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

Percentage Accurate: 87.7% → 98.0%

Time: 7.3s

Alternatives: 11

Speedup: 4.2×

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: 87.7% 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: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\varepsilon + x\right) \cdot 10, x, \left(5 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x, \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot \varepsilon, 5, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8.2e-53)
   (*
    (fma
     (* (fma (* (+ eps x) 10.0) x (* (* 5.0 eps) eps)) x)
     eps
     (* (pow x 4.0) 5.0))
    eps)
   (if (<= x 5e-60)
     (* (fma (/ x eps) 5.0 1.0) (pow eps 5.0))
     (*
      (fma (* (* x x) eps) 5.0 (* (* (* eps eps) (+ eps x)) 10.0))
      (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8.2e-53) {
		tmp = fma((fma(((eps + x) * 10.0), x, ((5.0 * eps) * eps)) * x), eps, (pow(x, 4.0) * 5.0)) * eps;
	} else if (x <= 5e-60) {
		tmp = fma((x / eps), 5.0, 1.0) * pow(eps, 5.0);
	} else {
		tmp = fma(((x * x) * eps), 5.0, (((eps * eps) * (eps + x)) * 10.0)) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8.2e-53)
		tmp = Float64(fma(Float64(fma(Float64(Float64(eps + x) * 10.0), x, Float64(Float64(5.0 * eps) * eps)) * x), eps, Float64((x ^ 4.0) * 5.0)) * eps);
	elseif (x <= 5e-60)
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * (eps ^ 5.0));
	else
		tmp = Float64(fma(Float64(Float64(x * x) * eps), 5.0, Float64(Float64(Float64(eps * eps) * Float64(eps + x)) * 10.0)) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8.2e-53], N[(N[(N[(N[(N[(N[(eps + x), $MachinePrecision] * 10.0), $MachinePrecision] * x + N[(N[(5.0 * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * eps + N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 5e-60], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.2000000000000001e-53

   1. Initial program 32.2%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x \cdot \left(5 \cdot {\varepsilon}^{2} + x \cdot \left(10 \cdot \varepsilon + 10 \cdot x\right)\right), \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon \]
   7. Step-by-step derivation

   8. Applied rewrites 95.0%

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

   if -8.2000000000000001e-53 < x < 5.0000000000000001e-60

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 5.0000000000000001e-60 < x

   1. Initial program 41.4%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   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 89.7%

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

   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 25.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 89.9%

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

Alternative 2: 98.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot \varepsilon, 5, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8.2e-53)
   (* (* (* (fma (* x x) 5.0 (* (* eps (+ eps x)) 10.0)) x) x) eps)
   (if (<= x 5e-60)
     (* (fma (/ x eps) 5.0 1.0) (pow eps 5.0))
     (*
      (fma (* (* x x) eps) 5.0 (* (* (* eps eps) (+ eps x)) 10.0))
      (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8.2e-53) {
		tmp = ((fma((x * x), 5.0, ((eps * (eps + x)) * 10.0)) * x) * x) * eps;
	} else if (x <= 5e-60) {
		tmp = fma((x / eps), 5.0, 1.0) * pow(eps, 5.0);
	} else {
		tmp = fma(((x * x) * eps), 5.0, (((eps * eps) * (eps + x)) * 10.0)) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8.2e-53)
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 5.0, Float64(Float64(eps * Float64(eps + x)) * 10.0)) * x) * x) * eps);
	elseif (x <= 5e-60)
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * (eps ^ 5.0));
	else
		tmp = Float64(fma(Float64(Float64(x * x) * eps), 5.0, Float64(Float64(Float64(eps * eps) * Float64(eps + x)) * 10.0)) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8.2e-53], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0 + N[(N[(eps * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 5e-60], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.2000000000000001e-53

   1. Initial program 32.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 94.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.9%

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

   if -8.2000000000000001e-53 < x < 5.0000000000000001e-60

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 5.0000000000000001e-60 < x

   1. Initial program 41.4%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   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 89.7%

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

   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 25.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 89.9%

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

Alternative 3: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot \varepsilon, 5, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8e-53)
   (* (* (* (fma (* x x) 5.0 (* (* eps (+ eps x)) 10.0)) x) x) eps)
   (if (<= x 5e-60)
     (pow eps 5.0)
     (*
      (fma (* (* x x) eps) 5.0 (* (* (* eps eps) (+ eps x)) 10.0))
      (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8e-53) {
		tmp = ((fma((x * x), 5.0, ((eps * (eps + x)) * 10.0)) * x) * x) * eps;
	} else if (x <= 5e-60) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = fma(((x * x) * eps), 5.0, (((eps * eps) * (eps + x)) * 10.0)) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8e-53)
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 5.0, Float64(Float64(eps * Float64(eps + x)) * 10.0)) * x) * x) * eps);
	elseif (x <= 5e-60)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(fma(Float64(Float64(x * x) * eps), 5.0, Float64(Float64(Float64(eps * eps) * Float64(eps + x)) * 10.0)) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8e-53], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0 + N[(N[(eps * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 5e-60], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.00000000000000025e-53

   1. Initial program 32.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 94.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.9%

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

   if -8.00000000000000025e-53 < x < 5.0000000000000001e-60

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 5.0000000000000001e-60 < x

   1. Initial program 41.4%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   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 89.7%

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

   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 25.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 89.9%

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

Alternative 4: 97.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot \varepsilon, 5, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8.2e-53)
   (* (* (* (fma (* x x) 5.0 (* (* eps (+ eps x)) 10.0)) x) x) eps)
   (if (<= x 5e-60)
     (* (* (* (fma (fma x 5.0 eps) eps (* (* 10.0 x) x)) eps) eps) eps)
     (*
      (fma (* (* x x) eps) 5.0 (* (* (* eps eps) (+ eps x)) 10.0))
      (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8.2e-53) {
		tmp = ((fma((x * x), 5.0, ((eps * (eps + x)) * 10.0)) * x) * x) * eps;
	} else if (x <= 5e-60) {
		tmp = ((fma(fma(x, 5.0, eps), eps, ((10.0 * x) * x)) * eps) * eps) * eps;
	} else {
		tmp = fma(((x * x) * eps), 5.0, (((eps * eps) * (eps + x)) * 10.0)) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8.2e-53)
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 5.0, Float64(Float64(eps * Float64(eps + x)) * 10.0)) * x) * x) * eps);
	elseif (x <= 5e-60)
		tmp = Float64(Float64(Float64(fma(fma(x, 5.0, eps), eps, Float64(Float64(10.0 * x) * x)) * eps) * eps) * eps);
	else
		tmp = Float64(fma(Float64(Float64(x * x) * eps), 5.0, Float64(Float64(Float64(eps * eps) * Float64(eps + x)) * 10.0)) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8.2e-53], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0 + N[(N[(eps * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 5e-60], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0 + N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.2000000000000001e-53

   1. Initial program 32.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 94.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.9%

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

   if -8.2000000000000001e-53 < x < 5.0000000000000001e-60

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + \color{blue}{{\varepsilon}^{5}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.9%

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

   if 5.0000000000000001e-60 < x

   1. Initial program 41.4%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   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 89.7%

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

   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 25.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 89.9%

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

Alternative 5: 97.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-53} \lor \neg \left(x \leq 5 \cdot 10^{-60}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -8.2e-53) (not (<= x 5e-60)))
   (* (* (* (fma (* x x) 5.0 (* (* eps (+ eps x)) 10.0)) x) x) eps)
   (* (* (* (fma (fma x 5.0 eps) eps (* (* 10.0 x) x)) eps) eps) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -8.2e-53) || !(x <= 5e-60)) {
		tmp = ((fma((x * x), 5.0, ((eps * (eps + x)) * 10.0)) * x) * x) * eps;
	} else {
		tmp = ((fma(fma(x, 5.0, eps), eps, ((10.0 * x) * x)) * eps) * eps) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -8.2e-53) || !(x <= 5e-60))
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 5.0, Float64(Float64(eps * Float64(eps + x)) * 10.0)) * x) * x) * eps);
	else
		tmp = Float64(Float64(Float64(fma(fma(x, 5.0, eps), eps, Float64(Float64(10.0 * x) * x)) * eps) * eps) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -8.2e-53], N[Not[LessEqual[x, 5e-60]], $MachinePrecision]], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0 + N[(N[(eps * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.2000000000000001e-53 or 5.0000000000000001e-60 < x

   1. Initial program 35.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 92.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.0%

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

   if -8.2000000000000001e-53 < x < 5.0000000000000001e-60

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + \color{blue}{{\varepsilon}^{5}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-53} \lor \neg \left(x \leq 5 \cdot 10^{-60}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot \left(\varepsilon + x\right)\right) \cdot 10\right)\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(t\_0 \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (* x x) 5.0 (* (* eps (+ eps x)) 10.0))))
   (if (<= x -8.2e-53)
     (* (* (* t_0 x) x) eps)
     (if (<= x 5e-60)
       (* (* (* (fma (fma x 5.0 eps) eps (* (* 10.0 x) x)) eps) eps) eps)
       (* (* t_0 (* x x)) eps)))))

C

double code(double x, double eps) {
	double t_0 = fma((x * x), 5.0, ((eps * (eps + x)) * 10.0));
	double tmp;
	if (x <= -8.2e-53) {
		tmp = ((t_0 * x) * x) * eps;
	} else if (x <= 5e-60) {
		tmp = ((fma(fma(x, 5.0, eps), eps, ((10.0 * x) * x)) * eps) * eps) * eps;
	} else {
		tmp = (t_0 * (x * x)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(Float64(x * x), 5.0, Float64(Float64(eps * Float64(eps + x)) * 10.0))
	tmp = 0.0
	if (x <= -8.2e-53)
		tmp = Float64(Float64(Float64(t_0 * x) * x) * eps);
	elseif (x <= 5e-60)
		tmp = Float64(Float64(Float64(fma(fma(x, 5.0, eps), eps, Float64(Float64(10.0 * x) * x)) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(t_0 * Float64(x * x)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 5.0 + N[(N[(eps * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.2e-53], N[(N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 5e-60], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.2000000000000001e-53

   1. Initial program 32.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 94.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.9%

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

   if -8.2000000000000001e-53 < x < 5.0000000000000001e-60

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + \color{blue}{{\varepsilon}^{5}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.9%

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

   if 5.0000000000000001e-60 < x

   1. Initial program 41.4%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   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 89.7%

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

   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 25.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 89.7%

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

Alternative 7: 97.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-53} \lor \neg \left(x \leq 5 \cdot 10^{-60}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -8.2e-53) (not (<= x 5e-60)))
   (* (* (* (fma (* x x) 5.0 (* (* eps x) 10.0)) x) x) eps)
   (* (* (* (fma (fma x 5.0 eps) eps (* (* 10.0 x) x)) eps) eps) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -8.2e-53) || !(x <= 5e-60)) {
		tmp = ((fma((x * x), 5.0, ((eps * x) * 10.0)) * x) * x) * eps;
	} else {
		tmp = ((fma(fma(x, 5.0, eps), eps, ((10.0 * x) * x)) * eps) * eps) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -8.2e-53) || !(x <= 5e-60))
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 5.0, Float64(Float64(eps * x) * 10.0)) * x) * x) * eps);
	else
		tmp = Float64(Float64(Float64(fma(fma(x, 5.0, eps), eps, Float64(Float64(10.0 * x) * x)) * eps) * eps) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -8.2e-53], N[Not[LessEqual[x, 5e-60]], $MachinePrecision]], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0 + N[(N[(eps * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(N[(x * 5.0 + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.2000000000000001e-53 or 5.0000000000000001e-60 < x

   1. Initial program 35.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 92.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 91.6%

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

   if -8.2000000000000001e-53 < x < 5.0000000000000001e-60

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + \color{blue}{{\varepsilon}^{5}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-53} \lor \neg \left(x \leq 5 \cdot 10^{-60}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 5, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(x \cdot x, 5, \left(\varepsilon \cdot x\right) \cdot 10\right) \cdot x\right) \cdot x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (* (fma (* x x) 5.0 (* (* eps x) 10.0)) x) x) eps))

C

double code(double x, double eps) {
	return ((fma((x * x), 5.0, ((eps * x) * 10.0)) * x) * x) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(Float64(x * x), 5.0, Float64(Float64(eps * x) * 10.0)) * x) * x) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0 + N[(N[(eps * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 87.7%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
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 84.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 84.3%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 83.9%

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

Alternative 9: 83.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (* (* (fma 10.0 eps (* 5.0 x)) x) x) x) eps))

C

double code(double x, double eps) {
	return (((fma(10.0, eps, (5.0 * x)) * x) * x) * x) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * x) * x) * x) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 87.7%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
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 84.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 84.3%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 83.9%

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

Alternative 10: 83.0% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
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 84.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 84.3%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 83.5%

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

Alternative 11: 70.8% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
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 84.2%

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

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 71.1%

   \[\leadsto \left(\left(\left(\left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]
9. Step-by-step derivation

10. Applied rewrites 71.1%

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

Reproduce

herbie shell --seed 2024343 
(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)))