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

Percentage Accurate: 88.8% → 99.4%

Time: 10.4s

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: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-312} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -4e-312) (not (<= t_0 0.0)))
     t_0
     (* (* eps 5.0) (pow x 4.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-312) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (eps * 5.0) * pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-4d-312)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -4e-312) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -4e-312) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -4e-312) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -4e-312) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (eps * 5.0) * (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-312], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -3.9999999999988e-312 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 97.7%

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

   if -3.9999999999988e-312 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 83.2%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

      3. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-312} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \]

Alternative 2: 98.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-52} \lor \neg \left(x \leq 6.2 \cdot 10^{-56}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right) + 10 \cdot \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.2e-52) (not (<= x 6.2e-56)))
   (+
    (* eps (* 5.0 (pow x 4.0)))
    (* 10.0 (* (* (* eps eps) (+ x eps)) (* x x))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.2e-52) || !(x <= 6.2e-56)) {
		tmp = (eps * (5.0 * pow(x, 4.0))) + (10.0 * (((eps * eps) * (x + eps)) * (x * x)));
	} 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 <= (-3.2d-52)) .or. (.not. (x <= 6.2d-56))) then
        tmp = (eps * (5.0d0 * (x ** 4.0d0))) + (10.0d0 * (((eps * eps) * (x + eps)) * (x * x)))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -3.2e-52) || !(x <= 6.2e-56)) {
		tmp = (eps * (5.0 * Math.pow(x, 4.0))) + (10.0 * (((eps * eps) * (x + eps)) * (x * x)));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.2e-52) or not (x <= 6.2e-56):
		tmp = (eps * (5.0 * math.pow(x, 4.0))) + (10.0 * (((eps * eps) * (x + eps)) * (x * x)))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.2e-52) || !(x <= 6.2e-56))
		tmp = Float64(Float64(eps * Float64(5.0 * (x ^ 4.0))) + Float64(10.0 * Float64(Float64(Float64(eps * eps) * Float64(x + eps)) * Float64(x * x))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -3.2e-52) || ~((x <= 6.2e-56)))
		tmp = (eps * (5.0 * (x ^ 4.0))) + (10.0 * (((eps * eps) * (x + eps)) * (x * x)));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.2e-52], N[Not[LessEqual[x, 6.2e-56]], $MachinePrecision]], N[(N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(10.0 * N[(N[(N[(eps * eps), $MachinePrecision] * N[(x + eps), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000001e-52 or 6.19999999999999975e-56 < x

   1. Initial program 33.5%

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

   2. Taylor expanded in x around inf 97.9%

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

      1. fma-def 97.9%

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

      2. distribute-lft1-in 97.9%

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

      3. metadata-eval 97.9%

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

      4. *-commutative 97.9%

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

      5. +-commutative 97.9%

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

      6. *-commutative 97.9%

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

      7. *-commutative 97.9%

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

      8. unpow3 97.9%

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

      9. unpow2 97.9%

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

      10. associate-*l* 97.9%

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

      11. distribute-lft-out 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in x around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. unpow2 97.9%

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

      3. associate-*r* 97.9%

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

      4. unpow2 97.9%

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

      5. associate-*r* 97.9%

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

      6. *-commutative 97.9%

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

      7. associate-*r* 97.9%

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

      8. cube-mult 97.9%

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

      9. associate-*r* 97.9%

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

      10. *-commutative 97.9%

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

      11. associate-*r* 97.9%

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

      12. distribute-rgt-in 97.9%

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

      13. *-commutative 97.9%

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

      14. associate-*l* 97.9%

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

      15. associate-*l* 97.9%

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

      16. unpow2 97.9%

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

   7. Simplified 97.9%

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

      1. fma-udef 97.9%

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

      2. associate-*r* 97.8%

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

      3. *-commutative 97.8%

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

      4. associate-*l* 97.8%

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

      5. cube-mult 97.8%

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

      6. distribute-rgt-out 97.8%

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

   9. Applied egg-rr 97.8%

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

   if -3.2000000000000001e-52 < x < 6.19999999999999975e-56

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-52} \lor \neg \left(x \leq 6.2 \cdot 10^{-56}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right) + 10 \cdot \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 3: 98.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-52} \lor \neg \left(x \leq 1.02 \cdot 10^{-56}\right):\\ \;\;\;\;x \cdot \left(\left(x \cdot 10\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\right)\right) + \left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.2e-52) (not (<= x 1.02e-56)))
   (+
    (* x (* (* x 10.0) (* (* eps eps) (+ x eps))))
    (* (* eps 5.0) (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.2e-52) || !(x <= 1.02e-56)) {
		tmp = (x * ((x * 10.0) * ((eps * eps) * (x + eps)))) + ((eps * 5.0) * pow(x, 4.0));
	} 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 <= (-3.2d-52)) .or. (.not. (x <= 1.02d-56))) then
        tmp = (x * ((x * 10.0d0) * ((eps * eps) * (x + eps)))) + ((eps * 5.0d0) * (x ** 4.0d0))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -3.2e-52) || !(x <= 1.02e-56)) {
		tmp = (x * ((x * 10.0) * ((eps * eps) * (x + eps)))) + ((eps * 5.0) * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.2e-52) or not (x <= 1.02e-56):
		tmp = (x * ((x * 10.0) * ((eps * eps) * (x + eps)))) + ((eps * 5.0) * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.2e-52) || !(x <= 1.02e-56))
		tmp = Float64(Float64(x * Float64(Float64(x * 10.0) * Float64(Float64(eps * eps) * Float64(x + eps)))) + Float64(Float64(eps * 5.0) * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -3.2e-52) || ~((x <= 1.02e-56)))
		tmp = (x * ((x * 10.0) * ((eps * eps) * (x + eps)))) + ((eps * 5.0) * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.2e-52], N[Not[LessEqual[x, 1.02e-56]], $MachinePrecision]], N[(N[(x * N[(N[(x * 10.0), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000001e-52 or 1.02e-56 < x

   1. Initial program 33.5%

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

   2. Taylor expanded in x around inf 97.9%

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

      1. fma-def 97.9%

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

      2. distribute-lft1-in 97.9%

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

      3. metadata-eval 97.9%

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

      4. *-commutative 97.9%

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

      5. +-commutative 97.9%

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

      6. *-commutative 97.9%

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

      7. *-commutative 97.9%

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

      8. unpow3 97.9%

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

      9. unpow2 97.9%

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

      10. associate-*l* 97.9%

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

      11. distribute-lft-out 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in x around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. unpow2 97.9%

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

      3. associate-*r* 97.9%

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

      4. unpow2 97.9%

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

      5. associate-*r* 97.9%

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

      6. *-commutative 97.9%

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

      7. associate-*r* 97.9%

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

      8. cube-mult 97.9%

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

      9. associate-*r* 97.9%

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

      10. *-commutative 97.9%

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

      11. associate-*r* 97.9%

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

      12. distribute-rgt-in 97.9%

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

      13. *-commutative 97.9%

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

      14. associate-*l* 97.9%

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

      15. associate-*l* 97.9%

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

      16. unpow2 97.9%

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

   7. Simplified 97.9%

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

      1. associate-*r* 97.9%

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

      2. distribute-lft-in 97.9%

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

      3. associate-*l* 97.9%

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

      4. associate-*l* 97.9%

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

      5. *-commutative 97.9%

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

      6. associate-*l* 97.9%

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

   9. Applied egg-rr 97.9%

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

       1. distribute-lft-out 97.9%

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

       2. unpow3 97.9%

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

       3. associate-*r* 97.9%

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

       4. distribute-lft-in 97.9%

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

       5. *-commutative 97.9%

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

       6. associate-*r* 97.9%

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

       7. *-commutative 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in eps around 0 97.6%

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

       1. +-commutative 97.6%

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

       2. associate-+r+ 97.6%

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

   14. Simplified 97.9%

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

   if -3.2000000000000001e-52 < x < 1.02e-56

   1. Initial program 99.8%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-52} \lor \neg \left(x \leq 1.02 \cdot 10^{-56}\right):\\ \;\;\;\;x \cdot \left(\left(x \cdot 10\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\right)\right) + \left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 4: 98.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-52} \lor \neg \left(x \leq 8.8 \cdot 10^{-56}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5\right) + \varepsilon \cdot \left(10 \cdot \left(x + \varepsilon\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.3e-52) (not (<= x 8.8e-56)))
   (* x (* x (* eps (+ (* x (* x 5.0)) (* eps (* 10.0 (+ x eps)))))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.3e-52) || !(x <= 8.8e-56)) {
		tmp = x * (x * (eps * ((x * (x * 5.0)) + (eps * (10.0 * (x + 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 <= (-3.3d-52)) .or. (.not. (x <= 8.8d-56))) then
        tmp = x * (x * (eps * ((x * (x * 5.0d0)) + (eps * (10.0d0 * (x + 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 <= -3.3e-52) || !(x <= 8.8e-56)) {
		tmp = x * (x * (eps * ((x * (x * 5.0)) + (eps * (10.0 * (x + eps))))));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.3e-52) or not (x <= 8.8e-56):
		tmp = x * (x * (eps * ((x * (x * 5.0)) + (eps * (10.0 * (x + eps))))))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.3e-52) || !(x <= 8.8e-56))
		tmp = Float64(x * Float64(x * Float64(eps * Float64(Float64(x * Float64(x * 5.0)) + Float64(eps * Float64(10.0 * Float64(x + eps)))))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -3.3e-52) || ~((x <= 8.8e-56)))
		tmp = x * (x * (eps * ((x * (x * 5.0)) + (eps * (10.0 * (x + eps))))));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.3e-52], N[Not[LessEqual[x, 8.8e-56]], $MachinePrecision]], N[(x * N[(x * N[(eps * N[(N[(x * N[(x * 5.0), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(10.0 * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.29999999999999995e-52 or 8.80000000000000017e-56 < x

   1. Initial program 33.5%

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

   2. Taylor expanded in x around inf 97.9%

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

      1. fma-def 97.9%

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

      2. distribute-lft1-in 97.9%

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

      3. metadata-eval 97.9%

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

      4. *-commutative 97.9%

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

      5. +-commutative 97.9%

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

      6. *-commutative 97.9%

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

      7. *-commutative 97.9%

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

      8. unpow3 97.9%

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

      9. unpow2 97.9%

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

      10. associate-*l* 97.9%

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

      11. distribute-lft-out 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in x around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. unpow2 97.9%

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

      3. associate-*r* 97.9%

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

      4. unpow2 97.9%

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

      5. associate-*r* 97.9%

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

      6. *-commutative 97.9%

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

      7. associate-*r* 97.9%

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

      8. cube-mult 97.9%

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

      9. associate-*r* 97.9%

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

      10. *-commutative 97.9%

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

      11. associate-*r* 97.9%

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

      12. distribute-rgt-in 97.9%

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

      13. *-commutative 97.9%

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

      14. associate-*l* 97.9%

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

      15. associate-*l* 97.9%

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

      16. unpow2 97.9%

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

   7. Simplified 97.9%

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

      1. associate-*r* 97.9%

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

      2. distribute-lft-in 97.9%

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

      3. associate-*l* 97.9%

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

      4. associate-*l* 97.9%

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

      5. *-commutative 97.9%

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

      6. associate-*l* 97.9%

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

   9. Applied egg-rr 97.9%

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

       1. distribute-lft-out 97.9%

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

       2. unpow3 97.9%

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

       3. associate-*r* 97.9%

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

       4. distribute-lft-in 97.9%

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

       5. *-commutative 97.9%

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

       6. associate-*r* 97.9%

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

       7. *-commutative 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in eps around 0 97.6%

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

   14. Simplified 97.6%

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

   if -3.29999999999999995e-52 < x < 8.80000000000000017e-56

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-52} \lor \neg \left(x \leq 8.8 \cdot 10^{-56}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5\right) + \varepsilon \cdot \left(10 \cdot \left(x + \varepsilon\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5: 83.1% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

2. Taylor expanded in x around inf 84.3%

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

   1. fma-def 84.3%

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

   2. distribute-lft1-in 84.3%

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

   3. metadata-eval 84.3%

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

   4. *-commutative 84.3%

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

   5. +-commutative 84.3%

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

   6. *-commutative 84.3%

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

   7. *-commutative 84.3%

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

   8. unpow3 84.3%

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

   9. unpow2 84.3%

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

   10. associate-*l* 84.3%

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

   11. distribute-lft-out 84.3%

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

4. Simplified 84.3%

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

5. Taylor expanded in x around 0 84.3%

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

   1. +-commutative 84.3%

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

   2. unpow2 84.3%

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

   3. associate-*r* 84.3%

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

   4. unpow2 84.3%

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

   5. associate-*r* 84.3%

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

   6. *-commutative 84.3%

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

   7. associate-*r* 84.3%

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

   8. cube-mult 84.3%

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

   9. associate-*r* 84.3%

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

   10. *-commutative 84.3%

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

   11. associate-*r* 84.3%

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

   12. distribute-rgt-in 84.3%

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

   13. *-commutative 84.3%

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

   14. associate-*l* 84.3%

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

   15. associate-*l* 84.3%

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

   16. unpow2 84.3%

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

7. Simplified 84.3%

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

   1. associate-*r* 84.3%

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

   2. distribute-lft-in 84.3%

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

   3. associate-*l* 84.3%

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

   4. associate-*l* 84.3%

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

   5. *-commutative 84.3%

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

   6. associate-*l* 84.3%

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

9. Applied egg-rr 84.3%

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

    1. distribute-lft-out 84.3%

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

    2. unpow3 84.3%

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

    3. associate-*r* 84.3%

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

    4. distribute-lft-in 84.3%

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

    5. *-commutative 84.3%

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

    6. associate-*r* 84.3%

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

    7. *-commutative 84.3%

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

11. Simplified 84.3%

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

12. Taylor expanded in eps around 0 84.2%

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

14. Simplified 84.2%

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

15. Final simplification 84.2%

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

Alternative 6: 82.8% 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 85.8%

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

2. Taylor expanded in eps around 0 83.7%

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

   1. distribute-lft1-in 83.7%

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

   2. metadata-eval 83.7%

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

   3. metadata-eval 83.7%

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

   4. pow-prod-up 83.7%

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

   5. pow2 83.7%

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

   6. pow2 83.7%

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

   7. associate-*r* 83.7%

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

4. Applied egg-rr 83.7%

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

5. Final simplification 83.7%

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

Alternative 7: 82.8% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

2. Taylor expanded in eps around 0 83.7%

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

   1. distribute-lft1-in 83.7%

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

   2. metadata-eval 83.7%

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

   3. metadata-eval 83.7%

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

   4. pow-prod-up 83.7%

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

   5. pow2 83.7%

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

   6. pow2 83.7%

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

   7. associate-*r* 83.7%

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

4. Applied egg-rr 83.7%

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

5. Taylor expanded in x around 0 83.7%

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

   1. unpow2 83.7%

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

   2. *-commutative 83.7%

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

   3. associate-*r* 83.7%

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

7. Simplified 83.7%

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

8. Final simplification 83.7%

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

Alternative 8: 82.8% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

2. Taylor expanded in x around inf 83.7%

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

   1. distribute-lft1-in 83.7%

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

   2. metadata-eval 83.7%

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

   3. *-commutative 83.7%

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

4. Simplified 83.7%

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

   1. add-cbrt-cube 73.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\right)\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\right)}} \]

   2. pow3 73.5%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\right)}^{3}}} \]

   3. *-commutative 73.5%

      \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(5 \cdot \varepsilon\right)} \cdot {x}^{4}\right)}^{3}} \]

   4. associate-*l* 73.5%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\right)}}^{3}} \]

6. Applied egg-rr 73.5%

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

   1. rem-cbrt-cube 83.7%

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

   2. associate-*r* 83.7%

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

   3. *-commutative 83.7%

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

   4. metadata-eval 83.7%

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

   5. pow-prod-up 83.7%

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

   6. pow2 83.7%

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

   7. pow2 83.7%

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

   8. associate-*r* 83.7%

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

8. Applied egg-rr 83.7%

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

9. Taylor expanded in eps around 0 83.7%

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

    1. unpow2 83.7%

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

11. Simplified 83.7%

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

12. Final simplification 83.7%

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

Alternative 9: 82.8% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

2. Taylor expanded in x around inf 83.7%

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

   1. distribute-lft1-in 83.7%

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

   2. metadata-eval 83.7%

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

   3. *-commutative 83.7%

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

4. Simplified 83.7%

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

   1. add-cbrt-cube 73.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\right)\right) \cdot \left(\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\right)}} \]

   2. pow3 73.5%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\right)}^{3}}} \]

   3. *-commutative 73.5%

      \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(5 \cdot \varepsilon\right)} \cdot {x}^{4}\right)}^{3}} \]

   4. associate-*l* 73.5%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\right)}}^{3}} \]

6. Applied egg-rr 73.5%

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

   1. rem-cbrt-cube 83.7%

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

   2. associate-*r* 83.7%

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

   3. *-commutative 83.7%

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

   4. metadata-eval 83.7%

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

   5. pow-prod-up 83.7%

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

   6. pow2 83.7%

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

   7. pow2 83.7%

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

   8. associate-*r* 83.7%

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

8. Applied egg-rr 83.7%

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

9. Final simplification 83.7%

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

Reproduce

herbie shell --seed 2023274 
(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)))