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

Percentage Accurate: 88.2% → 99.3%

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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-308} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-308) (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 <= -1e-308) || !(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 <= (-1d-308)) .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 <= -1e-308) || !(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 <= -1e-308) 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 <= -1e-308) || !(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 <= -1e-308) || ~((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, -1e-308], 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))) < -9.9999999999999991e-309 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.4%

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

   if -9.9999999999999991e-309 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 84.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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. distribute-rgt1-in 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

      1. metadata-eval 99.9%

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

      2. pow-prod-up 99.8%

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

      3. unpow2 99.8%

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

      4. associate-*r* 99.9%

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

      5. unpow2 99.9%

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

      6. unpow3 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-308} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-48} \lor \neg \left(x \leq 7.6 \cdot 10^{-55}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \left(x \cdot x\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5e-48) (not (<= x 7.6e-55)))
   (* eps (+ (* 5.0 (pow x 4.0)) (* (* x x) (* 10.0 (* eps (+ x eps))))))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-48) || !(x <= 7.6e-55)) {
		tmp = eps * ((5.0 * pow(x, 4.0)) + ((x * x) * (10.0 * (eps * (x + eps)))));
	} 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 <= (-5d-48)) .or. (.not. (x <= 7.6d-55))) then
        tmp = eps * ((5.0d0 * (x ** 4.0d0)) + ((x * x) * (10.0d0 * (eps * (x + eps)))))
    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 <= -5e-48) || !(x <= 7.6e-55)) {
		tmp = eps * ((5.0 * Math.pow(x, 4.0)) + ((x * x) * (10.0 * (eps * (x + eps)))));
	} else {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -5e-48) or not (x <= 7.6e-55):
		tmp = eps * ((5.0 * math.pow(x, 4.0)) + ((x * x) * (10.0 * (eps * (x + eps)))))
	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 <= -5e-48) || !(x <= 7.6e-55))
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(Float64(x * x) * Float64(10.0 * Float64(eps * Float64(x + eps))))));
	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 <= -5e-48) || ~((x <= 7.6e-55)))
		tmp = eps * ((5.0 * (x ^ 4.0)) + ((x * x) * (10.0 * (eps * (x + eps)))));
	else
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5e-48], N[Not[LessEqual[x, 7.6e-55]], $MachinePrecision]], N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(10.0 * N[(eps * N[(x + eps), $MachinePrecision]), $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 < -4.9999999999999999e-48 or 7.5999999999999993e-55 < x

   1. Initial program 43.7%

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

   3. Taylor expanded in eps around 0 95.3%

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

      1. +-commutative 95.3%

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

      2. associate-+r+ 95.3%

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

      3. distribute-lft1-in 95.3%

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

      4. metadata-eval 95.3%

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

      5. +-commutative 95.3%

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

   5. Simplified 95.3%

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

   6. Taylor expanded in x around 0 95.3%

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

      1. distribute-lft-out 95.3%

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

      2. unpow2 95.3%

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

      3. distribute-lft-out 95.3%

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

      4. +-commutative 95.3%

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

   8. Simplified 95.3%

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

      1. unpow2 95.3%

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

   10. Applied egg-rr 95.3%

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

   if -4.9999999999999999e-48 < x < 7.5999999999999993e-55

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 99.7%

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

      1. distribute-lft1-in 99.7%

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

      2. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-48} \lor \neg \left(x \leq 7.6 \cdot 10^{-55}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \left(x \cdot x\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-53}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.5e-43)
   (* eps (* (pow x 4.0) (+ 5.0 (* 10.0 (/ eps x)))))
   (if (<= x 5.8e-53)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* (* eps 5.0) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.5e-43) {
		tmp = eps * (pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	} else if (x <= 5.8e-53) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = (eps * 5.0) * pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-4.5d-43)) then
        tmp = eps * ((x ** 4.0d0) * (5.0d0 + (10.0d0 * (eps / x))))
    else if (x <= 5.8d-53) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -4.5e-43) {
		tmp = eps * (Math.pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	} else if (x <= 5.8e-53) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4.5e-43:
		tmp = eps * (math.pow(x, 4.0) * (5.0 + (10.0 * (eps / x))))
	elif x <= 5.8e-53:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.5e-43)
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	elseif (x <= 5.8e-53)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4.5e-43)
		tmp = eps * ((x ^ 4.0) * (5.0 + (10.0 * (eps / x))));
	elseif (x <= 5.8e-53)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = (eps * 5.0) * (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.5e-43], 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, 5.8e-53], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.50000000000000025e-43

   1. Initial program 32.3%

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

   3. Taylor expanded in eps around 0 96.7%

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

      1. +-commutative 96.7%

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

      2. associate-+r+ 96.7%

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

      3. distribute-lft1-in 96.7%

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

      4. metadata-eval 96.7%

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

      5. +-commutative 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in x around inf 95.8%

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

   if -4.50000000000000025e-43 < x < 5.7999999999999996e-53

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 99.2%

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

      1. distribute-lft1-in 99.2%

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

      2. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

   if 5.7999999999999996e-53 < x

   1. Initial program 55.1%

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

   3. Taylor expanded in x around inf 96.4%

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

      1. distribute-rgt1-in 96.4%

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

      2. metadata-eval 96.4%

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

   5. Simplified 96.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-43}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-53}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-48}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-54}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.3e-48)
   (* (* x (pow x 3.0)) (* eps 5.0))
   (if (<= x 1.8e-54)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* (* eps 5.0) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.3e-48) {
		tmp = (x * pow(x, 3.0)) * (eps * 5.0);
	} else if (x <= 1.8e-54) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = (eps * 5.0) * pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.3d-48)) then
        tmp = (x * (x ** 3.0d0)) * (eps * 5.0d0)
    else if (x <= 1.8d-54) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.3e-48) {
		tmp = (x * Math.pow(x, 3.0)) * (eps * 5.0);
	} else if (x <= 1.8e-54) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.3e-48:
		tmp = (x * math.pow(x, 3.0)) * (eps * 5.0)
	elif x <= 1.8e-54:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.3e-48)
		tmp = Float64(Float64(x * (x ^ 3.0)) * Float64(eps * 5.0));
	elseif (x <= 1.8e-54)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.3e-48)
		tmp = (x * (x ^ 3.0)) * (eps * 5.0);
	elseif (x <= 1.8e-54)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = (eps * 5.0) * (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.3e-48], N[(N[(x * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e-54], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3000000000000001e-48

   1. Initial program 34.4%

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

   3. Taylor expanded in x around inf 92.5%

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

      1. distribute-rgt1-in 92.5%

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

      2. metadata-eval 92.5%

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

   5. Simplified 92.5%

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

      1. metadata-eval 92.5%

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

      2. pow-prod-up 92.1%

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

      3. unpow2 92.1%

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

      4. associate-*r* 92.5%

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

      5. unpow2 92.5%

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

      6. unpow3 92.6%

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

   7. Applied egg-rr 92.6%

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

   if -2.3000000000000001e-48 < x < 1.79999999999999988e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 99.7%

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

      1. distribute-lft1-in 99.7%

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

      2. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

   if 1.79999999999999988e-54 < x

   1. Initial program 55.1%

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

   3. Taylor expanded in x around inf 96.4%

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

      1. distribute-rgt1-in 96.4%

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

      2. metadata-eval 96.4%

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

   5. Simplified 96.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-48}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-54}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-48}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-54}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.1e-48)
   (* (* x (pow x 3.0)) (* eps 5.0))
   (if (<= x 1.25e-54)
     (* (pow eps 4.0) (+ eps (* x 5.0)))
     (* (* eps 5.0) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.1e-48) {
		tmp = (x * pow(x, 3.0)) * (eps * 5.0);
	} else if (x <= 1.25e-54) {
		tmp = pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = (eps * 5.0) * pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.1d-48)) then
        tmp = (x * (x ** 3.0d0)) * (eps * 5.0d0)
    else if (x <= 1.25d-54) then
        tmp = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    else
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.1e-48) {
		tmp = (x * Math.pow(x, 3.0)) * (eps * 5.0);
	} else if (x <= 1.25e-54) {
		tmp = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.1e-48:
		tmp = (x * math.pow(x, 3.0)) * (eps * 5.0)
	elif x <= 1.25e-54:
		tmp = math.pow(eps, 4.0) * (eps + (x * 5.0))
	else:
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.1e-48)
		tmp = Float64(Float64(x * (x ^ 3.0)) * Float64(eps * 5.0));
	elseif (x <= 1.25e-54)
		tmp = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)));
	else
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.1e-48)
		tmp = (x * (x ^ 3.0)) * (eps * 5.0);
	elseif (x <= 1.25e-54)
		tmp = (eps ^ 4.0) * (eps + (x * 5.0));
	else
		tmp = (eps * 5.0) * (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.1e-48], N[(N[(x * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-54], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.09999999999999989e-48

   1. Initial program 34.4%

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

   3. Taylor expanded in x around inf 92.5%

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

      1. distribute-rgt1-in 92.5%

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

      2. metadata-eval 92.5%

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

   5. Simplified 92.5%

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

      1. metadata-eval 92.5%

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

      2. pow-prod-up 92.1%

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

      3. unpow2 92.1%

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

      4. associate-*r* 92.5%

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

      5. unpow2 92.5%

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

      6. unpow3 92.6%

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

   7. Applied egg-rr 92.6%

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

   if -2.09999999999999989e-48 < x < 1.25000000000000004e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. fma-define 99.7%

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

      2. distribute-lft1-in 99.7%

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

      3. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in eps around 0 99.6%

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

   if 1.25000000000000004e-54 < x

   1. Initial program 55.1%

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

   3. Taylor expanded in x around inf 96.4%

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

      1. distribute-rgt1-in 96.4%

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

      2. metadata-eval 96.4%

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

   5. Simplified 96.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-48}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-54}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-46} \lor \neg \left(x \leq 3.7 \cdot 10^{-54}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5.8e-46) (not (<= x 3.7e-54)))
   (* (* eps 5.0) (pow x 4.0))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5.8e-46) || !(x <= 3.7e-54)) {
		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 <= (-5.8d-46)) .or. (.not. (x <= 3.7d-54))) 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 <= -5.8e-46) || !(x <= 3.7e-54)) {
		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 <= -5.8e-46) or not (x <= 3.7e-54):
		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 <= -5.8e-46) || !(x <= 3.7e-54))
		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 <= -5.8e-46) || ~((x <= 3.7e-54)))
		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, -5.8e-46], N[Not[LessEqual[x, 3.7e-54]], $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 < -5.80000000000000009e-46 or 3.7000000000000003e-54 < x

   1. Initial program 43.7%

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

   3. Taylor expanded in x around inf 94.2%

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

      1. distribute-rgt1-in 94.2%

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

      2. metadata-eval 94.2%

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

   5. Simplified 94.2%

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

   if -5.80000000000000009e-46 < x < 3.7000000000000003e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-46} \lor \neg \left(x \leq 3.7 \cdot 10^{-54}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-48} \lor \neg \left(x \leq 1.9 \cdot 10^{-54}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.1e-48) (not (<= x 1.9e-54)))
   (* eps (* 5.0 (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.1e-48) || !(x <= 1.9e-54)) {
		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 <= (-2.1d-48)) .or. (.not. (x <= 1.9d-54))) 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 <= -2.1e-48) || !(x <= 1.9e-54)) {
		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 <= -2.1e-48) or not (x <= 1.9e-54):
		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 <= -2.1e-48) || !(x <= 1.9e-54))
		tmp = Float64(eps * Float64(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 <= -2.1e-48) || ~((x <= 1.9e-54)))
		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, -2.1e-48], N[Not[LessEqual[x, 1.9e-54]], $MachinePrecision]], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999989e-48 or 1.9000000000000001e-54 < x

   1. Initial program 43.7%

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

   3. Taylor expanded in x around inf 94.2%

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

      1. *-commutative 94.2%

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

      2. distribute-rgt1-in 94.2%

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

      3. metadata-eval 94.2%

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

      4. *-commutative 94.2%

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

      5. associate-*r* 94.2%

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

   5. Simplified 94.2%

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

   if -2.09999999999999989e-48 < x < 1.9000000000000001e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-48} \lor \neg \left(x \leq 1.9 \cdot 10^{-54}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-48} \lor \neg \left(x \leq 1.75 \cdot 10^{-54}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.3e-48) (not (<= x 1.75e-54)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.3e-48) || !(x <= 1.75e-54)) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.3d-48)) .or. (.not. (x <= 1.75d-54))) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.3e-48) || !(x <= 1.75e-54)) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.3e-48) or not (x <= 1.75e-54):
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.3e-48) || !(x <= 1.75e-54))
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.3e-48) || ~((x <= 1.75e-54)))
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.3e-48], N[Not[LessEqual[x, 1.75e-54]], $MachinePrecision]], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3000000000000001e-48 or 1.74999999999999991e-54 < x

   1. Initial program 43.7%

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

   3. Taylor expanded in x around inf 94.2%

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

      1. distribute-rgt1-in 94.2%

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

      2. metadata-eval 94.2%

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

   5. Simplified 94.2%

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

   6. Taylor expanded in x around 0 94.2%

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

   if -2.3000000000000001e-48 < x < 1.74999999999999991e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-48} \lor \neg \left(x \leq 1.75 \cdot 10^{-54}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-48}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-53}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.2e-48)
   (* (* x (pow x 3.0)) (* eps 5.0))
   (if (<= x 1.4e-53) (pow eps 5.0) (* (* eps 5.0) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.2e-48) {
		tmp = (x * pow(x, 3.0)) * (eps * 5.0);
	} else if (x <= 1.4e-53) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (eps * 5.0) * pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-4.2d-48)) then
        tmp = (x * (x ** 3.0d0)) * (eps * 5.0d0)
    else if (x <= 1.4d-53) then
        tmp = eps ** 5.0d0
    else
        tmp = (eps * 5.0d0) * (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -4.2e-48) {
		tmp = (x * Math.pow(x, 3.0)) * (eps * 5.0);
	} else if (x <= 1.4e-53) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = (eps * 5.0) * Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4.2e-48:
		tmp = (x * math.pow(x, 3.0)) * (eps * 5.0)
	elif x <= 1.4e-53:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = (eps * 5.0) * math.pow(x, 4.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.2e-48)
		tmp = Float64(Float64(x * (x ^ 3.0)) * Float64(eps * 5.0));
	elseif (x <= 1.4e-53)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(eps * 5.0) * (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4.2e-48)
		tmp = (x * (x ^ 3.0)) * (eps * 5.0);
	elseif (x <= 1.4e-53)
		tmp = eps ^ 5.0;
	else
		tmp = (eps * 5.0) * (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.2e-48], N[(N[(x * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-53], N[Power[eps, 5.0], $MachinePrecision], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.19999999999999977e-48

   1. Initial program 34.4%

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

   3. Taylor expanded in x around inf 92.5%

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

      1. distribute-rgt1-in 92.5%

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

      2. metadata-eval 92.5%

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

   5. Simplified 92.5%

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

      1. metadata-eval 92.5%

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

      2. pow-prod-up 92.1%

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

      3. unpow2 92.1%

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

      4. associate-*r* 92.5%

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

      5. unpow2 92.5%

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

      6. unpow3 92.6%

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

   7. Applied egg-rr 92.6%

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

   if -4.19999999999999977e-48 < x < 1.39999999999999993e-53

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.4%

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

   if 1.39999999999999993e-53 < x

   1. Initial program 55.1%

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

   3. Taylor expanded in x around inf 96.4%

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

      1. distribute-rgt1-in 96.4%

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

      2. metadata-eval 96.4%

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

   5. Simplified 96.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-48}:\\ \;\;\;\;\left(x \cdot {x}^{3}\right) \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-53}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.3% 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 87.2%

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

3. Taylor expanded in x around 0 85.7%

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

Reproduce

herbie shell --seed 2024152 
(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)))