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

Percentage Accurate: 88.6% → 99.5%

Time: 10.1s

Alternatives: 13

Speedup: 5.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-322}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ eps x) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-322)
     t_0
     (if (<= t_0 0.0)
       (* (* (* (fma (* 5.0 x) x (* (* 10.0 x) eps)) eps) x) x)
       t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((eps + x), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = ((fma((5.0 * x), x, ((10.0 * x) * eps)) * eps) * x) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(eps + x) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(fma(Float64(5.0 * x), x, Float64(Float64(10.0 * x) * eps)) * eps) * x) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-322], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * x + N[(N[(10.0 * x), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.97626e-322 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 -1.97626e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 90.1%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.9%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-322}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(\varepsilon + x\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;{\left(\varepsilon + x\right)}^{5} - {x}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (fma (* 5.0 x) x (* (* 10.0 eps) (+ eps x))) eps) x) x)))
   (if (<= x -1.95e-62)
     t_0
     (if (<= x 6.5e-38) (* (pow eps 5.0) (fma (/ x eps) 5.0 1.0)) t_0))))

C

double code(double x, double eps) {
	double t_0 = ((fma((5.0 * x), x, ((10.0 * eps) * (eps + x))) * eps) * x) * x;
	double tmp;
	if (x <= -1.95e-62) {
		tmp = t_0;
	} else if (x <= 6.5e-38) {
		tmp = pow(eps, 5.0) * fma((x / eps), 5.0, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(fma(Float64(5.0 * x), x, Float64(Float64(10.0 * eps) * Float64(eps + x))) * eps) * x) * x)
	tmp = 0.0
	if (x <= -1.95e-62)
		tmp = t_0;
	elseif (x <= 6.5e-38)
		tmp = Float64((eps ^ 5.0) * fma(Float64(x / eps), 5.0, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * x + N[(N[(10.0 * eps), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.95e-62], t$95$0, If[LessEqual[x, 6.5e-38], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9500000000000002e-62 or 6.49999999999999949e-38 < x

   1. Initial program 41.6%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 31.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 92.7%

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

   if -1.9500000000000002e-62 < x < 6.49999999999999949e-38

   1. Initial program 99.6%

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

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

   5. Applied rewrites 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (fma (* 5.0 x) x (* (* 10.0 eps) (+ eps x))) eps) x) x)))
   (if (<= x -1.95e-62)
     t_0
     (if (<= x 6.5e-38) (* (pow eps 4.0) (fma 5.0 x eps)) t_0))))

C

double code(double x, double eps) {
	double t_0 = ((fma((5.0 * x), x, ((10.0 * eps) * (eps + x))) * eps) * x) * x;
	double tmp;
	if (x <= -1.95e-62) {
		tmp = t_0;
	} else if (x <= 6.5e-38) {
		tmp = pow(eps, 4.0) * fma(5.0, x, eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(fma(Float64(5.0 * x), x, Float64(Float64(10.0 * eps) * Float64(eps + x))) * eps) * x) * x)
	tmp = 0.0
	if (x <= -1.95e-62)
		tmp = t_0;
	elseif (x <= 6.5e-38)
		tmp = Float64((eps ^ 4.0) * fma(5.0, x, eps));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * x + N[(N[(10.0 * eps), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.95e-62], t$95$0, If[LessEqual[x, 6.5e-38], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9500000000000002e-62 or 6.49999999999999949e-38 < x

   1. Initial program 41.6%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 31.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 92.7%

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

   if -1.9500000000000002e-62 < x < 6.49999999999999949e-38

   1. Initial program 99.6%

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

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

   5. Applied rewrites 99.6%

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

   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 99.6

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

   8. Applied rewrites 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (fma (* 5.0 x) x (* (* 10.0 eps) (+ eps x))) eps) x) x)))
   (if (<= x -1.95e-62)
     t_0
     (if (<= x 6.5e-38)
       (* (* (* (fma (fma 5.0 x eps) eps (* (* 10.0 x) x)) eps) eps) eps)
       t_0))))

C

double code(double x, double eps) {
	double t_0 = ((fma((5.0 * x), x, ((10.0 * eps) * (eps + x))) * eps) * x) * x;
	double tmp;
	if (x <= -1.95e-62) {
		tmp = t_0;
	} else if (x <= 6.5e-38) {
		tmp = ((fma(fma(5.0, x, eps), eps, ((10.0 * x) * x)) * eps) * eps) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(fma(Float64(5.0 * x), x, Float64(Float64(10.0 * eps) * Float64(eps + x))) * eps) * x) * x)
	tmp = 0.0
	if (x <= -1.95e-62)
		tmp = t_0;
	elseif (x <= 6.5e-38)
		tmp = Float64(Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(10.0 * x) * x)) * eps) * eps) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * x + N[(N[(10.0 * eps), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.95e-62], t$95$0, If[LessEqual[x, 6.5e-38], N[(N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9500000000000002e-62 or 6.49999999999999949e-38 < x

   1. Initial program 41.6%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 31.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 92.7%

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

   if -1.9500000000000002e-62 < x < 6.49999999999999949e-38

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

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

   10. Applied rewrites 99.5%

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

   12. Applied rewrites 99.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.95e-62)
   (* (* x x) (* (* (fma 10.0 eps (* 5.0 x)) x) eps))
   (if (<= x 6.5e-38)
     (* (* (* (fma (fma 5.0 x eps) eps (* (* 10.0 x) x)) eps) eps) eps)
     (* (* (* (fma (* 5.0 x) x (* (* 10.0 x) eps)) eps) x) x))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.95e-62) {
		tmp = (x * x) * ((fma(10.0, eps, (5.0 * x)) * x) * eps);
	} else if (x <= 6.5e-38) {
		tmp = ((fma(fma(5.0, x, eps), eps, ((10.0 * x) * x)) * eps) * eps) * eps;
	} else {
		tmp = ((fma((5.0 * x), x, ((10.0 * x) * eps)) * eps) * x) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.95e-62)
		tmp = Float64(Float64(x * x) * Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * x) * eps));
	elseif (x <= 6.5e-38)
		tmp = Float64(Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(10.0 * x) * x)) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(fma(Float64(5.0 * x), x, Float64(Float64(10.0 * x) * eps)) * eps) * x) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.95e-62], N[(N[(x * x), $MachinePrecision] * N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e-38], N[(N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * x + N[(N[(10.0 * x), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9500000000000002e-62

   1. Initial program 56.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   7. Applied rewrites 86.7%

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

   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 87.0%

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

   if -1.9500000000000002e-62 < x < 6.49999999999999949e-38

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

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

   10. Applied rewrites 99.5%

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

   12. Applied rewrites 99.5%

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

   if 6.49999999999999949e-38 < x

   1. Initial program 17.2%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 13.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.6%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 98.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.95e-62)
   (* (* x x) (* (* (fma 10.0 eps (* 5.0 x)) x) eps))
   (if (<= x 6.5e-38)
     (* (* (* eps eps) (* eps eps)) (fma 5.0 x eps))
     (* (* (* (fma (* 5.0 x) x (* (* 10.0 x) eps)) eps) x) x))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.95e-62) {
		tmp = (x * x) * ((fma(10.0, eps, (5.0 * x)) * x) * eps);
	} else if (x <= 6.5e-38) {
		tmp = ((eps * eps) * (eps * eps)) * fma(5.0, x, eps);
	} else {
		tmp = ((fma((5.0 * x), x, ((10.0 * x) * eps)) * eps) * x) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.95e-62)
		tmp = Float64(Float64(x * x) * Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * x) * eps));
	elseif (x <= 6.5e-38)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * fma(5.0, x, eps));
	else
		tmp = Float64(Float64(Float64(fma(Float64(5.0 * x), x, Float64(Float64(10.0 * x) * eps)) * eps) * x) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.95e-62], N[(N[(x * x), $MachinePrecision] * N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e-38], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * x + N[(N[(10.0 * x), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9500000000000002e-62

   1. Initial program 56.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   7. Applied rewrites 86.7%

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

   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 87.0%

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

   if -1.9500000000000002e-62 < x < 6.49999999999999949e-38

   1. Initial program 99.6%

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

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

   5. Applied rewrites 99.6%

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

   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 99.6

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.5%

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

   if 6.49999999999999949e-38 < x

   1. Initial program 17.2%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 13.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.6%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 98.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(5 \cdot x, x, \left(10 \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) (* (* (fma 10.0 eps (* 5.0 x)) x) eps))))
   (if (<= x -1.95e-62)
     t_0
     (if (<= x 6.5e-38) (* (* (* eps eps) (* eps eps)) (fma 5.0 x eps)) t_0))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * ((fma(10.0, eps, (5.0 * x)) * x) * eps);
	double tmp;
	if (x <= -1.95e-62) {
		tmp = t_0;
	} else if (x <= 6.5e-38) {
		tmp = ((eps * eps) * (eps * eps)) * fma(5.0, x, eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * x) * eps))
	tmp = 0.0
	if (x <= -1.95e-62)
		tmp = t_0;
	elseif (x <= 6.5e-38)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * fma(5.0, x, eps));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.95e-62], t$95$0, If[LessEqual[x, 6.5e-38], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9500000000000002e-62 or 6.49999999999999949e-38 < x

   1. Initial program 41.6%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 90.9%

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

   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 91.2%

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

   if -1.9500000000000002e-62 < x < 6.49999999999999949e-38

   1. Initial program 99.6%

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

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

   5. Applied rewrites 99.6%

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

   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 99.6

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(5 \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;t\_0 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* 5.0 x) x) eps)))
   (if (<= x -1.95e-62)
     (* t_0 (* x x))
     (if (<= x 6.5e-38)
       (* (* (* eps eps) (* eps eps)) (fma 5.0 x eps))
       (* (* t_0 x) x)))))

C

double code(double x, double eps) {
	double t_0 = ((5.0 * x) * x) * eps;
	double tmp;
	if (x <= -1.95e-62) {
		tmp = t_0 * (x * x);
	} else if (x <= 6.5e-38) {
		tmp = ((eps * eps) * (eps * eps)) * fma(5.0, x, eps);
	} else {
		tmp = (t_0 * x) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(5.0 * x) * x) * eps)
	tmp = 0.0
	if (x <= -1.95e-62)
		tmp = Float64(t_0 * Float64(x * x));
	elseif (x <= 6.5e-38)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * fma(5.0, x, eps));
	else
		tmp = Float64(Float64(t_0 * x) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(5.0 * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -1.95e-62], N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e-38], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9500000000000002e-62

   1. Initial program 56.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   7. Applied rewrites 86.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 86.8%

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

   if -1.9500000000000002e-62 < x < 6.49999999999999949e-38

   1. Initial program 99.6%

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

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

   5. Applied rewrites 99.6%

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

   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 99.6

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.5%

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

   if 6.49999999999999949e-38 < x

   1. Initial program 17.2%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 95.6%

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

   12. Applied rewrites 96.1%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(\left(\left(5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(5 \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;t\_0 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* 5.0 x) x) eps)))
   (if (<= x -1.95e-62)
     (* t_0 (* x x))
     (if (<= x 6.5e-38)
       (* (* (* eps eps) (fma 5.0 x eps)) (* eps eps))
       (* (* t_0 x) x)))))

C

double code(double x, double eps) {
	double t_0 = ((5.0 * x) * x) * eps;
	double tmp;
	if (x <= -1.95e-62) {
		tmp = t_0 * (x * x);
	} else if (x <= 6.5e-38) {
		tmp = ((eps * eps) * fma(5.0, x, eps)) * (eps * eps);
	} else {
		tmp = (t_0 * x) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(5.0 * x) * x) * eps)
	tmp = 0.0
	if (x <= -1.95e-62)
		tmp = Float64(t_0 * Float64(x * x));
	elseif (x <= 6.5e-38)
		tmp = Float64(Float64(Float64(eps * eps) * fma(5.0, x, eps)) * Float64(eps * eps));
	else
		tmp = Float64(Float64(t_0 * x) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(5.0 * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -1.95e-62], N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e-38], N[(N[(N[(eps * eps), $MachinePrecision] * N[(5.0 * x + eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9500000000000002e-62

   1. Initial program 56.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   7. Applied rewrites 86.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 86.8%

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

   if -1.9500000000000002e-62 < x < 6.49999999999999949e-38

   1. Initial program 99.6%

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

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

   5. Applied rewrites 99.6%

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

   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 99.6

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.5%

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

   if 6.49999999999999949e-38 < x

   1. Initial program 17.2%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 95.6%

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

   12. Applied rewrites 96.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(\left(\left(5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(5, x, \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(5 \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;t\_0 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* 5.0 x) x) eps)))
   (if (<= x -1.95e-62)
     (* t_0 (* x x))
     (if (<= x 6.5e-38) (* (* (* eps eps) (* eps eps)) eps) (* (* t_0 x) x)))))

C

double code(double x, double eps) {
	double t_0 = ((5.0 * x) * x) * eps;
	double tmp;
	if (x <= -1.95e-62) {
		tmp = t_0 * (x * x);
	} else if (x <= 6.5e-38) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = (t_0 * x) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((5.0d0 * x) * x) * eps
    if (x <= (-1.95d-62)) then
        tmp = t_0 * (x * x)
    else if (x <= 6.5d-38) then
        tmp = ((eps * eps) * (eps * eps)) * eps
    else
        tmp = (t_0 * x) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((5.0 * x) * x) * eps;
	double tmp;
	if (x <= -1.95e-62) {
		tmp = t_0 * (x * x);
	} else if (x <= 6.5e-38) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = (t_0 * x) * x;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((5.0 * x) * x) * eps
	tmp = 0
	if x <= -1.95e-62:
		tmp = t_0 * (x * x)
	elif x <= 6.5e-38:
		tmp = ((eps * eps) * (eps * eps)) * eps
	else:
		tmp = (t_0 * x) * x
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(5.0 * x) * x) * eps)
	tmp = 0.0
	if (x <= -1.95e-62)
		tmp = Float64(t_0 * Float64(x * x));
	elseif (x <= 6.5e-38)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps);
	else
		tmp = Float64(Float64(t_0 * x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((5.0 * x) * x) * eps;
	tmp = 0.0;
	if (x <= -1.95e-62)
		tmp = t_0 * (x * x);
	elseif (x <= 6.5e-38)
		tmp = ((eps * eps) * (eps * eps)) * eps;
	else
		tmp = (t_0 * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(5.0 * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -1.95e-62], N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e-38], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9500000000000002e-62

   1. Initial program 56.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   7. Applied rewrites 86.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 86.8%

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

   if -1.9500000000000002e-62 < x < 6.49999999999999949e-38

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.3%

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

   if 6.49999999999999949e-38 < x

   1. Initial program 17.2%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 95.6%

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

   12. Applied rewrites 96.1%

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

Alternative 11: 97.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(\left(\left(5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.95e-62)
   (* (* (* (* 5.0 x) x) eps) (* x x))
   (if (<= x 6.5e-38)
     (* (* (* eps eps) (* eps eps)) eps)
     (* (* (* (* x x) eps) 5.0) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.95e-62) {
		tmp = (((5.0 * x) * x) * eps) * (x * x);
	} else if (x <= 6.5e-38) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.95d-62)) then
        tmp = (((5.0d0 * x) * x) * eps) * (x * x)
    else if (x <= 6.5d-38) then
        tmp = ((eps * eps) * (eps * eps)) * eps
    else
        tmp = (((x * x) * eps) * 5.0d0) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.95e-62) {
		tmp = (((5.0 * x) * x) * eps) * (x * x);
	} else if (x <= 6.5e-38) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.95e-62:
		tmp = (((5.0 * x) * x) * eps) * (x * x)
	elif x <= 6.5e-38:
		tmp = ((eps * eps) * (eps * eps)) * eps
	else:
		tmp = (((x * x) * eps) * 5.0) * (x * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.95e-62)
		tmp = Float64(Float64(Float64(Float64(5.0 * x) * x) * eps) * Float64(x * x));
	elseif (x <= 6.5e-38)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps);
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * eps) * 5.0) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.95e-62)
		tmp = (((5.0 * x) * x) * eps) * (x * x);
	elseif (x <= 6.5e-38)
		tmp = ((eps * eps) * (eps * eps)) * eps;
	else
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.95e-62], N[(N[(N[(N[(5.0 * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e-38], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9500000000000002e-62

   1. Initial program 56.5%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   7. Applied rewrites 86.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 86.8%

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

   if -1.9500000000000002e-62 < x < 6.49999999999999949e-38

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.3%

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

   if 6.49999999999999949e-38 < x

   1. Initial program 17.2%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 95.6%

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

   12. Applied rewrites 95.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(\left(\left(5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (* x x) eps) 5.0) (* x x))))
   (if (<= x -1.95e-62)
     t_0
     (if (<= x 6.5e-38) (* (* (* eps eps) (* eps eps)) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = (((x * x) * eps) * 5.0) * (x * x);
	double tmp;
	if (x <= -1.95e-62) {
		tmp = t_0;
	} else if (x <= 6.5e-38) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) * eps) * 5.0d0) * (x * x)
    if (x <= (-1.95d-62)) then
        tmp = t_0
    else if (x <= 6.5d-38) then
        tmp = ((eps * eps) * (eps * eps)) * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (((x * x) * eps) * 5.0) * (x * x);
	double tmp;
	if (x <= -1.95e-62) {
		tmp = t_0;
	} else if (x <= 6.5e-38) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (((x * x) * eps) * 5.0) * (x * x)
	tmp = 0
	if x <= -1.95e-62:
		tmp = t_0
	elif x <= 6.5e-38:
		tmp = ((eps * eps) * (eps * eps)) * eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(x * x) * eps) * 5.0) * Float64(x * x))
	tmp = 0.0
	if (x <= -1.95e-62)
		tmp = t_0;
	elseif (x <= 6.5e-38)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (((x * x) * eps) * 5.0) * (x * x);
	tmp = 0.0;
	if (x <= -1.95e-62)
		tmp = t_0;
	elseif (x <= 6.5e-38)
		tmp = ((eps * eps) * (eps * eps)) * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.95e-62], t$95$0, If[LessEqual[x, 6.5e-38], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9500000000000002e-62 or 6.49999999999999949e-38 < x

   1. Initial program 41.6%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 90.1%

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

   12. Applied rewrites 90.1%

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

   if -1.9500000000000002e-62 < x < 6.49999999999999949e-38

   1. Initial program 99.6%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-62}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.5% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

3. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \left(4 \cdot \frac{x}{\varepsilon} + \left(8 \cdot \frac{{x}^{2}}{{\varepsilon}^{2}} + \frac{x}{\varepsilon}\right)\right)\right)\right) \cdot {\varepsilon}^{5}} \]

5. Applied rewrites 83.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, 1\right) + \mathsf{fma}\left(8 \cdot \frac{x}{\varepsilon}, \frac{x}{\varepsilon}, \frac{x}{\varepsilon} \cdot 5\right)\right) \cdot {\varepsilon}^{5}} \]

6. Taylor expanded in eps around 0

   \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(2 \cdot {x}^{2} + \left(8 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon + 5 \cdot x\right)\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 90.1%

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

10. Applied rewrites 90.1%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 89.8%

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

Reproduce

herbie shell --seed 2024332 
(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)))