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

Percentage Accurate: 88.4% → 99.4%

Time: 7.1s

Alternatives: 9

Speedup: 5.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-320} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(10, \frac{\varepsilon \cdot \varepsilon}{x}, 4 \cdot \varepsilon\right), {x}^{4}, \varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -4e-320) (not (<= t_0 0.0)))
     t_0
     (fma
      (fma 10.0 (/ (* eps eps) x) (* 4.0 eps))
      (pow x 4.0)
      (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-320) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = fma(fma(10.0, ((eps * eps) / x), (4.0 * eps)), pow(x, 4.0), (eps * pow(x, 4.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -4e-320) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = fma(fma(10.0, Float64(Float64(eps * eps) / x), Float64(4.0 * eps)), (x ^ 4.0), Float64(eps * (x ^ 4.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-320], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$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[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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))) < -3.99996e-320 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.6%

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

   if -3.99996e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.5%

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

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-320} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(10, \frac{\varepsilon \cdot \varepsilon}{x}, 4 \cdot \varepsilon\right), {x}^{4}, \varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-320} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -4e-320) (not (<= t_0 0.0)))
     t_0
     (* (* (pow x 4.0) eps) 5.0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-320) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (pow(x, 4.0) * eps) * 5.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-4d-320)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = ((x ** 4.0d0) * eps) * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-320) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (Math.pow(x, 4.0) * eps) * 5.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -4e-320) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (math.pow(x, 4.0) * eps) * 5.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -4e-320) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64((x ^ 4.0) * eps) * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -4e-320) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = ((x ^ 4.0) * eps) * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-320], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision]]]

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))) < -3.99996e-320 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 96.6%

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

   if -3.99996e-320 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 99.9

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-320} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-64}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot \mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right)\right) \cdot \left(\varepsilon \cdot x\right)\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.5e-64)
   (* (* (* x (fma 10.0 eps (* 5.0 x))) eps) (* x x))
   (if (<= x 1.32e-64)
     (* (fma (/ x eps) 5.0 1.0) (pow eps 5.0))
     (* (* (* (* x (fma (/ eps x) 10.0 5.0)) (* eps x)) x) x))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.5e-64) {
		tmp = ((x * fma(10.0, eps, (5.0 * x))) * eps) * (x * x);
	} else if (x <= 1.32e-64) {
		tmp = fma((x / eps), 5.0, 1.0) * pow(eps, 5.0);
	} else {
		tmp = (((x * fma((eps / x), 10.0, 5.0)) * (eps * x)) * x) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.5e-64)
		tmp = Float64(Float64(Float64(x * fma(10.0, eps, Float64(5.0 * x))) * eps) * Float64(x * x));
	elseif (x <= 1.32e-64)
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * (eps ^ 5.0));
	else
		tmp = Float64(Float64(Float64(Float64(x * fma(Float64(eps / x), 10.0, 5.0)) * Float64(eps * x)) * x) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.5e-64], N[(N[(N[(x * N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e-64], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision]), $MachinePrecision] * N[(eps * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e-64

   1. Initial program 61.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \frac{{\varepsilon}^{2}}{x} + \left(4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right)} \]
   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 92.3

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

   5. Applied rewrites 92.3%

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

   7. Applied rewrites 92.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 92.4%

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

   if -1.5e-64 < x < 1.32e-64

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 1.32e-64 < x

   1. Initial program 35.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \frac{{\varepsilon}^{2}}{x} + \left(4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right)} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.3%

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

   12. Applied rewrites 99.6%

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

Alternative 4: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-64}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot \mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right)\right) \cdot \left(\varepsilon \cdot x\right)\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.5e-64)
   (* (* (* x (fma 10.0 eps (* 5.0 x))) eps) (* x x))
   (if (<= x 1.32e-64)
     (* (fma 5.0 x eps) (pow eps 4.0))
     (* (* (* (* x (fma (/ eps x) 10.0 5.0)) (* eps x)) x) x))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.5e-64) {
		tmp = ((x * fma(10.0, eps, (5.0 * x))) * eps) * (x * x);
	} else if (x <= 1.32e-64) {
		tmp = fma(5.0, x, eps) * pow(eps, 4.0);
	} else {
		tmp = (((x * fma((eps / x), 10.0, 5.0)) * (eps * x)) * x) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.5e-64)
		tmp = Float64(Float64(Float64(x * fma(10.0, eps, Float64(5.0 * x))) * eps) * Float64(x * x));
	elseif (x <= 1.32e-64)
		tmp = Float64(fma(5.0, x, eps) * (eps ^ 4.0));
	else
		tmp = Float64(Float64(Float64(Float64(x * fma(Float64(eps / x), 10.0, 5.0)) * Float64(eps * x)) * x) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.5e-64], N[(N[(N[(x * N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e-64], N[(N[(5.0 * x + eps), $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision]), $MachinePrecision] * N[(eps * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e-64

   1. Initial program 61.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \frac{{\varepsilon}^{2}}{x} + \left(4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right)} \]
   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 92.3

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

   5. Applied rewrites 92.3%

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

   7. Applied rewrites 92.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 92.4%

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

   if -1.5e-64 < x < 1.32e-64

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 1.32e-64 < x

   1. Initial program 35.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \frac{{\varepsilon}^{2}}{x} + \left(4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right)} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.3%

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

   12. Applied rewrites 99.6%

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

Alternative 5: 97.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-64}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-64}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot \mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right)\right) \cdot \left(\varepsilon \cdot x\right)\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.5e-64)
   (* (* (* x (fma 10.0 eps (* 5.0 x))) eps) (* x x))
   (if (<= x 1.32e-64)
     (* (* (* (* (fma 5.0 x eps) eps) eps) eps) eps)
     (* (* (* (* x (fma (/ eps x) 10.0 5.0)) (* eps x)) x) x))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.5e-64) {
		tmp = ((x * fma(10.0, eps, (5.0 * x))) * eps) * (x * x);
	} else if (x <= 1.32e-64) {
		tmp = (((fma(5.0, x, eps) * eps) * eps) * eps) * eps;
	} else {
		tmp = (((x * fma((eps / x), 10.0, 5.0)) * (eps * x)) * x) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.5e-64)
		tmp = Float64(Float64(Float64(x * fma(10.0, eps, Float64(5.0 * x))) * eps) * Float64(x * x));
	elseif (x <= 1.32e-64)
		tmp = Float64(Float64(Float64(Float64(fma(5.0, x, eps) * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(Float64(x * fma(Float64(eps / x), 10.0, 5.0)) * Float64(eps * x)) * x) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.5e-64], N[(N[(N[(x * N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e-64], N[(N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(x * N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision]), $MachinePrecision] * N[(eps * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e-64

   1. Initial program 61.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \frac{{\varepsilon}^{2}}{x} + \left(4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right)} \]
   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 92.3

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

   5. Applied rewrites 92.3%

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

   7. Applied rewrites 92.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 92.4%

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

   if -1.5e-64 < x < 1.32e-64

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.9%

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

   if 1.32e-64 < x

   1. Initial program 35.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \frac{{\varepsilon}^{2}}{x} + \left(4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right)} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.3%

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

   12. Applied rewrites 99.6%

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

Alternative 6: 97.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-64} \lor \neg \left(x \leq 1.32 \cdot 10^{-64}\right):\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.5e-64) (not (<= x 1.32e-64)))
   (* (* (* x (fma 10.0 eps (* 5.0 x))) eps) (* x x))
   (* (* (* (* (fma 5.0 x eps) eps) eps) eps) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.5e-64) || !(x <= 1.32e-64)) {
		tmp = ((x * fma(10.0, eps, (5.0 * x))) * eps) * (x * x);
	} else {
		tmp = (((fma(5.0, x, eps) * eps) * eps) * eps) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.5e-64) || !(x <= 1.32e-64))
		tmp = Float64(Float64(Float64(x * fma(10.0, eps, Float64(5.0 * x))) * eps) * Float64(x * x));
	else
		tmp = Float64(Float64(Float64(Float64(fma(5.0, x, eps) * eps) * eps) * eps) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.5e-64], N[Not[LessEqual[x, 1.32e-64]], $MachinePrecision]], N[(N[(N[(x * N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e-64 or 1.32e-64 < x

   1. Initial program 47.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 96.1

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

   5. Applied rewrites 96.1%

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

   7. Applied rewrites 95.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 96.1%

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

   if -1.5e-64 < x < 1.32e-64

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-64} \lor \neg \left(x \leq 1.32 \cdot 10^{-64}\right):\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right)\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-64}:\\ \;\;\;\;\left(\left(\left(5 \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-64}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot x\right) \cdot 5\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.8e-64)
   (* (* (* (* 5.0 eps) x) x) (* x x))
   (if (<= x 1.32e-64)
     (* (* (* (* (fma 5.0 x eps) eps) eps) eps) eps)
     (* (* (* (* eps x) 5.0) x) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.8e-64) {
		tmp = (((5.0 * eps) * x) * x) * (x * x);
	} else if (x <= 1.32e-64) {
		tmp = (((fma(5.0, x, eps) * eps) * eps) * eps) * eps;
	} else {
		tmp = (((eps * x) * 5.0) * x) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.8e-64)
		tmp = Float64(Float64(Float64(Float64(5.0 * eps) * x) * x) * Float64(x * x));
	elseif (x <= 1.32e-64)
		tmp = Float64(Float64(Float64(Float64(fma(5.0, x, eps) * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(Float64(eps * x) * 5.0) * x) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.8e-64], N[(N[(N[(N[(5.0 * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e-64], N[(N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999997e-64

   1. Initial program 59.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \frac{{\varepsilon}^{2}}{x} + \left(4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right)} \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

   7. Applied rewrites 92.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 92.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 91.5%

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

   if -4.79999999999999997e-64 < x < 1.32e-64

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-pow.f64 99.8

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

   8. Applied rewrites 99.8%

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

   10. Applied rewrites 99.7%

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

   12. Applied rewrites 99.8%

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

   if 1.32e-64 < x

   1. Initial program 35.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \frac{{\varepsilon}^{2}}{x} + \left(4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right)} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.3%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 98.0%

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

Alternative 8: 81.6% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \frac{{\varepsilon}^{2}}{x} + \left(4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right)} \]
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 83.5

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

5. Applied rewrites 83.5%

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

7. Applied rewrites 83.5%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 83.5%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 83.2%

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

Alternative 9: 81.6% accurate, 8.0× speedup

Math

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

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) * x) * Float64(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] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(2 \cdot \frac{{\varepsilon}^{2}}{x} + \left(4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right)} \]
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 83.5

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

5. Applied rewrites 83.5%

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

7. Applied rewrites 83.5%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 83.5%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 83.2%

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

Reproduce

herbie shell --seed 2024317 
(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)))