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

Percentage Accurate: 88.4% → 97.3%

Time: 5.9s

Alternatives: 9

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% 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: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(x \cdot x\right) \cdot \left({\varepsilon}^{3} \cdot 10 + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -7.4e-37)
   (fma
    (* eps 5.0)
    (pow x 4.0)
    (* (* x x) (+ (* (pow eps 3.0) 10.0) (* x (* eps (* eps 10.0))))))
   (if (<= x 6.4e-66) (pow eps 5.0) (* (* eps 5.0) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -7.4e-37) {
		tmp = fma((eps * 5.0), pow(x, 4.0), ((x * x) * ((pow(eps, 3.0) * 10.0) + (x * (eps * (eps * 10.0))))));
	} else if (x <= 6.4e-66) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (eps * 5.0) * pow(x, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -7.4e-37)
		tmp = fma(Float64(eps * 5.0), (x ^ 4.0), Float64(Float64(x * x) * Float64(Float64((eps ^ 3.0) * 10.0) + Float64(x * Float64(eps * Float64(eps * 10.0))))));
	elseif (x <= 6.4e-66)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -7.4e-37], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(N[Power[eps, 3.0], $MachinePrecision] * 10.0), $MachinePrecision] + N[(x * N[(eps * N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.4e-66], N[Power[eps, 5.0], $MachinePrecision], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.4e-37

   1. Initial program 17.2%

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

   2. Taylor expanded in x around inf 99.4%

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

      1. fma-def 99.5%

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

      2. distribute-lft1-in 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. *-commutative 99.5%

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

      7. *-commutative 99.5%

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

      8. unpow3 99.5%

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

      9. unpow2 99.5%

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

      10. associate-*l* 99.5%

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

      11. distribute-lft-out 99.5%

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

   4. Simplified 99.5%

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

   if -7.4e-37 < x < 6.39999999999999963e-66

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

   if 6.39999999999999963e-66 < x

   1. Initial program 52.6%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. distribute-lft1-in 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-commutative 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(x \cdot x\right) \cdot \left({\varepsilon}^{3} \cdot 10 + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \]

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-288} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -2e-288) (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 <= -2e-288) || !(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 <= (-2d-288)) .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 <= -2e-288) || !(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 <= -2e-288) 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 <= -2e-288) || !(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 <= -2e-288) || ~((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, -2e-288], 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)) < -2.00000000000000012e-288 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.2%

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

   if -2.00000000000000012e-288 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 85.0%

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

   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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-288} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \]

Alternative 3: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-37}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.4e-37)
   (* eps (+ (* 5.0 (pow x 4.0)) (* eps (* 10.0 (pow x 3.0)))))
   (if (<= x 6.4e-66) (pow eps 5.0) (* (* eps 5.0) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.4e-37) {
		tmp = eps * ((5.0 * pow(x, 4.0)) + (eps * (10.0 * pow(x, 3.0))));
	} else if (x <= 6.4e-66) {
		tmp = pow(eps, 5.0);
	} 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) :: tmp
    if (x <= (-6.4d-37)) then
        tmp = eps * ((5.0d0 * (x ** 4.0d0)) + (eps * (10.0d0 * (x ** 3.0d0))))
    else if (x <= 6.4d-66) then
        tmp = eps ** 5.0d0
    else
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.4e-37) {
		tmp = eps * ((5.0 * Math.pow(x, 4.0)) + (eps * (10.0 * Math.pow(x, 3.0))));
	} else if (x <= 6.4e-66) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.4e-37:
		tmp = eps * ((5.0 * math.pow(x, 4.0)) + (eps * (10.0 * math.pow(x, 3.0))))
	elif x <= 6.4e-66:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.4e-37)
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(eps * Float64(10.0 * (x ^ 3.0)))));
	elseif (x <= 6.4e-66)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.4e-37)
		tmp = eps * ((5.0 * (x ^ 4.0)) + (eps * (10.0 * (x ^ 3.0))));
	elseif (x <= 6.4e-66)
		tmp = eps ^ 5.0;
	else
		tmp = (eps * 5.0) * (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.4e-37], N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(10.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.4e-66], N[Power[eps, 5.0], $MachinePrecision], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.3999999999999998e-37

   1. Initial program 17.2%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. unpow2 99.3%

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

      3. associate-*l* 99.3%

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

      4. distribute-lft-out 99.3%

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

      5. distribute-lft1-in 99.3%

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

      6. metadata-eval 99.3%

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

      7. *-commutative 99.3%

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

      8. *-commutative 99.3%

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

      9. distribute-rgt-out 99.3%

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

      10. associate-*r* 99.3%

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

   4. Simplified 99.3%

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

   if -6.3999999999999998e-37 < x < 6.39999999999999963e-66

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

   if 6.39999999999999963e-66 < x

   1. Initial program 52.6%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. distribute-lft1-in 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-commutative 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-37}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \]

Alternative 4: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-37} \lor \neg \left(x \leq 6.4 \cdot 10^{-66}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -6.4e-37) (not (<= x 6.4e-66)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -6.4e-37) || !(x <= 6.4e-66)) {
		tmp = 5.0 * (eps * 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 <= (-6.4d-37)) .or. (.not. (x <= 6.4d-66))) then
        tmp = 5.0d0 * (eps * (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 <= -6.4e-37) || !(x <= 6.4e-66)) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -6.4e-37) or not (x <= 6.4e-66):
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -6.4e-37) || !(x <= 6.4e-66))
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -6.4e-37) || ~((x <= 6.4e-66)))
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -6.4e-37], N[Not[LessEqual[x, 6.4e-66]], $MachinePrecision]], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.3999999999999998e-37 or 6.39999999999999963e-66 < x

   1. Initial program 41.5%

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

   2. Taylor expanded in x around inf 98.4%

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

      1. distribute-lft1-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*l* 98.4%

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

   4. Simplified 98.4%

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

   if -6.3999999999999998e-37 < x < 6.39999999999999963e-66

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-37} \lor \neg \left(x \leq 6.4 \cdot 10^{-66}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-37}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.4e-37)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 6.4e-66) (pow eps 5.0) (* (* eps 5.0) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.4e-37) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 6.4e-66) {
		tmp = pow(eps, 5.0);
	} 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) :: tmp
    if (x <= (-6.4d-37)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 6.4d-66) then
        tmp = eps ** 5.0d0
    else
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.4e-37) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 6.4e-66) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.4e-37:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 6.4e-66:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.4e-37)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 6.4e-66)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.4e-37)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 6.4e-66)
		tmp = eps ^ 5.0;
	else
		tmp = (eps * 5.0) * (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.4e-37], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.4e-66], N[Power[eps, 5.0], $MachinePrecision], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.3999999999999998e-37

   1. Initial program 17.2%

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

   2. Taylor expanded in x around inf 95.5%

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

      1. distribute-lft1-in 95.5%

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

      2. metadata-eval 95.5%

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

      3. associate-*l* 95.6%

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

   4. Simplified 95.6%

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

   if -6.3999999999999998e-37 < x < 6.39999999999999963e-66

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

   if 6.39999999999999963e-66 < x

   1. Initial program 52.6%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. distribute-lft1-in 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-commutative 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-37}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \]

Alternative 6: 97.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-37} \lor \neg \left(x \leq 6.4 \cdot 10^{-66}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -6.4e-37) (not (<= x 6.4e-66)))
   (* eps (* (* x x) (* 5.0 (* x x))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -6.4e-37) || !(x <= 6.4e-66)) {
		tmp = eps * ((x * x) * (5.0 * (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 <= (-6.4d-37)) .or. (.not. (x <= 6.4d-66))) then
        tmp = eps * ((x * x) * (5.0d0 * (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 <= -6.4e-37) || !(x <= 6.4e-66)) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -6.4e-37) or not (x <= 6.4e-66):
		tmp = eps * ((x * x) * (5.0 * (x * x)))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -6.4e-37) || !(x <= 6.4e-66))
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(5.0 * Float64(x * x))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -6.4e-37) || ~((x <= 6.4e-66)))
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -6.4e-37], N[Not[LessEqual[x, 6.4e-66]], $MachinePrecision]], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.3999999999999998e-37 or 6.39999999999999963e-66 < x

   1. Initial program 41.5%

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

   2. Taylor expanded in eps around 0 98.4%

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

      1. distribute-lft1-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. *-commutative 98.4%

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

      4. sqr-pow 98.2%

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

      5. associate-*l* 98.3%

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

      6. metadata-eval 98.3%

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

      7. pow2 98.3%

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

      8. metadata-eval 98.3%

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

      9. pow2 98.3%

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

   4. Applied egg-rr 98.3%

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

   if -6.3999999999999998e-37 < x < 6.39999999999999963e-66

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-37} \lor \neg \left(x \leq 6.4 \cdot 10^{-66}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 7: 81.9% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

2. Taylor expanded in x around inf 82.9%

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

   1. distribute-lft1-in 82.9%

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

   2. metadata-eval 82.9%

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

   3. associate-*l* 82.8%

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

4. Simplified 82.8%

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

   1. add-sqr-sqrt 76.4%

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

   2. pow2 76.4%

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

   3. sqrt-prod 39.5%

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

   4. sqrt-pow1 39.5%

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

   5. metadata-eval 39.5%

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

   6. pow2 39.5%

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

6. Applied egg-rr 39.5%

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

   1. unpow2 39.5%

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

   2. *-commutative 39.5%

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

   3. associate-*l* 39.5%

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

   4. associate-*l* 39.5%

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

   5. add-sqr-sqrt 82.8%

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

8. Applied egg-rr 82.8%

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

9. Final simplification 82.8%

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

Alternative 8: 82.0% 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 87.5%

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

2. Taylor expanded in eps around 0 82.8%

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

   1. distribute-lft1-in 82.8%

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

   2. metadata-eval 82.8%

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

   3. *-commutative 82.8%

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

   4. sqr-pow 82.8%

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

   5. associate-*l* 82.8%

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

   6. metadata-eval 82.8%

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

   7. pow2 82.8%

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

   8. metadata-eval 82.8%

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

   9. pow2 82.8%

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

4. Applied egg-rr 82.8%

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

5. Taylor expanded in x around 0 82.8%

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

   1. unpow2 82.8%

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

   2. *-commutative 82.8%

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

   3. associate-*r* 82.8%

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

7. Simplified 82.8%

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

8. Final simplification 82.8%

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

Alternative 9: 82.0% 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 87.5%

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

2. Taylor expanded in eps around 0 82.8%

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

   1. distribute-lft1-in 82.8%

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

   2. metadata-eval 82.8%

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

   3. *-commutative 82.8%

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

   4. sqr-pow 82.8%

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

   5. associate-*l* 82.8%

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

   6. metadata-eval 82.8%

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

   7. pow2 82.8%

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

   8. metadata-eval 82.8%

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

   9. pow2 82.8%

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

4. Applied egg-rr 82.8%

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

5. Final simplification 82.8%

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

Reproduce

herbie shell --seed 2023230 
(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)))