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

Percentage Accurate: 88.3% → 98.6%

Time: 7.9s

Alternatives: 9

Speedup: 2.0×

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: 88.3% 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: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-296}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-296)
     (pow eps 5.0)
     (if (<= t_0 0.0) (* eps (* 5.0 (pow x 4.0))) t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-296) {
		tmp = pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} 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 + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-5d-296)) then
        tmp = eps ** 5.0d0
    else if (t_0 <= 0.0d0) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-296) {
		tmp = Math.pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -5e-296:
		tmp = math.pow(eps, 5.0)
	elif t_0 <= 0.0:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-296)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -5e-296)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-296], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -5.0000000000000003e-296

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if -5.0000000000000003e-296 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 88.3%

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

   2. Taylor expanded in eps around 0 99.9%

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

   3. Taylor expanded in x around 0 99.9%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 97.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-296}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternative 2: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-47}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-71}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(10 \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8.4e-47)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 2.85e-71)
     (pow eps 5.0)
     (*
      x
      (+
       (* x (* eps (* 5.0 (* x x))))
       (* x (* 10.0 (* (+ x eps) (* eps eps)))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8.4e-47) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 2.85e-71) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-8.4d-47)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 2.85d-71) then
        tmp = eps ** 5.0d0
    else
        tmp = x * ((x * (eps * (5.0d0 * (x * x)))) + (x * (10.0d0 * ((x + eps) * (eps * eps)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -8.4e-47) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 2.85e-71) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -8.4e-47:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 2.85e-71:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8.4e-47)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 2.85e-71)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(x * Float64(Float64(x * Float64(eps * Float64(5.0 * Float64(x * x)))) + Float64(x * Float64(10.0 * Float64(Float64(x + eps) * Float64(eps * eps))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -8.4e-47)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 2.85e-71)
		tmp = eps ^ 5.0;
	else
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8.4e-47], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.85e-71], N[Power[eps, 5.0], $MachinePrecision], N[(x * N[(N[(x * N[(eps * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(10.0 * N[(N[(x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.4000000000000003e-47

   1. Initial program 34.4%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. distribute-lft1-in 95.7%

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

      2. metadata-eval 95.7%

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

      3. associate-*l* 95.4%

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

   4. Simplified 95.4%

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

   if -8.4000000000000003e-47 < x < 2.8500000000000001e-71

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.5%

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

   if 2.8500000000000001e-71 < x

   1. Initial program 52.8%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. fma-def 99.6%

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

      2. distribute-lft1-in 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      3. metadata-eval 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      4. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      5. +-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      6. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]

      7. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]

      8. unpow3 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      9. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      10. associate-*l* 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

      11. distribute-lft-out 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   4. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)}\right) \]

      2. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left({\varepsilon}^{3} \cdot \color{blue}{\left(x \cdot x\right)}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      3. associate-*r* 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right)} + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      4. associate-*r* 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      5. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left({\varepsilon}^{2} \cdot 10\right)} \cdot {x}^{3}\right) \]

      6. unpow2 99.6%

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

      7. associate-*r* 99.6%

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

      8. cube-mult 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.6%

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

      12. associate-*l* 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-*r* 99.6%

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

      15. unpow2 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \left(\color{blue}{{\varepsilon}^{2}} \cdot 10\right) \cdot x\right)\right)\right) \]

      16. *-commutative 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right)} \cdot x\right)\right)\right) \]

      17. associate-*r* 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot x\right)}\right)\right)\right) \]

   7. Simplified 99.6%

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

   8. Taylor expanded in eps around 0 99.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-out 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \color{blue}{\left(10 \cdot \left({\varepsilon}^{3} + {\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]

      2. unpow2 99.6%

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

      3. associate-*r* 99.6%

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

      4. unpow3 99.6%

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

      5. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left(\color{blue}{{\varepsilon}^{2}} \cdot \varepsilon + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right) \]

      6. associate-*r* 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot \varepsilon + \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot x}\right)\right)\right)\right) \]

      7. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot \varepsilon + \color{blue}{{\varepsilon}^{2}} \cdot x\right)\right)\right)\right) \]

      8. distribute-lft-out 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\varepsilon + x\right)\right)}\right)\right)\right) \]

      9. unpow2 99.6%

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

   10. Simplified 99.6%

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

       1. fma-udef 99.5%

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

       2. metadata-eval 99.5%

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

       3. pow-prod-up 99.5%

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

       4. pow2 99.5%

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

       5. pow2 99.5%

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

       6. associate-*l* 99.3%

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

       7. associate-*r* 99.6%

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

       8. *-commutative 99.6%

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

       9. distribute-rgt-out 99.7%

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

       10. associate-*l* 99.7%

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

   12. Applied egg-rr 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-47}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-71}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(10 \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-71}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(10 \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9.2e-47)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 2.85e-71)
     (pow eps 5.0)
     (*
      x
      (+
       (* x (* eps (* 5.0 (* x x))))
       (* x (* 10.0 (* (+ x eps) (* eps eps)))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9.2e-47) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 2.85e-71) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-9.2d-47)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 2.85d-71) then
        tmp = eps ** 5.0d0
    else
        tmp = x * ((x * (eps * (5.0d0 * (x * x)))) + (x * (10.0d0 * ((x + eps) * (eps * eps)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -9.2e-47) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 2.85e-71) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9.2e-47:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 2.85e-71:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9.2e-47)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 2.85e-71)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(x * Float64(Float64(x * Float64(eps * Float64(5.0 * Float64(x * x)))) + Float64(x * Float64(10.0 * Float64(Float64(x + eps) * Float64(eps * eps))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9.2e-47)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 2.85e-71)
		tmp = eps ^ 5.0;
	else
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9.2e-47], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.85e-71], N[Power[eps, 5.0], $MachinePrecision], N[(x * N[(N[(x * N[(eps * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(10.0 * N[(N[(x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.19999999999999928e-47

   1. Initial program 34.4%

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

   2. Taylor expanded in eps around 0 95.5%

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

   3. Taylor expanded in x around 0 95.5%

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

   if -9.19999999999999928e-47 < x < 2.8500000000000001e-71

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.5%

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

   if 2.8500000000000001e-71 < x

   1. Initial program 52.8%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. fma-def 99.6%

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

      2. distribute-lft1-in 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      3. metadata-eval 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      4. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      5. +-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      6. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]

      7. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]

      8. unpow3 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      9. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      10. associate-*l* 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

      11. distribute-lft-out 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   4. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)}\right) \]

      2. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left({\varepsilon}^{3} \cdot \color{blue}{\left(x \cdot x\right)}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      3. associate-*r* 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right)} + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      4. associate-*r* 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      5. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left({\varepsilon}^{2} \cdot 10\right)} \cdot {x}^{3}\right) \]

      6. unpow2 99.6%

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

      7. associate-*r* 99.6%

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

      8. cube-mult 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.6%

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

      12. associate-*l* 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-*r* 99.6%

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

      15. unpow2 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \left(\color{blue}{{\varepsilon}^{2}} \cdot 10\right) \cdot x\right)\right)\right) \]

      16. *-commutative 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right)} \cdot x\right)\right)\right) \]

      17. associate-*r* 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot x\right)}\right)\right)\right) \]

   7. Simplified 99.6%

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

   8. Taylor expanded in eps around 0 99.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-out 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \color{blue}{\left(10 \cdot \left({\varepsilon}^{3} + {\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]

      2. unpow2 99.6%

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

      3. associate-*r* 99.6%

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

      4. unpow3 99.6%

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

      5. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left(\color{blue}{{\varepsilon}^{2}} \cdot \varepsilon + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right) \]

      6. associate-*r* 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot \varepsilon + \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot x}\right)\right)\right)\right) \]

      7. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot \varepsilon + \color{blue}{{\varepsilon}^{2}} \cdot x\right)\right)\right)\right) \]

      8. distribute-lft-out 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\varepsilon + x\right)\right)}\right)\right)\right) \]

      9. unpow2 99.6%

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

   10. Simplified 99.6%

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

       1. fma-udef 99.5%

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

       2. metadata-eval 99.5%

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

       3. pow-prod-up 99.5%

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

       4. pow2 99.5%

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

       5. pow2 99.5%

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

       6. associate-*l* 99.3%

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

       7. associate-*r* 99.6%

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

       8. *-commutative 99.6%

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

       9. distribute-rgt-out 99.7%

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

       10. associate-*l* 99.7%

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

   12. Applied egg-rr 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-71}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(10 \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \]

Alternative 4: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-47}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-71}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(10 \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.5e-47)
   (* (pow x 4.0) (* eps 5.0))
   (if (<= x 2.8e-71)
     (pow eps 5.0)
     (*
      x
      (+
       (* x (* eps (* 5.0 (* x x))))
       (* x (* 10.0 (* (+ x eps) (* eps eps)))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.5e-47) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 2.8e-71) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6.5d-47)) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    else if (x <= 2.8d-71) then
        tmp = eps ** 5.0d0
    else
        tmp = x * ((x * (eps * (5.0d0 * (x * x)))) + (x * (10.0d0 * ((x + eps) * (eps * eps)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.5e-47) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 2.8e-71) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.5e-47:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	elif x <= 2.8e-71:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.5e-47)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	elseif (x <= 2.8e-71)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(x * Float64(Float64(x * Float64(eps * Float64(5.0 * Float64(x * x)))) + Float64(x * Float64(10.0 * Float64(Float64(x + eps) * Float64(eps * eps))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.5e-47)
		tmp = (x ^ 4.0) * (eps * 5.0);
	elseif (x <= 2.8e-71)
		tmp = eps ^ 5.0;
	else
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.5e-47], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-71], N[Power[eps, 5.0], $MachinePrecision], N[(x * N[(N[(x * N[(eps * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(10.0 * N[(N[(x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5000000000000004e-47

   1. Initial program 34.4%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. distribute-lft1-in 95.7%

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

      2. metadata-eval 95.7%

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

      3. *-commutative 95.7%

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

   4. Simplified 95.7%

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

   if -6.5000000000000004e-47 < x < 2.8e-71

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.5%

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

   if 2.8e-71 < x

   1. Initial program 52.8%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. fma-def 99.6%

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

      2. distribute-lft1-in 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      3. metadata-eval 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      4. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      5. +-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      6. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]

      7. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]

      8. unpow3 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      9. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      10. associate-*l* 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

      11. distribute-lft-out 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   4. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)}\right) \]

      2. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left({\varepsilon}^{3} \cdot \color{blue}{\left(x \cdot x\right)}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      3. associate-*r* 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right)} + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      4. associate-*r* 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      5. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left({\varepsilon}^{2} \cdot 10\right)} \cdot {x}^{3}\right) \]

      6. unpow2 99.6%

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

      7. associate-*r* 99.6%

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

      8. cube-mult 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.6%

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

      12. associate-*l* 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-*r* 99.6%

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

      15. unpow2 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \left(\color{blue}{{\varepsilon}^{2}} \cdot 10\right) \cdot x\right)\right)\right) \]

      16. *-commutative 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right)} \cdot x\right)\right)\right) \]

      17. associate-*r* 99.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot x\right)}\right)\right)\right) \]

   7. Simplified 99.6%

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

   8. Taylor expanded in eps around 0 99.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-out 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \color{blue}{\left(10 \cdot \left({\varepsilon}^{3} + {\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]

      2. unpow2 99.6%

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

      3. associate-*r* 99.6%

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

      4. unpow3 99.6%

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

      5. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left(\color{blue}{{\varepsilon}^{2}} \cdot \varepsilon + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right) \]

      6. associate-*r* 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot \varepsilon + \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot x}\right)\right)\right)\right) \]

      7. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot \varepsilon + \color{blue}{{\varepsilon}^{2}} \cdot x\right)\right)\right)\right) \]

      8. distribute-lft-out 99.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\varepsilon + x\right)\right)}\right)\right)\right) \]

      9. unpow2 99.6%

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

   10. Simplified 99.6%

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

       1. fma-udef 99.5%

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

       2. metadata-eval 99.5%

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

       3. pow-prod-up 99.5%

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

       4. pow2 99.5%

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

       5. pow2 99.5%

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

       6. associate-*l* 99.3%

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

       7. associate-*r* 99.6%

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

       8. *-commutative 99.6%

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

       9. distribute-rgt-out 99.7%

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

       10. associate-*l* 99.7%

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

   12. Applied egg-rr 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-47}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-71}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(10 \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-47} \lor \neg \left(x \leq 2.85 \cdot 10^{-71}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(10 \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -6.5e-47) (not (<= x 2.85e-71)))
   (*
    x
    (+ (* x (* eps (* 5.0 (* x x)))) (* x (* 10.0 (* (+ x eps) (* eps eps))))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -6.5e-47) || !(x <= 2.85e-71)) {
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-6.5d-47)) .or. (.not. (x <= 2.85d-71))) then
        tmp = x * ((x * (eps * (5.0d0 * (x * x)))) + (x * (10.0d0 * ((x + eps) * (eps * eps)))))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -6.5e-47) || !(x <= 2.85e-71)) {
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -6.5e-47) or not (x <= 2.85e-71):
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -6.5e-47) || !(x <= 2.85e-71))
		tmp = Float64(x * Float64(Float64(x * Float64(eps * Float64(5.0 * Float64(x * x)))) + Float64(x * Float64(10.0 * Float64(Float64(x + eps) * Float64(eps * eps))))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -6.5e-47) || ~((x <= 2.85e-71)))
		tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -6.5e-47], N[Not[LessEqual[x, 2.85e-71]], $MachinePrecision]], N[(x * N[(N[(x * N[(eps * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(10.0 * N[(N[(x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5000000000000004e-47 or 2.8500000000000001e-71 < x

   1. Initial program 43.6%

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

   2. Taylor expanded in x around inf 97.6%

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

      1. fma-def 97.6%

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

      2. distribute-lft1-in 97.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      3. metadata-eval 97.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      4. *-commutative 97.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

      5. +-commutative 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      6. *-commutative 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]

      7. *-commutative 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]

      8. unpow3 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      9. unpow2 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

      10. associate-*l* 97.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

      11. distribute-lft-out 97.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   4. Simplified 97.6%

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

   5. Taylor expanded in x around 0 97.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)}\right) \]

      2. unpow2 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left({\varepsilon}^{3} \cdot \color{blue}{\left(x \cdot x\right)}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      3. associate-*r* 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right)} + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

      4. associate-*r* 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

      5. *-commutative 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left({\varepsilon}^{2} \cdot 10\right)} \cdot {x}^{3}\right) \]

      6. unpow2 97.6%

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

      7. associate-*r* 97.6%

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

      8. cube-mult 97.6%

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

      9. associate-*r* 97.6%

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

      10. *-commutative 97.6%

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

      11. distribute-rgt-out 97.6%

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

      12. associate-*l* 97.6%

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

      13. *-commutative 97.6%

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

      14. associate-*r* 97.6%

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

      15. unpow2 97.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \left(\color{blue}{{\varepsilon}^{2}} \cdot 10\right) \cdot x\right)\right)\right) \]

      16. *-commutative 97.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right)} \cdot x\right)\right)\right) \]

      17. associate-*r* 97.6%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot x\right)}\right)\right)\right) \]

   7. Simplified 97.6%

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

   8. Taylor expanded in eps around 0 97.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-out 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \color{blue}{\left(10 \cdot \left({\varepsilon}^{3} + {\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]

      2. unpow2 97.6%

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

      3. associate-*r* 97.6%

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

      4. unpow3 97.6%

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

      5. unpow2 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left(\color{blue}{{\varepsilon}^{2}} \cdot \varepsilon + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right) \]

      6. associate-*r* 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot \varepsilon + \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot x}\right)\right)\right)\right) \]

      7. unpow2 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot \varepsilon + \color{blue}{{\varepsilon}^{2}} \cdot x\right)\right)\right)\right) \]

      8. distribute-lft-out 97.6%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\varepsilon + x\right)\right)}\right)\right)\right) \]

      9. unpow2 97.6%

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

   10. Simplified 97.6%

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

       1. fma-udef 97.6%

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

       2. metadata-eval 97.6%

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

       3. pow-prod-up 97.4%

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

       4. pow2 97.4%

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

       5. pow2 97.4%

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

       6. associate-*l* 97.2%

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

       7. associate-*r* 97.5%

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

       8. *-commutative 97.5%

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

       9. distribute-rgt-out 97.5%

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

       10. associate-*l* 97.5%

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

   12. Applied egg-rr 97.5%

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

   if -6.5000000000000004e-47 < x < 2.8500000000000001e-71

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.5%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-47} \lor \neg \left(x \leq 2.85 \cdot 10^{-71}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(10 \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 6: 83.0% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x * ((x * (eps * (5.0d0 * (x * x)))) + (x * (10.0d0 * ((x + eps) * (eps * eps)))))
end function

Java

public static double code(double x, double eps) {
	return x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
}

Python

def code(x, eps):
	return x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = x * ((x * (eps * (5.0 * (x * x)))) + (x * (10.0 * ((x + eps) * (eps * eps)))));
end

Wolfram

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

Derivation

1. Initial program 90.4%

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

2. Taylor expanded in x around inf 81.5%

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

   1. fma-def 81.5%

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

   2. distribute-lft1-in 81.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

   3. metadata-eval 81.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

   4. *-commutative 81.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

   5. +-commutative 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

   6. *-commutative 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]

   7. *-commutative 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]

   8. unpow3 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

   9. unpow2 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

   10. associate-*l* 81.5%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   11. distribute-lft-out 81.5%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

4. Simplified 81.5%

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

5. Taylor expanded in x around 0 81.5%

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. +-commutative 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)}\right) \]

   2. unpow2 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left({\varepsilon}^{3} \cdot \color{blue}{\left(x \cdot x\right)}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

   3. associate-*r* 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right)} + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

   4. associate-*r* 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

   5. *-commutative 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left({\varepsilon}^{2} \cdot 10\right)} \cdot {x}^{3}\right) \]

   6. unpow2 81.5%

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

   7. associate-*r* 81.5%

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

   8. cube-mult 81.5%

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

   9. associate-*r* 81.5%

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

   10. *-commutative 81.5%

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

   11. distribute-rgt-out 81.5%

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

   12. associate-*l* 81.5%

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

   13. *-commutative 81.5%

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

   14. associate-*r* 81.5%

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

   15. unpow2 81.5%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \left(\color{blue}{{\varepsilon}^{2}} \cdot 10\right) \cdot x\right)\right)\right) \]

   16. *-commutative 81.5%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right)} \cdot x\right)\right)\right) \]

   17. associate-*r* 81.5%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot x\right)}\right)\right)\right) \]

7. Simplified 81.5%

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

8. Taylor expanded in eps around 0 81.5%

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
9. Step-by-step derivation

   1. distribute-lft-out 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \color{blue}{\left(10 \cdot \left({\varepsilon}^{3} + {\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]

   2. unpow2 81.5%

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

   3. associate-*r* 81.5%

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

   4. unpow3 81.5%

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

   5. unpow2 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left(\color{blue}{{\varepsilon}^{2}} \cdot \varepsilon + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right) \]

   6. associate-*r* 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot \varepsilon + \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot x}\right)\right)\right)\right) \]

   7. unpow2 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot \varepsilon + \color{blue}{{\varepsilon}^{2}} \cdot x\right)\right)\right)\right) \]

   8. distribute-lft-out 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\varepsilon + x\right)\right)}\right)\right)\right) \]

   9. unpow2 81.5%

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

10. Simplified 81.5%

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

    1. fma-udef 81.5%

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

    2. metadata-eval 81.5%

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

    3. pow-prod-up 81.5%

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

    4. pow2 81.5%

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

    5. pow2 81.5%

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

    6. associate-*l* 81.4%

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

    7. associate-*r* 81.5%

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

    8. *-commutative 81.5%

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

    9. distribute-rgt-out 81.5%

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

    10. associate-*l* 81.5%

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

12. Applied egg-rr 81.5%

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

13. Final simplification 81.5%

    \[\leadsto x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(10 \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right) \]

Alternative 7: 83.0% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (x * x) * ((eps * (5.0d0 * (x * x))) + (10.0d0 * ((x + eps) * (eps * eps))))
end function

Java

public static double code(double x, double eps) {
	return (x * x) * ((eps * (5.0 * (x * x))) + (10.0 * ((x + eps) * (eps * eps))));
}

Python

def code(x, eps):
	return (x * x) * ((eps * (5.0 * (x * x))) + (10.0 * ((x + eps) * (eps * eps))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (x * x) * ((eps * (5.0 * (x * x))) + (10.0 * ((x + eps) * (eps * eps))));
end

Wolfram

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

Derivation

1. Initial program 90.4%

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

2. Taylor expanded in x around inf 81.5%

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

   1. fma-def 81.5%

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

   2. distribute-lft1-in 81.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

   3. metadata-eval 81.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

   4. *-commutative 81.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]

   5. +-commutative 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

   6. *-commutative 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]

   7. *-commutative 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]

   8. unpow3 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

   9. unpow2 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]

   10. associate-*l* 81.5%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   11. distribute-lft-out 81.5%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

4. Simplified 81.5%

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

5. Taylor expanded in x around 0 81.5%

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. +-commutative 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)}\right) \]

   2. unpow2 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left({\varepsilon}^{3} \cdot \color{blue}{\left(x \cdot x\right)}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

   3. associate-*r* 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right)} + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]

   4. associate-*r* 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]

   5. *-commutative 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left({\varepsilon}^{2} \cdot 10\right)} \cdot {x}^{3}\right) \]

   6. unpow2 81.5%

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

   7. associate-*r* 81.5%

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

   8. cube-mult 81.5%

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

   9. associate-*r* 81.5%

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

   10. *-commutative 81.5%

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

   11. distribute-rgt-out 81.5%

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

   12. associate-*l* 81.5%

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

   13. *-commutative 81.5%

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

   14. associate-*r* 81.5%

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

   15. unpow2 81.5%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \left(\color{blue}{{\varepsilon}^{2}} \cdot 10\right) \cdot x\right)\right)\right) \]

   16. *-commutative 81.5%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right)} \cdot x\right)\right)\right) \]

   17. associate-*r* 81.5%

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot x\right)}\right)\right)\right) \]

7. Simplified 81.5%

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

8. Taylor expanded in eps around 0 81.5%

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
9. Step-by-step derivation

   1. distribute-lft-out 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \color{blue}{\left(10 \cdot \left({\varepsilon}^{3} + {\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]

   2. unpow2 81.5%

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

   3. associate-*r* 81.5%

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

   4. unpow3 81.5%

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

   5. unpow2 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left(\color{blue}{{\varepsilon}^{2}} \cdot \varepsilon + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right) \]

   6. associate-*r* 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot \varepsilon + \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot x}\right)\right)\right)\right) \]

   7. unpow2 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot \varepsilon + \color{blue}{{\varepsilon}^{2}} \cdot x\right)\right)\right)\right) \]

   8. distribute-lft-out 81.5%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(x \cdot \left(10 \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\varepsilon + x\right)\right)}\right)\right)\right) \]

   9. unpow2 81.5%

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

10. Simplified 81.5%

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

    1. fma-udef 81.5%

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

    2. metadata-eval 81.5%

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

    3. pow-prod-up 81.5%

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

    4. pow2 81.5%

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

    5. pow2 81.5%

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

    6. associate-*l* 81.4%

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

    7. *-commutative 81.4%

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

    8. associate-*r* 81.4%

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

    9. distribute-lft-out 81.5%

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

    10. associate-*l* 81.5%

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

12. Applied egg-rr 81.5%

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

13. Final simplification 81.5%

    \[\leadsto \left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right) + 10 \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

Alternative 8: 82.6% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

2. Taylor expanded in eps around 0 81.2%

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

   1. add-sqr-sqrt 81.1%

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

   2. sqrt-unprod 79.0%

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

   3. pow2 79.0%

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

   4. distribute-lft1-in 79.0%

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

   5. metadata-eval 79.0%

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

4. Applied egg-rr 79.0%

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

   1. unpow2 79.0%

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

   2. *-commutative 79.0%

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

   3. *-commutative 79.0%

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

   4. swap-sqr 79.0%

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

   5. pow-sqr 79.0%

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

   6. metadata-eval 79.0%

      \[\leadsto \varepsilon \cdot \sqrt{{x}^{\color{blue}{8}} \cdot \left(5 \cdot 5\right)} \]

   7. metadata-eval 79.0%

      \[\leadsto \varepsilon \cdot \sqrt{{x}^{8} \cdot \color{blue}{25}} \]

6. Simplified 79.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\sqrt{{x}^{8} \cdot 25}} \]
7. Step-by-step derivation

   1. *-commutative 79.0%

      \[\leadsto \varepsilon \cdot \sqrt{\color{blue}{25 \cdot {x}^{8}}} \]

   2. sqrt-prod 79.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\sqrt{25} \cdot \sqrt{{x}^{8}}\right)} \]

   3. metadata-eval 79.0%

      \[\leadsto \varepsilon \cdot \left(\color{blue}{5} \cdot \sqrt{{x}^{8}}\right) \]

   4. sqrt-pow1 81.2%

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

   5. metadata-eval 81.2%

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

   6. metadata-eval 81.2%

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

   7. pow-prod-up 81.1%

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

   8. pow2 81.1%

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

   9. pow2 81.1%

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

   10. associate-*r* 81.1%

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

8. Applied egg-rr 81.1%

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

9. Final simplification 81.1%

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

Alternative 9: 70.8% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 90.4%

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

   1. sqr-pow 41.2%

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

   2. metadata-eval 41.2%

      \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{2.5}} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)} - {x}^{5} \]

   3. metadata-eval 41.2%

      \[\leadsto {\left(x + \varepsilon\right)}^{2.5} \cdot {\left(x + \varepsilon\right)}^{\color{blue}{2.5}} - {x}^{5} \]

3. Applied egg-rr 41.2%

   \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{2.5} \cdot {\left(x + \varepsilon\right)}^{2.5}} - {x}^{5} \]
4. Step-by-step derivation

   1. pow2 41.2%

      \[\leadsto \color{blue}{{\left({\left(x + \varepsilon\right)}^{2.5}\right)}^{2}} - {x}^{5} \]

   2. +-commutative 41.2%

      \[\leadsto {\left({\color{blue}{\left(\varepsilon + x\right)}}^{2.5}\right)}^{2} - {x}^{5} \]

5. Applied egg-rr 41.2%

   \[\leadsto \color{blue}{{\left({\left(\varepsilon + x\right)}^{2.5}\right)}^{2}} - {x}^{5} \]

6. Taylor expanded in x around inf 71.4%

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

   1. distribute-rgt1-in 71.4%

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

   2. metadata-eval 71.4%

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

   3. mul0-lft 71.4%

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

8. Simplified 71.4%

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

9. Final simplification 71.4%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023240 
(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)))