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

Percentage Accurate: 88.0% → 97.5%

Time: 11.7s

Alternatives: 6

Speedup: 1.8×

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: 88.0% 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: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{-49} \lor \neg \left(x \leq 4 \cdot 10^{-35}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)}{x \cdot x} \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.15e-49) (not (<= x 4e-35)))
   (*
    (pow x 4.0)
    (- (* eps 5.0) (* (/ (* (* eps eps) (+ x eps)) (* x x)) -10.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.15e-49) || !(x <= 4e-35)) {
		tmp = pow(x, 4.0) * ((eps * 5.0) - ((((eps * eps) * (x + eps)) / (x * x)) * -10.0));
	} 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 <= (-3.15d-49)) .or. (.not. (x <= 4d-35))) then
        tmp = (x ** 4.0d0) * ((eps * 5.0d0) - ((((eps * eps) * (x + eps)) / (x * x)) * (-10.0d0)))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -3.15e-49) || !(x <= 4e-35)) {
		tmp = Math.pow(x, 4.0) * ((eps * 5.0) - ((((eps * eps) * (x + eps)) / (x * x)) * -10.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.15e-49) or not (x <= 4e-35):
		tmp = math.pow(x, 4.0) * ((eps * 5.0) - ((((eps * eps) * (x + eps)) / (x * x)) * -10.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.15e-49) || !(x <= 4e-35))
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) - Float64(Float64(Float64(Float64(eps * eps) * Float64(x + eps)) / Float64(x * x)) * -10.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -3.15e-49) || ~((x <= 4e-35)))
		tmp = (x ^ 4.0) * ((eps * 5.0) - ((((eps * eps) * (x + eps)) / (x * x)) * -10.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.15e-49], N[Not[LessEqual[x, 4e-35]], $MachinePrecision]], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(N[(eps * eps), $MachinePrecision] * N[(x + eps), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1499999999999998e-49 or 4.00000000000000003e-35 < x

   1. Initial program 25.8%

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

   3. Taylor expanded in x around -inf 98.6%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + \left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -1 \cdot \frac{4 \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

   5. Simplified 98.6%

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

      1. cube-mult 98.6%

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in x around 0 98.6%

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

      1. distribute-lft-in 98.6%

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

      2. unpow2 98.6%

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

      3. associate-/l* 98.6%

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

      4. unpow2 98.6%

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

      5. unpow3 98.6%

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

      6. distribute-lft-out 98.6%

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

   10. Simplified 98.6%

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

   if -3.1499999999999998e-49 < x < 4.00000000000000003e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{-49} \lor \neg \left(x \leq 4 \cdot 10^{-35}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)}{x \cdot x} \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-50} \lor \neg \left(x \leq 4 \cdot 10^{-35}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10 - \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{10}{x}}{x}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.85e-50) (not (<= x 4e-35)))
   (*
    (-
     (* eps 5.0)
     (/ (- (* (* eps eps) -10.0) (* (* eps (* eps eps)) (/ 10.0 x))) x))
    (* (* x x) (* x x)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.85e-50) || !(x <= 4e-35)) {
		tmp = ((eps * 5.0) - ((((eps * eps) * -10.0) - ((eps * (eps * eps)) * (10.0 / x))) / x)) * ((x * x) * (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 <= (-1.85d-50)) .or. (.not. (x <= 4d-35))) then
        tmp = ((eps * 5.0d0) - ((((eps * eps) * (-10.0d0)) - ((eps * (eps * eps)) * (10.0d0 / x))) / x)) * ((x * x) * (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 <= -1.85e-50) || !(x <= 4e-35)) {
		tmp = ((eps * 5.0) - ((((eps * eps) * -10.0) - ((eps * (eps * eps)) * (10.0 / x))) / x)) * ((x * x) * (x * x));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.85e-50) or not (x <= 4e-35):
		tmp = ((eps * 5.0) - ((((eps * eps) * -10.0) - ((eps * (eps * eps)) * (10.0 / x))) / x)) * ((x * x) * (x * x))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.85e-50) || !(x <= 4e-35))
		tmp = Float64(Float64(Float64(eps * 5.0) - Float64(Float64(Float64(Float64(eps * eps) * -10.0) - Float64(Float64(eps * Float64(eps * eps)) * Float64(10.0 / x))) / x)) * Float64(Float64(x * x) * Float64(x * x)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.85e-50) || ~((x <= 4e-35)))
		tmp = ((eps * 5.0) - ((((eps * eps) * -10.0) - ((eps * (eps * eps)) * (10.0 / x))) / x)) * ((x * x) * (x * x));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.85e-50], N[Not[LessEqual[x, 4e-35]], $MachinePrecision]], N[(N[(N[(eps * 5.0), $MachinePrecision] - N[(N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] - N[(N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(10.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85e-50 or 4.00000000000000003e-35 < x

   1. Initial program 25.8%

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

   3. Taylor expanded in x around -inf 98.6%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + \left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -1 \cdot \frac{4 \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

   5. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-/l* 98.6%

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

      3. cube-mult 98.6%

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

      4. metadata-eval 98.6%

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

      5. pow-prod-up 98.4%

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

      6. pow2 98.4%

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

      7. pow2 98.4%

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

   7. Applied egg-rr 98.4%

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

   if -1.85e-50 < x < 4.00000000000000003e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-50} \lor \neg \left(x \leq 4 \cdot 10^{-35}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5 - \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot -10 - \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{10}{x}}{x}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 10.9× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (* x x) (* x x)) (- (* eps 5.0) (* (/ (* eps eps) x) -10.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

3. Taylor expanded in x around -inf 86.5%

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

   1. +-commutative 86.5%

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

   2. associate-+r+ 86.5%

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

   3. mul-1-neg 86.5%

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

   4. unsub-neg 86.5%

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

   5. distribute-rgt1-in 86.5%

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

   6. metadata-eval 86.5%

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

   7. *-commutative 86.5%

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

5. Simplified 86.5%

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

   1. *-commutative 86.5%

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

   2. *-commutative 86.5%

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

   3. *-un-lft-identity 86.5%

      \[\leadsto \left(\varepsilon \cdot 5 - \frac{-10 \cdot \left(\varepsilon \cdot \varepsilon\right)}{\color{blue}{1 \cdot x}}\right) \cdot {x}^{4} \]

   4. times-frac 86.5%

      \[\leadsto \left(\varepsilon \cdot 5 - \color{blue}{\frac{-10}{1} \cdot \frac{\varepsilon \cdot \varepsilon}{x}}\right) \cdot {x}^{4} \]

   5. metadata-eval 86.5%

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

   6. metadata-eval 86.5%

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

   7. pow-prod-up 86.5%

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

   8. pow2 86.5%

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

   9. pow2 86.5%

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

7. Applied egg-rr 86.5%

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

8. Final simplification 86.5%

   \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot 5 - \frac{\varepsilon \cdot \varepsilon}{x} \cdot -10\right) \]
9. Add Preprocessing

Alternative 4: 82.9% 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 86.7%

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

3. Taylor expanded in eps around 0 86.3%

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

   1. *-commutative 86.3%

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

   2. distribute-lft1-in 86.3%

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

   3. metadata-eval 86.3%

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

   4. metadata-eval 86.3%

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

   5. pow-prod-up 86.3%

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

   6. pow2 86.3%

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

   7. pow2 86.3%

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

5. Applied egg-rr 86.3%

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

   1. associate-*r* 86.3%

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

7. Applied egg-rr 86.3%

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

8. Final simplification 86.3%

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

Alternative 5: 82.9% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

3. Taylor expanded in eps around 0 86.3%

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

   1. *-commutative 86.3%

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

   2. distribute-lft1-in 86.3%

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

   3. metadata-eval 86.3%

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

   4. metadata-eval 86.3%

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

   5. pow-prod-up 86.3%

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

   6. pow2 86.3%

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

   7. pow2 86.3%

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

5. Applied egg-rr 86.3%

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

6. Final simplification 86.3%

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

Alternative 6: 83.0% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

3. Taylor expanded in x around inf 86.3%

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

   1. distribute-rgt1-in 86.3%

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

   2. metadata-eval 86.3%

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

   3. *-commutative 86.3%

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

5. Simplified 86.3%

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

   1. metadata-eval 86.3%

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

   2. pow-prod-up 86.3%

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

   3. pow2 86.3%

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

   4. pow2 86.3%

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

   5. associate-*l* 86.2%

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

7. Applied egg-rr 86.2%

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

   1. associate-*l* 86.3%

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

9. Applied egg-rr 86.3%

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

10. Final simplification 86.3%

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

Reproduce

herbie shell --seed 2024097 
(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)))