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

Percentage Accurate: 88.2% → 99.1%

Time: 9.3s

Alternatives: 10

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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-302} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-302) (not (<= t_0 0.0)))
     t_0
     (* (* x (pow x 3.0)) (* eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-302) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (x * pow(x, 3.0)) * (eps * 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-5d-302)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (x * (x ** 3.0d0)) * (eps * 5.0d0)
    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 <= -5e-302) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (x * Math.pow(x, 3.0)) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -5e-302) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (x * math.pow(x, 3.0)) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-302) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(x * (x ^ 3.0)) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -5e-302) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (x * (x ^ 3.0)) * (eps * 5.0);
	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, -5e-302], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(x * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0), $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))) < -5.00000000000000033e-302 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.9%

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

   if -5.00000000000000033e-302 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 89.6%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. distribute-rgt1-in 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

      1. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 98.8%

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

      3. pow-prod-up 98.8%

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

      4. metadata-eval 98.8%

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

   7. Applied egg-rr 98.8%

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

      1. sqrt-pow1 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. pow-prod-up 99.9%

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

      5. pow2 99.9%

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

      6. associate-*r* 100.0%

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

      7. pow2 100.0%

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

      8. pow3 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-302} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-48} \lor \neg \left(x \leq 8.2 \cdot 10^{-63}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(\left(x + \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot 10\right) + 5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -6.2e-48) (not (<= x 8.2e-63)))
   (* eps (+ (* (* (+ x eps) (* x x)) (* eps 10.0)) (* 5.0 (pow x 4.0))))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -6.2e-48) || !(x <= 8.2e-63)) {
		tmp = eps * ((((x + eps) * (x * x)) * (eps * 10.0)) + (5.0 * pow(x, 4.0)));
	} else {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-6.2d-48)) .or. (.not. (x <= 8.2d-63))) then
        tmp = eps * ((((x + eps) * (x * x)) * (eps * 10.0d0)) + (5.0d0 * (x ** 4.0d0)))
    else
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -6.2e-48) || !(x <= 8.2e-63)) {
		tmp = eps * ((((x + eps) * (x * x)) * (eps * 10.0)) + (5.0 * Math.pow(x, 4.0)));
	} else {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -6.2e-48) or not (x <= 8.2e-63):
		tmp = eps * ((((x + eps) * (x * x)) * (eps * 10.0)) + (5.0 * math.pow(x, 4.0)))
	else:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -6.2e-48) || !(x <= 8.2e-63))
		tmp = Float64(eps * Float64(Float64(Float64(Float64(x + eps) * Float64(x * x)) * Float64(eps * 10.0)) + Float64(5.0 * (x ^ 4.0))));
	else
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -6.2e-48) || ~((x <= 8.2e-63)))
		tmp = eps * ((((x + eps) * (x * x)) * (eps * 10.0)) + (5.0 * (x ^ 4.0)));
	else
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -6.2e-48], N[Not[LessEqual[x, 8.2e-63]], $MachinePrecision]], N[(eps * N[(N[(N[(N[(x + eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps * 10.0), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000033e-48 or 8.1999999999999995e-63 < x

   1. Initial program 49.8%

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

   3. Taylor expanded in x around -inf 97.3%

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

   5. Simplified 97.2%

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

   6. Taylor expanded in eps around 0 95.5%

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

      1. fma-define 95.4%

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

      2. distribute-lft-out 95.4%

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

      3. cube-mult 95.4%

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

      4. unpow2 95.4%

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

      5. distribute-rgt-out 95.4%

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

   8. Simplified 95.4%

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

      1. fma-undefine 95.5%

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

      2. +-commutative 95.5%

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

      3. associate-*r* 95.5%

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

      4. *-commutative 95.5%

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

      5. +-commutative 95.5%

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

   10. Applied egg-rr 95.5%

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

       1. unpow2 95.5%

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

   12. Applied egg-rr 95.5%

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

   if -6.20000000000000033e-48 < x < 8.1999999999999995e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 100.0%

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

      1. distribute-lft1-in 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-48} \lor \neg \left(x \leq 8.2 \cdot 10^{-63}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(\left(x + \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot 10\right) + 5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-49} \lor \neg \left(x \leq 8.2 \cdot 10^{-63}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -7e-49) (not (<= x 8.2e-63)))
   (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x)))))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -7e-49) || !(x <= 8.2e-63)) {
		tmp = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	} else {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-7d-49)) .or. (.not. (x <= 8.2d-63))) then
        tmp = (x ** 4.0d0) * (eps * (5.0d0 + (10.0d0 * (eps / x))))
    else
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -7e-49) || !(x <= 8.2e-63)) {
		tmp = Math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	} else {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -7e-49) or not (x <= 8.2e-63):
		tmp = math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))))
	else:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -7e-49) || !(x <= 8.2e-63))
		tmp = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	else
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -7e-49) || ~((x <= 8.2e-63)))
		tmp = (x ^ 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	else
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -7e-49], N[Not[LessEqual[x, 8.2e-63]], $MachinePrecision]], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.00000000000000012e-49 or 8.1999999999999995e-63 < x

   1. Initial program 49.8%

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

   3. Taylor expanded in x around -inf 94.0%

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

      1. +-commutative 94.0%

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

      2. associate-+r+ 94.0%

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

      3. mul-1-neg 94.0%

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

      4. unsub-neg 94.0%

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

      5. distribute-rgt1-in 94.0%

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

      6. metadata-eval 94.0%

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

      7. *-commutative 94.0%

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

   5. Simplified 94.0%

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

   6. Taylor expanded in eps around 0 94.0%

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

   if -7.00000000000000012e-49 < x < 8.1999999999999995e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 100.0%

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

      1. distribute-lft1-in 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-49} \lor \neg \left(x \leq 8.2 \cdot 10^{-63}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-49} \lor \neg \left(x \leq 8.2 \cdot 10^{-63}\right):\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -6e-49) (not (<= x 8.2e-63)))
   (* eps (* (pow x 3.0) (+ (* eps 10.0) (* x 5.0))))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -6e-49) || !(x <= 8.2e-63)) {
		tmp = eps * (pow(x, 3.0) * ((eps * 10.0) + (x * 5.0)));
	} else {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-6d-49)) .or. (.not. (x <= 8.2d-63))) then
        tmp = eps * ((x ** 3.0d0) * ((eps * 10.0d0) + (x * 5.0d0)))
    else
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -6e-49) || !(x <= 8.2e-63)) {
		tmp = eps * (Math.pow(x, 3.0) * ((eps * 10.0) + (x * 5.0)));
	} else {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -6e-49) or not (x <= 8.2e-63):
		tmp = eps * (math.pow(x, 3.0) * ((eps * 10.0) + (x * 5.0)))
	else:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -6e-49) || !(x <= 8.2e-63))
		tmp = Float64(eps * Float64((x ^ 3.0) * Float64(Float64(eps * 10.0) + Float64(x * 5.0))));
	else
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -6e-49) || ~((x <= 8.2e-63)))
		tmp = eps * ((x ^ 3.0) * ((eps * 10.0) + (x * 5.0)));
	else
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -6e-49], N[Not[LessEqual[x, 8.2e-63]], $MachinePrecision]], N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(eps * 10.0), $MachinePrecision] + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6e-49 or 8.1999999999999995e-63 < x

   1. Initial program 49.8%

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

   3. Taylor expanded in x around -inf 97.3%

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

   5. Simplified 97.2%

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

   6. Taylor expanded in eps around 0 95.5%

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

      1. fma-define 95.4%

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

      2. distribute-lft-out 95.4%

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

      3. cube-mult 95.4%

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

      4. unpow2 95.4%

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

      5. distribute-rgt-out 95.4%

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

   8. Simplified 95.4%

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

      1. fma-undefine 95.5%

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

      2. +-commutative 95.5%

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

      3. associate-*r* 95.5%

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

      4. *-commutative 95.5%

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

      5. +-commutative 95.5%

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

   10. Applied egg-rr 95.5%

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

   11. Taylor expanded in eps around 0 93.8%

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

       1. *-commutative 93.8%

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

       2. metadata-eval 93.8%

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

       3. pow-plus 93.9%

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

       4. associate-*l* 93.8%

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

       5. *-commutative 93.8%

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

       6. associate-*r* 93.8%

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

       7. *-commutative 93.8%

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

       8. distribute-lft-out 93.8%

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

       9. *-commutative 93.8%

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

   13. Simplified 93.8%

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

   if -6e-49 < x < 8.1999999999999995e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 100.0%

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

      1. distribute-lft1-in 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-49} \lor \neg \left(x \leq 8.2 \cdot 10^{-63}\right):\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-63}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-48)
   (* eps (* (pow x 4.0) (+ 5.0 (* 10.0 (/ eps x)))))
   (if (<= x 8.2e-63)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* eps (* (pow x 3.0) (+ (* eps 10.0) (* x 5.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-48) {
		tmp = eps * (pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	} else if (x <= 8.2e-63) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * (pow(x, 3.0) * ((eps * 10.0) + (x * 5.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.2d-48)) then
        tmp = eps * ((x ** 4.0d0) * (5.0d0 + (10.0d0 * (eps / x))))
    else if (x <= 8.2d-63) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = eps * ((x ** 3.0d0) * ((eps * 10.0d0) + (x * 5.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-48) {
		tmp = eps * (Math.pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	} else if (x <= 8.2e-63) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * (Math.pow(x, 3.0) * ((eps * 10.0) + (x * 5.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.2e-48:
		tmp = eps * (math.pow(x, 4.0) * (5.0 + (10.0 * (eps / x))))
	elif x <= 8.2e-63:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = eps * (math.pow(x, 3.0) * ((eps * 10.0) + (x * 5.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-48)
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	elseif (x <= 8.2e-63)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64(eps * Float64((x ^ 3.0) * Float64(Float64(eps * 10.0) + Float64(x * 5.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.2e-48)
		tmp = eps * ((x ^ 4.0) * (5.0 + (10.0 * (eps / x))));
	elseif (x <= 8.2e-63)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = eps * ((x ^ 3.0) * ((eps * 10.0) + (x * 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-48], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e-63], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(eps * 10.0), $MachinePrecision] + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1999999999999998e-48

   1. Initial program 52.7%

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

   3. Taylor expanded in x around -inf 94.3%

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

   5. Simplified 94.2%

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

   6. Taylor expanded in eps around 0 92.2%

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

      1. fma-define 92.1%

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

      2. distribute-lft-out 92.1%

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

      3. cube-mult 92.1%

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

      4. unpow2 92.1%

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

      5. distribute-rgt-out 92.1%

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

   8. Simplified 92.1%

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

   9. Taylor expanded in x around inf 90.3%

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

   if -3.1999999999999998e-48 < x < 8.1999999999999995e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 100.0%

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

      1. distribute-lft1-in 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   if 8.1999999999999995e-63 < x

   1. Initial program 48.0%

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

   3. Taylor expanded in x around -inf 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in eps around 0 97.3%

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

      1. fma-define 97.3%

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

      2. distribute-lft-out 97.3%

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

      3. cube-mult 97.3%

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

      4. unpow2 97.3%

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

      5. distribute-rgt-out 97.3%

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

   8. Simplified 97.3%

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

      1. fma-undefine 97.3%

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

      2. +-commutative 97.3%

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

      3. associate-*r* 97.3%

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

      4. *-commutative 97.3%

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

      5. +-commutative 97.3%

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

   10. Applied egg-rr 97.3%

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

   11. Taylor expanded in eps around 0 95.9%

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

       1. *-commutative 95.9%

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

       2. metadata-eval 95.9%

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

       3. pow-plus 96.0%

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

       4. associate-*l* 96.0%

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

       5. *-commutative 96.0%

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

       6. associate-*r* 96.0%

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

       7. *-commutative 96.0%

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

       8. distribute-lft-out 96.0%

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

       9. *-commutative 96.0%

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

   13. Simplified 96.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-63}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-48}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-63}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.4e-48)
   (* (* eps 5.0) (pow x 4.0))
   (if (<= x 8.2e-63)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* (* x (pow x 3.0)) (* eps 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.4e-48) {
		tmp = (eps * 5.0) * pow(x, 4.0);
	} else if (x <= 8.2e-63) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = (x * pow(x, 3.0)) * (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 (x <= (-4.4d-48)) then
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    else if (x <= 8.2d-63) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = (x * (x ** 3.0d0)) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -4.4e-48) {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	} else if (x <= 8.2e-63) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = (x * Math.pow(x, 3.0)) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4.4e-48:
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	elif x <= 8.2e-63:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = (x * math.pow(x, 3.0)) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.4e-48)
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	elseif (x <= 8.2e-63)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64(Float64(x * (x ^ 3.0)) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4.4e-48)
		tmp = (eps * 5.0) * (x ^ 4.0);
	elseif (x <= 8.2e-63)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = (x * (x ^ 3.0)) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.4e-48], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e-63], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.40000000000000025e-48

   1. Initial program 52.7%

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

   3. Taylor expanded in x around inf 87.0%

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

      1. distribute-rgt1-in 87.0%

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

      2. metadata-eval 87.0%

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

   5. Simplified 87.0%

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

   if -4.40000000000000025e-48 < x < 8.1999999999999995e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 100.0%

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

      1. distribute-lft1-in 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   if 8.1999999999999995e-63 < x

   1. Initial program 48.0%

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

   3. Taylor expanded in x around inf 94.7%

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

      1. distribute-rgt1-in 94.7%

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

      2. metadata-eval 94.7%

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

   5. Simplified 94.7%

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

      1. add-sqr-sqrt 94.5%

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

      2. sqrt-unprod 88.7%

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

      3. pow-prod-up 88.7%

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

      4. metadata-eval 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. sqrt-pow1 94.7%

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

      2. metadata-eval 94.7%

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

      3. metadata-eval 94.7%

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

      4. pow-prod-up 94.6%

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

      5. pow2 94.6%

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

      6. associate-*r* 94.7%

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

      7. pow2 94.7%

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

      8. pow3 94.7%

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

   9. Applied egg-rr 94.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-48}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-63}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-48}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-63}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.05e-48)
   (* (* eps 5.0) (pow x 4.0))
   (if (<= x 8.2e-63) (pow eps 5.0) (* (* x (pow x 3.0)) (* eps 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.05e-48) {
		tmp = (eps * 5.0) * pow(x, 4.0);
	} else if (x <= 8.2e-63) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (x * pow(x, 3.0)) * (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 (x <= (-1.05d-48)) then
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    else if (x <= 8.2d-63) then
        tmp = eps ** 5.0d0
    else
        tmp = (x * (x ** 3.0d0)) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.05e-48) {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	} else if (x <= 8.2e-63) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = (x * Math.pow(x, 3.0)) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.05e-48:
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	elif x <= 8.2e-63:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = (x * math.pow(x, 3.0)) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.05e-48)
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	elseif (x <= 8.2e-63)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(x * (x ^ 3.0)) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.05e-48)
		tmp = (eps * 5.0) * (x ^ 4.0);
	elseif (x <= 8.2e-63)
		tmp = eps ^ 5.0;
	else
		tmp = (x * (x ^ 3.0)) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.05e-48], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e-63], N[Power[eps, 5.0], $MachinePrecision], N[(N[(x * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.04999999999999994e-48

   1. Initial program 52.7%

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

   3. Taylor expanded in x around inf 87.0%

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

      1. distribute-rgt1-in 87.0%

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

      2. metadata-eval 87.0%

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

   5. Simplified 87.0%

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

   if -1.04999999999999994e-48 < x < 8.1999999999999995e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.9%

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

   if 8.1999999999999995e-63 < x

   1. Initial program 48.0%

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

   3. Taylor expanded in x around inf 94.7%

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

      1. distribute-rgt1-in 94.7%

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

      2. metadata-eval 94.7%

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

   5. Simplified 94.7%

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

      1. add-sqr-sqrt 94.5%

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

      2. sqrt-unprod 88.7%

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

      3. pow-prod-up 88.7%

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

      4. metadata-eval 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. sqrt-pow1 94.7%

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

      2. metadata-eval 94.7%

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

      3. metadata-eval 94.7%

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

      4. pow-prod-up 94.6%

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

      5. pow2 94.6%

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

      6. associate-*r* 94.7%

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

      7. pow2 94.7%

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

      8. pow3 94.7%

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

   9. Applied egg-rr 94.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-48}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-63}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-48} \lor \neg \left(x \leq 8.2 \cdot 10^{-63}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.2e-48) (not (<= x 8.2e-63)))
   (* (* eps 5.0) (pow x 4.0))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.2e-48) || !(x <= 8.2e-63)) {
		tmp = (eps * 5.0) * pow(x, 4.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-1.2d-48)) .or. (.not. (x <= 8.2d-63))) then
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -1.2e-48) || !(x <= 8.2e-63)) {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.2e-48) or not (x <= 8.2e-63):
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.2e-48) || !(x <= 8.2e-63))
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.2e-48) || ~((x <= 8.2e-63)))
		tmp = (eps * 5.0) * (x ^ 4.0);
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.2e-48], N[Not[LessEqual[x, 8.2e-63]], $MachinePrecision]], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e-48 or 8.1999999999999995e-63 < x

   1. Initial program 49.8%

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

   3. Taylor expanded in x around inf 91.8%

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

      1. distribute-rgt1-in 91.8%

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

      2. metadata-eval 91.8%

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

   5. Simplified 91.8%

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

   if -1.2e-48 < x < 8.1999999999999995e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-48} \lor \neg \left(x \leq 8.2 \cdot 10^{-63}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-49} \lor \neg \left(x \leq 8.2 \cdot 10^{-63}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -7e-49) (not (<= x 8.2e-63)))
   (* eps (* 5.0 (* (* x x) (* x x))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -7e-49) || !(x <= 8.2e-63)) {
		tmp = eps * (5.0 * ((x * x) * (x * x)));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-7d-49)) .or. (.not. (x <= 8.2d-63))) then
        tmp = eps * (5.0d0 * ((x * x) * (x * x)))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -7e-49) || !(x <= 8.2e-63)) {
		tmp = eps * (5.0 * ((x * x) * (x * x)));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -7e-49) or not (x <= 8.2e-63):
		tmp = eps * (5.0 * ((x * x) * (x * x)))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -7e-49) || !(x <= 8.2e-63))
		tmp = Float64(eps * Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -7e-49) || ~((x <= 8.2e-63)))
		tmp = eps * (5.0 * ((x * x) * (x * x)));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -7e-49], N[Not[LessEqual[x, 8.2e-63]], $MachinePrecision]], N[(eps * N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.00000000000000012e-49 or 8.1999999999999995e-63 < x

   1. Initial program 49.8%

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

   3. Taylor expanded in x around inf 91.8%

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

      1. *-commutative 91.8%

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

      2. distribute-rgt1-in 91.8%

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

      3. metadata-eval 91.8%

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

      4. *-commutative 91.8%

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

      5. associate-*r* 91.6%

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

   5. Simplified 91.6%

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

      1. metadata-eval 91.6%

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

      2. pow-prod-up 91.7%

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

   7. Applied egg-rr 91.7%

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

      1. unpow2 95.5%

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

   9. Applied egg-rr 91.7%

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

       1. unpow2 95.5%

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

   11. Applied egg-rr 91.7%

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

   if -7.00000000000000012e-49 < x < 8.1999999999999995e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-49} \lor \neg \left(x \leq 8.2 \cdot 10^{-63}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.3% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

3. Taylor expanded in x around inf 76.6%

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

   1. *-commutative 76.6%

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

   2. distribute-rgt1-in 76.6%

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

   3. metadata-eval 76.6%

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

   4. *-commutative 76.6%

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

   5. associate-*r* 76.5%

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

5. Simplified 76.5%

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

   1. metadata-eval 76.5%

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

   2. pow-prod-up 76.6%

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

7. Applied egg-rr 76.6%

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

   1. unpow2 77.2%

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

9. Applied egg-rr 76.6%

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

    1. unpow2 77.2%

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

11. Applied egg-rr 76.6%

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

Reproduce

herbie shell --seed 2024177 
(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)))