94.5

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

   4. Applied rewrites 94.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-plus N/A

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

      9. metadata-eval N/A

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

      10. lift-pow.f64 N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 94.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 94.4

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

   6. Applied rewrites 94.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, -x, {\left(\varepsilon + x\right)}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-x\right), x, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ eps x) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -1e-321)
     t_1
     (if (<= t_1 0.0)
       (* (* (pow x 4.0) eps) 5.0)
       (fma (* (* (* x x) x) (- x)) x t_0)))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (pow(x, 4.0) * eps) * 5.0;
	} else {
		tmp = fma((((x * x) * x) * -x), x, t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps + x) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * eps) * 5.0);
	else
		tmp = fma(Float64(Float64(Float64(x * x) * x) * Float64(-x)), x, t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-321], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * (-x)), $MachinePrecision] * x + 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))) < -9.98013e-322

   1. Initial program 96.2%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 94.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 94.5

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

   4. Applied rewrites 94.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 94.3

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 94.3

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

   6. Applied rewrites 94.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 94.3

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

   8. Applied rewrites 94.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-x\right), x, {\left(\varepsilon + x\right)}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-x\right), x, t\_0\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ eps x) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (fma (* (* (* x x) x) (- x)) x t_0)))
   (if (<= t_1 -1e-321)
     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((eps + x), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double t_2 = fma((((x * x) * x) * -x), x, t_0);
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (pow(x, 4.0) * eps) * 5.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps + x) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = fma(Float64(Float64(Float64(x * x) * x) * Float64(-x)), x, t_0)
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * eps) * 5.0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * (-x)), $MachinePrecision] * x + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-321], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * 5.0), $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))) < -9.98013e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 95.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 95.4

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

   4. Applied rewrites 95.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 95.3

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 95.3

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

   6. Applied rewrites 95.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 95.3

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

   8. Applied rewrites 95.3%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-x\right), x, {\left(\varepsilon + x\right)}^{5}\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-x\right), x, {\left(\varepsilon + x\right)}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 10\\ t_1 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, t\_0\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_0, \varepsilon + x, \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) 10.0)) (t_1 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_1 -1e-321)
     (* (* (fma (fma 5.0 x eps) eps t_0) eps) (* eps eps))
     (if (<= t_1 0.0)
       (* (* (pow x 4.0) eps) 5.0)
       (* (* (fma t_0 (+ eps x) (* (* (fma 5.0 x eps) eps) eps)) eps) eps)))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * 10.0;
	double t_1 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = (fma(fma(5.0, x, eps), eps, t_0) * eps) * (eps * eps);
	} else if (t_1 <= 0.0) {
		tmp = (pow(x, 4.0) * eps) * 5.0;
	} else {
		tmp = (fma(t_0, (eps + x), ((fma(5.0, x, eps) * eps) * eps)) * eps) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * 10.0)
	t_1 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = Float64(Float64(fma(fma(5.0, x, eps), eps, t_0) * eps) * Float64(eps * eps));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * eps) * 5.0);
	else
		tmp = Float64(Float64(fma(t_0, Float64(eps + x), Float64(Float64(fma(5.0, x, eps) * eps) * eps)) * eps) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-321], N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[(eps + x), $MachinePrecision] + N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $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))) < -9.98013e-322

   1. Initial program 96.2%

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

   3. Taylor expanded in eps around -inf

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 88.1%

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

   10. Applied rewrites 88.2%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 89.0%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 94.3%

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

   3. Taylor expanded in eps around -inf

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 90.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 10, \varepsilon + x, \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 10\\ t_1 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, t\_0\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_0, \varepsilon + x, \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) 10.0)) (t_1 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_1 -1e-321)
     (* (* (fma (fma 5.0 x eps) eps t_0) eps) (* eps eps))
     (if (<= t_1 0.0)
       (* (* (* (* eps x) 5.0) (* x x)) x)
       (* (* (fma t_0 (+ eps x) (* (* (fma 5.0 x eps) eps) eps)) eps) eps)))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * 10.0;
	double t_1 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = (fma(fma(5.0, x, eps), eps, t_0) * eps) * (eps * eps);
	} else if (t_1 <= 0.0) {
		tmp = (((eps * x) * 5.0) * (x * x)) * x;
	} else {
		tmp = (fma(t_0, (eps + x), ((fma(5.0, x, eps) * eps) * eps)) * eps) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * 10.0)
	t_1 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = Float64(Float64(fma(fma(5.0, x, eps), eps, t_0) * eps) * Float64(eps * eps));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(eps * x) * 5.0) * Float64(x * x)) * x);
	else
		tmp = Float64(Float64(fma(t_0, Float64(eps + x), Float64(Float64(fma(5.0, x, eps) * eps) * eps)) * eps) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-321], N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(t$95$0 * N[(eps + x), $MachinePrecision] + N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $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))) < -9.98013e-322

   1. Initial program 96.2%

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

   3. Taylor expanded in eps around -inf

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 88.1%

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

   10. Applied rewrites 88.2%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 89.0%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 88.5

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in eps around 0

       \[\leadsto \left(\left(5 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 99.9%

       \[\leadsto \left(\left(\left(x \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot x \]

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

   1. Initial program 94.3%

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

   3. Taylor expanded in eps around -inf

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 90.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 10, \varepsilon + x, \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ t_1 := \left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0)))
        (t_1
         (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) eps) (* eps eps))))
   (if (<= t_0 -1e-321)
     t_1
     (if (<= t_0 0.0) (* (* (* (* eps x) 5.0) (* x x)) x) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double t_1 = (fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * eps) * (eps * eps);
	double tmp;
	if (t_0 <= -1e-321) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (((eps * x) * 5.0) * (x * x)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * eps) * Float64(eps * eps))
	tmp = 0.0
	if (t_0 <= -1e-321)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(eps * x) * 5.0) * Float64(x * x)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-321], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $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))) < -9.98013e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 95.3%

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

   3. Taylor expanded in eps around -inf

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 89.2%

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

   10. Applied rewrites 89.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 89.2%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 88.5

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in eps around 0

       \[\leadsto \left(\left(5 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 99.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ t_1 := \left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0)))
        (t_1 (* (* (* (fma 5.0 x eps) eps) eps) (* eps eps))))
   (if (<= t_0 -1e-321)
     t_1
     (if (<= t_0 0.0) (* (* (* (* eps x) 5.0) (* x x)) x) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double t_1 = ((fma(5.0, x, eps) * eps) * eps) * (eps * eps);
	double tmp;
	if (t_0 <= -1e-321) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (((eps * x) * 5.0) * (x * x)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(Float64(fma(5.0, x, eps) * eps) * eps) * Float64(eps * eps))
	tmp = 0.0
	if (t_0 <= -1e-321)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(eps * x) * 5.0) * Float64(x * x)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-321], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $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))) < -9.98013e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 95.3%

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

   3. Taylor expanded in eps around -inf

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 89.2%

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

   10. Applied rewrites 89.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 87.5%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 88.5

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in eps around 0

       \[\leadsto \left(\left(5 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.0% accurate, 8.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* (* (* (* eps x) 5.0) (* x x)) x))

C

double code(double x, double eps) {
	return (((eps * x) * 5.0) * (x * x)) * x;
}

Fortran

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

Java

public static double code(double x, double eps) {
	return (((eps * x) * 5.0) * (x * x)) * x;
}

Python

def code(x, eps):
	return (((eps * x) * 5.0) * (x * x)) * x

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(eps * x) * 5.0) * Float64(x * x)) * x)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((eps * x) * 5.0) * (x * x)) * x;
end

Wolfram

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

Derivation

1. Initial program 89.8%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 88.4

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

5. Applied rewrites 88.4%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 83.7%

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

10. Applied rewrites 83.7%

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

11. Taylor expanded in eps around 0

    \[\leadsto \left(\left(5 \cdot \left(\varepsilon \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot x \]
12. Step-by-step derivation

13. Applied rewrites 83.6%

    \[\leadsto \left(\left(\left(x \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot x \]

14. Final simplification 83.6%

    \[\leadsto \left(\left(\left(\varepsilon \cdot x\right) \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot x \]
15. Add Preprocessing

Alternative 12: 83.0% accurate, 8.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* (* (* (* x x) (* x x)) 5.0) eps))

C

double code(double x, double eps) {
	return (((x * x) * (x * x)) * 5.0) * eps;
}

Fortran

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

Java

public static double code(double x, double eps) {
	return (((x * x) * (x * x)) * 5.0) * eps;
}

Python

def code(x, eps):
	return (((x * x) * (x * x)) * 5.0) * eps

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * 5.0) * eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((x * x) * (x * x)) * 5.0) * eps;
end

Wolfram

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

Derivation

1. Initial program 89.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 83.6

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

5. Applied rewrites 83.6%

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

7. Applied rewrites 83.5%

   \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

8. Final simplification 83.5%

   \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon \]
9. Add Preprocessing

Alternative 13: 71.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(\varepsilon \cdot x\right) \cdot 10\right) \cdot x\right) \cdot \varepsilon\right) \cdot x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* (* (* (* eps x) 10.0) x) eps) x))

C

double code(double x, double eps) {
	return ((((eps * x) * 10.0) * x) * eps) * x;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((((eps * x) * 10.0d0) * x) * eps) * x
end function

Java

public static double code(double x, double eps) {
	return ((((eps * x) * 10.0) * x) * eps) * x;
}

Python

def code(x, eps):
	return ((((eps * x) * 10.0) * x) * eps) * x

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(Float64(eps * x) * 10.0) * x) * eps) * x)
end

MATLAB

function tmp = code(x, eps)
	tmp = ((((eps * x) * 10.0) * x) * eps) * x;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(eps * x), $MachinePrecision] * 10.0), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 89.8%

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

3. Taylor expanded in eps around -inf

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

5. Applied rewrites 71.6%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 73.3%

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

9. Taylor expanded in eps around 0

   \[\leadsto \left(10 \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right)\right) \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 73.1%

    \[\leadsto \left(\left(\left(\left(x \cdot \varepsilon\right) \cdot 10\right) \cdot x\right) \cdot \varepsilon\right) \cdot x \]

12. Final simplification 73.1%

    \[\leadsto \left(\left(\left(\left(\varepsilon \cdot x\right) \cdot 10\right) \cdot x\right) \cdot \varepsilon\right) \cdot x \]
13. Add Preprocessing

Alternative 14: 71.5% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 89.8%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. pow2 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 87.5

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

4. Applied rewrites 87.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-neg-in N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 82.7

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 82.7

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

6. Applied rewrites 82.7%

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

7. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{-1 \cdot {x}^{5} + {x}^{5}} \]
8. Step-by-step derivation

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot {x}^{5} \]

   3. mul0-lft 73.0

      \[\leadsto \color{blue}{0} \]

9. Applied rewrites 73.0%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024277 
(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)))