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

Percentage Accurate: 88.2% → 98.9%

Time: 8.1s

Alternatives: 9

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: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(\frac{x \cdot 5 - \frac{{x}^{2} \cdot \left(-10 \cdot \frac{x}{\varepsilon} - 10\right)}{\varepsilon}}{\varepsilon} - -1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-309)
     (*
      (pow eps 5.0)
      (-
       (/
        (- (* x 5.0) (/ (* (pow x 2.0) (- (* -10.0 (/ x eps)) 10.0)) eps))
        eps)
       -1.0))
     (if (<= t_0 0.0) (* eps (* (* x 5.0) (pow x 3.0))) t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-309) {
		tmp = pow(eps, 5.0) * ((((x * 5.0) - ((pow(x, 2.0) * ((-10.0 * (x / eps)) - 10.0)) / eps)) / eps) - -1.0);
	} else if (t_0 <= 0.0) {
		tmp = eps * ((x * 5.0) * pow(x, 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-1d-309)) then
        tmp = (eps ** 5.0d0) * ((((x * 5.0d0) - (((x ** 2.0d0) * (((-10.0d0) * (x / eps)) - 10.0d0)) / eps)) / eps) - (-1.0d0))
    else if (t_0 <= 0.0d0) then
        tmp = eps * ((x * 5.0d0) * (x ** 3.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-309) {
		tmp = Math.pow(eps, 5.0) * ((((x * 5.0) - ((Math.pow(x, 2.0) * ((-10.0 * (x / eps)) - 10.0)) / eps)) / eps) - -1.0);
	} else if (t_0 <= 0.0) {
		tmp = eps * ((x * 5.0) * Math.pow(x, 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -1e-309:
		tmp = math.pow(eps, 5.0) * ((((x * 5.0) - ((math.pow(x, 2.0) * ((-10.0 * (x / eps)) - 10.0)) / eps)) / eps) - -1.0)
	elif t_0 <= 0.0:
		tmp = eps * ((x * 5.0) * math.pow(x, 3.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-309)
		tmp = Float64((eps ^ 5.0) * Float64(Float64(Float64(Float64(x * 5.0) - Float64(Float64((x ^ 2.0) * Float64(Float64(-10.0 * Float64(x / eps)) - 10.0)) / eps)) / eps) - -1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(Float64(x * 5.0) * (x ^ 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -1e-309)
		tmp = (eps ^ 5.0) * ((((x * 5.0) - (((x ^ 2.0) * ((-10.0 * (x / eps)) - 10.0)) / eps)) / eps) - -1.0);
	elseif (t_0 <= 0.0)
		tmp = eps * ((x * 5.0) * (x ^ 3.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-309], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(N[(N[(N[(x * 5.0), $MachinePrecision] - N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(-10.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(N[(x * 5.0), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.000000000000002e-309

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around -inf 99.9%

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + \left(-1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right) + -1 \cdot \frac{4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\left(-1 - \frac{5 \cdot x - \frac{{x}^{2} \cdot -10 - \frac{{x}^{3} \cdot 10}{\varepsilon}}{\varepsilon}}{\varepsilon}\right) \cdot \left(-{\varepsilon}^{5}\right)} \]

   6. Taylor expanded in x around 0 99.9%

      \[\leadsto \left(-1 - \frac{5 \cdot x - \frac{\color{blue}{{x}^{2} \cdot \left(-10 \cdot \frac{x}{\varepsilon} - 10\right)}}{\varepsilon}}{\varepsilon}\right) \cdot \left(-{\varepsilon}^{5}\right) \]

   if -1.000000000000002e-309 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

   5. Simplified 99.9%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. swap-sqr 99.1%

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

      6. pow-prod-up 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. sqrt-prod 99.1%

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

      2. sqrt-pow1 99.9%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{{x}^{\left(\frac{8}{2}\right)}} \cdot \sqrt{25}\right) \]

      3. metadata-eval 99.9%

         \[\leadsto \varepsilon \cdot \left({x}^{\color{blue}{4}} \cdot \sqrt{25}\right) \]

      4. metadata-eval 99.9%

         \[\leadsto \varepsilon \cdot \left({x}^{\color{blue}{\left(3 + 1\right)}} \cdot \sqrt{25}\right) \]

      5. pow-plus 99.9%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left({x}^{3} \cdot x\right)} \cdot \sqrt{25}\right) \]

      6. metadata-eval 99.9%

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

      7. associate-*l* 99.9%

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

   9. Applied egg-rr 99.9%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-309}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(\frac{x \cdot 5 - \frac{{x}^{2} \cdot \left(-10 \cdot \frac{x}{\varepsilon} - 10\right)}{\varepsilon}}{\varepsilon} - -1\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-309} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-309) (not (<= t_0 0.0)))
     t_0
     (* eps (* (* x 5.0) (pow x 3.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -1e-309) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * ((x * 5.0) * pow(x, 3.0));
	}
	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 <= (-1d-309)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = eps * ((x * 5.0d0) * (x ** 3.0d0))
    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 <= -1e-309) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * ((x * 5.0) * Math.pow(x, 3.0));
	}
	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 <= -1e-309) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = eps * ((x * 5.0) * math.pow(x, 3.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -1e-309) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(eps * Float64(Float64(x * 5.0) * (x ^ 3.0)));
	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 <= -1e-309) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = eps * ((x * 5.0) * (x ^ 3.0));
	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, -1e-309], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(eps * N[(N[(x * 5.0), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.000000000000002e-309 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.9%

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

   if -1.000000000000002e-309 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

   5. Simplified 99.9%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. swap-sqr 99.1%

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

      6. pow-prod-up 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. sqrt-prod 99.1%

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

      2. sqrt-pow1 99.9%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{{x}^{\left(\frac{8}{2}\right)}} \cdot \sqrt{25}\right) \]

      3. metadata-eval 99.9%

         \[\leadsto \varepsilon \cdot \left({x}^{\color{blue}{4}} \cdot \sqrt{25}\right) \]

      4. metadata-eval 99.9%

         \[\leadsto \varepsilon \cdot \left({x}^{\color{blue}{\left(3 + 1\right)}} \cdot \sqrt{25}\right) \]

      5. pow-plus 99.9%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left({x}^{3} \cdot x\right)} \cdot \sqrt{25}\right) \]

      6. metadata-eval 99.9%

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

      7. associate-*l* 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-309} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 2.8 \cdot 10^{-63}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3e-64) (not (<= eps 2.8e-63)))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
   (* eps (* (* x 5.0) (pow x 3.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3e-64) || !(eps <= 2.8e-63)) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * ((x * 5.0) * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3d-64)) .or. (.not. (eps <= 2.8d-63))) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = eps * ((x * 5.0d0) * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3e-64) || !(eps <= 2.8e-63)) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * ((x * 5.0) * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3e-64) or not (eps <= 2.8e-63):
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = eps * ((x * 5.0) * math.pow(x, 3.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3e-64) || !(eps <= 2.8e-63))
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64(eps * Float64(Float64(x * 5.0) * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3e-64) || ~((eps <= 2.8e-63)))
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = eps * ((x * 5.0) * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3e-64], N[Not[LessEqual[eps, 2.8e-63]], $MachinePrecision]], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(x * 5.0), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.0000000000000001e-64 or 2.8000000000000002e-63 < eps

   1. Initial program 94.1%

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

   3. Taylor expanded in eps around inf 91.9%

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

      1. distribute-lft1-in 91.9%

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

      2. metadata-eval 91.9%

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

   5. Simplified 91.9%

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

   if -3.0000000000000001e-64 < eps < 2.8000000000000002e-63

   1. Initial program 87.8%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-rgt1-in 99.8%

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

      3. metadata-eval 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*r* 99.8%

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

   5. Simplified 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.0%

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

      3. *-commutative 99.0%

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

      4. *-commutative 99.0%

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

      5. swap-sqr 99.0%

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

      6. pow-prod-up 99.0%

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

      7. metadata-eval 99.0%

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

      8. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. sqrt-prod 99.0%

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

      2. sqrt-pow1 99.8%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{{x}^{\left(\frac{8}{2}\right)}} \cdot \sqrt{25}\right) \]

      3. metadata-eval 99.8%

         \[\leadsto \varepsilon \cdot \left({x}^{\color{blue}{4}} \cdot \sqrt{25}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \varepsilon \cdot \left({x}^{\color{blue}{\left(3 + 1\right)}} \cdot \sqrt{25}\right) \]

      5. pow-plus 99.8%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left({x}^{3} \cdot x\right)} \cdot \sqrt{25}\right) \]

      6. metadata-eval 99.8%

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

      7. associate-*l* 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 2.8 \cdot 10^{-63}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon + x \cdot 5\\ \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{4} \cdot t\_0\\ \mathbf{elif}\;\varepsilon \leq 2.2 \cdot 10^{-63}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\varepsilon \cdot {\varepsilon}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ eps (* x 5.0))))
   (if (<= eps -4.5e-64)
     (* (pow eps 4.0) t_0)
     (if (<= eps 2.2e-63)
       (* eps (* (* x 5.0) (pow x 3.0)))
       (* t_0 (* eps (pow eps 3.0)))))))

C

double code(double x, double eps) {
	double t_0 = eps + (x * 5.0);
	double tmp;
	if (eps <= -4.5e-64) {
		tmp = pow(eps, 4.0) * t_0;
	} else if (eps <= 2.2e-63) {
		tmp = eps * ((x * 5.0) * pow(x, 3.0));
	} else {
		tmp = t_0 * (eps * pow(eps, 3.0));
	}
	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 = eps + (x * 5.0d0)
    if (eps <= (-4.5d-64)) then
        tmp = (eps ** 4.0d0) * t_0
    else if (eps <= 2.2d-63) then
        tmp = eps * ((x * 5.0d0) * (x ** 3.0d0))
    else
        tmp = t_0 * (eps * (eps ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps + (x * 5.0);
	double tmp;
	if (eps <= -4.5e-64) {
		tmp = Math.pow(eps, 4.0) * t_0;
	} else if (eps <= 2.2e-63) {
		tmp = eps * ((x * 5.0) * Math.pow(x, 3.0));
	} else {
		tmp = t_0 * (eps * Math.pow(eps, 3.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps + (x * 5.0)
	tmp = 0
	if eps <= -4.5e-64:
		tmp = math.pow(eps, 4.0) * t_0
	elif eps <= 2.2e-63:
		tmp = eps * ((x * 5.0) * math.pow(x, 3.0))
	else:
		tmp = t_0 * (eps * math.pow(eps, 3.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps + Float64(x * 5.0))
	tmp = 0.0
	if (eps <= -4.5e-64)
		tmp = Float64((eps ^ 4.0) * t_0);
	elseif (eps <= 2.2e-63)
		tmp = Float64(eps * Float64(Float64(x * 5.0) * (x ^ 3.0)));
	else
		tmp = Float64(t_0 * Float64(eps * (eps ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps + (x * 5.0);
	tmp = 0.0;
	if (eps <= -4.5e-64)
		tmp = (eps ^ 4.0) * t_0;
	elseif (eps <= 2.2e-63)
		tmp = eps * ((x * 5.0) * (x ^ 3.0));
	else
		tmp = t_0 * (eps * (eps ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.5e-64], N[(N[Power[eps, 4.0], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[eps, 2.2e-63], N[(eps * N[(N[(x * 5.0), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(eps * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.5000000000000001e-64

   1. Initial program 88.8%

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

   3. Taylor expanded in eps around inf 87.5%

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

      1. distribute-lft1-in 87.5%

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

      2. metadata-eval 87.5%

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

   5. Simplified 87.5%

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

   6. Taylor expanded in eps around 0 87.2%

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

   if -4.5000000000000001e-64 < eps < 2.2e-63

   1. Initial program 87.8%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-rgt1-in 99.8%

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

      3. metadata-eval 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*r* 99.8%

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

   5. Simplified 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.0%

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

      3. *-commutative 99.0%

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

      4. *-commutative 99.0%

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

      5. swap-sqr 99.0%

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

      6. pow-prod-up 99.0%

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

      7. metadata-eval 99.0%

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

      8. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. sqrt-prod 99.0%

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

      2. sqrt-pow1 99.8%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{{x}^{\left(\frac{8}{2}\right)}} \cdot \sqrt{25}\right) \]

      3. metadata-eval 99.8%

         \[\leadsto \varepsilon \cdot \left({x}^{\color{blue}{4}} \cdot \sqrt{25}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \varepsilon \cdot \left({x}^{\color{blue}{\left(3 + 1\right)}} \cdot \sqrt{25}\right) \]

      5. pow-plus 99.8%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left({x}^{3} \cdot x\right)} \cdot \sqrt{25}\right) \]

      6. metadata-eval 99.8%

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

      7. associate-*l* 99.8%

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

   9. Applied egg-rr 99.8%

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

   if 2.2e-63 < eps

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf 96.7%

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

      1. distribute-lft1-in 96.7%

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

      2. metadata-eval 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in eps around 0 96.5%

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

      1. metadata-eval 96.5%

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

      2. pow-prod-up 96.4%

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

      3. unpow2 96.4%

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

      4. associate-*r* 96.4%

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

      5. unpow2 96.4%

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

      6. pow3 96.6%

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

   8. Applied egg-rr 96.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{elif}\;\varepsilon \leq 2.2 \cdot 10^{-63}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + x \cdot 5\right) \cdot \left(\varepsilon \cdot {\varepsilon}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 2.2 \cdot 10^{-63}\right):\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.4e-64) (not (<= eps 2.2e-63)))
   (* (pow eps 4.0) (+ eps (* x 5.0)))
   (* eps (* (* x 5.0) (pow x 3.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.4e-64) || !(eps <= 2.2e-63)) {
		tmp = pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = eps * ((x * 5.0) * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.4d-64)) .or. (.not. (eps <= 2.2d-63))) then
        tmp = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    else
        tmp = eps * ((x * 5.0d0) * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.4e-64) || !(eps <= 2.2e-63)) {
		tmp = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = eps * ((x * 5.0) * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.4e-64) or not (eps <= 2.2e-63):
		tmp = math.pow(eps, 4.0) * (eps + (x * 5.0))
	else:
		tmp = eps * ((x * 5.0) * math.pow(x, 3.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.4e-64) || !(eps <= 2.2e-63))
		tmp = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)));
	else
		tmp = Float64(eps * Float64(Float64(x * 5.0) * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.4e-64) || ~((eps <= 2.2e-63)))
		tmp = (eps ^ 4.0) * (eps + (x * 5.0));
	else
		tmp = eps * ((x * 5.0) * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.4e-64], N[Not[LessEqual[eps, 2.2e-63]], $MachinePrecision]], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(x * 5.0), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.39999999999999998e-64 or 2.2e-63 < eps

   1. Initial program 94.1%

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

   3. Taylor expanded in eps around inf 91.9%

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

      1. distribute-lft1-in 91.9%

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

      2. metadata-eval 91.9%

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

   5. Simplified 91.9%

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

   6. Taylor expanded in eps around 0 91.7%

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

   if -2.39999999999999998e-64 < eps < 2.2e-63

   1. Initial program 87.8%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-rgt1-in 99.8%

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

      3. metadata-eval 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*r* 99.8%

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

   5. Simplified 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.0%

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

      3. *-commutative 99.0%

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

      4. *-commutative 99.0%

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

      5. swap-sqr 99.0%

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

      6. pow-prod-up 99.0%

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

      7. metadata-eval 99.0%

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

      8. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. sqrt-prod 99.0%

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

      2. sqrt-pow1 99.8%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{{x}^{\left(\frac{8}{2}\right)}} \cdot \sqrt{25}\right) \]

      3. metadata-eval 99.8%

         \[\leadsto \varepsilon \cdot \left({x}^{\color{blue}{4}} \cdot \sqrt{25}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \varepsilon \cdot \left({x}^{\color{blue}{\left(3 + 1\right)}} \cdot \sqrt{25}\right) \]

      5. pow-plus 99.8%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left({x}^{3} \cdot x\right)} \cdot \sqrt{25}\right) \]

      6. metadata-eval 99.8%

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

      7. associate-*l* 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 2.2 \cdot 10^{-63}\right):\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.4 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 2.2 \cdot 10^{-63}\right):\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -5.4e-64) (not (<= eps 2.2e-63)))
   (pow eps 5.0)
   (* eps (* (* x 5.0) (pow x 3.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.4e-64) || !(eps <= 2.2e-63)) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * ((x * 5.0) * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-5.4d-64)) .or. (.not. (eps <= 2.2d-63))) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * ((x * 5.0d0) * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.4e-64) || !(eps <= 2.2e-63)) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * ((x * 5.0) * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -5.4e-64) or not (eps <= 2.2e-63):
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * ((x * 5.0) * math.pow(x, 3.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -5.4e-64) || !(eps <= 2.2e-63))
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(Float64(x * 5.0) * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -5.4e-64) || ~((eps <= 2.2e-63)))
		tmp = eps ^ 5.0;
	else
		tmp = eps * ((x * 5.0) * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -5.4e-64], N[Not[LessEqual[eps, 2.2e-63]], $MachinePrecision]], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(N[(x * 5.0), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.39999999999999971e-64 or 2.2e-63 < eps

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0 89.5%

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

   if -5.39999999999999971e-64 < eps < 2.2e-63

   1. Initial program 87.8%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-rgt1-in 99.8%

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

      3. metadata-eval 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*r* 99.8%

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

   5. Simplified 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.0%

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

      3. *-commutative 99.0%

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

      4. *-commutative 99.0%

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

      5. swap-sqr 99.0%

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

      6. pow-prod-up 99.0%

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

      7. metadata-eval 99.0%

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

      8. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. sqrt-prod 99.0%

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

      2. sqrt-pow1 99.8%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{{x}^{\left(\frac{8}{2}\right)}} \cdot \sqrt{25}\right) \]

      3. metadata-eval 99.8%

         \[\leadsto \varepsilon \cdot \left({x}^{\color{blue}{4}} \cdot \sqrt{25}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \varepsilon \cdot \left({x}^{\color{blue}{\left(3 + 1\right)}} \cdot \sqrt{25}\right) \]

      5. pow-plus 99.8%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left({x}^{3} \cdot x\right)} \cdot \sqrt{25}\right) \]

      6. metadata-eval 99.8%

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

      7. associate-*l* 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.4 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 2.2 \cdot 10^{-63}\right):\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot 5\right) \cdot {x}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.35 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 4.4 \cdot 10^{-63}\right):\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.35e-64) (not (<= eps 4.4e-63)))
   (pow eps 5.0)
   (* (pow x 4.0) (* eps 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.35e-64) || !(eps <= 4.4e-63)) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = pow(x, 4.0) * (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 ((eps <= (-2.35d-64)) .or. (.not. (eps <= 4.4d-63))) then
        tmp = eps ** 5.0d0
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.35e-64) || !(eps <= 4.4e-63)) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.35e-64) or not (eps <= 4.4e-63):
		tmp = math.pow(eps, 5.0)
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.35e-64) || !(eps <= 4.4e-63))
		tmp = eps ^ 5.0;
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.35e-64) || ~((eps <= 4.4e-63)))
		tmp = eps ^ 5.0;
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.35e-64], N[Not[LessEqual[eps, 4.4e-63]], $MachinePrecision]], N[Power[eps, 5.0], $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.3499999999999999e-64 or 4.3999999999999999e-63 < eps

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0 89.5%

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

   if -2.3499999999999999e-64 < eps < 4.3999999999999999e-63

   1. Initial program 87.8%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. distribute-rgt1-in 99.8%

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

      2. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.35 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 4.4 \cdot 10^{-63}\right):\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-63}\right):\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.5e-64) (not (<= eps 5.5e-63)))
   (pow eps 5.0)
   (* 5.0 (* eps (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.5e-64) || !(eps <= 5.5e-63)) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * 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 ((eps <= (-3.5d-64)) .or. (.not. (eps <= 5.5d-63))) then
        tmp = eps ** 5.0d0
    else
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.5e-64) || !(eps <= 5.5e-63)) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.5e-64) or not (eps <= 5.5e-63):
		tmp = math.pow(eps, 5.0)
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.5e-64) || !(eps <= 5.5e-63))
		tmp = eps ^ 5.0;
	else
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.5e-64) || ~((eps <= 5.5e-63)))
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.5e-64], N[Not[LessEqual[eps, 5.5e-63]], $MachinePrecision]], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.5000000000000003e-64 or 5.50000000000000043e-63 < eps

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0 89.5%

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

   if -3.5000000000000003e-64 < eps < 5.50000000000000043e-63

   1. Initial program 87.8%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-rgt1-in 99.8%

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

      3. metadata-eval 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*r* 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in eps around 0 99.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-63}\right):\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\varepsilon}^{5} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return eps ^ 5.0
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Power[eps, 5.0], $MachinePrecision]

Derivation

1. Initial program 89.1%

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

3. Taylor expanded in x around 0 88.2%

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(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)))