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

Percentage Accurate: 88.3% → 98.8%

Time: 17.6s

Alternatives: 5

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: 88.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := 2 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-280} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot \left(4 \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(t\_1 + 4 \cdot {\varepsilon}^{2}\right)\right) + \left({x}^{3} \cdot \left(t\_1 + {\varepsilon}^{2} \cdot 8\right) + {x}^{4} \cdot \left(\varepsilon + \varepsilon \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))) (t_1 (* 2.0 (pow eps 2.0))))
   (if (or (<= t_0 -5e-280) (not (<= t_0 0.0)))
     t_0
     (+
      (*
       (pow x 2.0)
       (+ (* 4.0 (pow eps 3.0)) (* eps (+ t_1 (* 4.0 (pow eps 2.0))))))
      (+
       (* (pow x 3.0) (+ t_1 (* (pow eps 2.0) 8.0)))
       (* (pow x 4.0) (+ eps (* eps 4.0))))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = 2.0 * pow(eps, 2.0);
	double tmp;
	if ((t_0 <= -5e-280) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (pow(x, 2.0) * ((4.0 * pow(eps, 3.0)) + (eps * (t_1 + (4.0 * pow(eps, 2.0)))))) + ((pow(x, 3.0) * (t_1 + (pow(eps, 2.0) * 8.0))) + (pow(x, 4.0) * (eps + (eps * 4.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    t_1 = 2.0d0 * (eps ** 2.0d0)
    if ((t_0 <= (-5d-280)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = ((x ** 2.0d0) * ((4.0d0 * (eps ** 3.0d0)) + (eps * (t_1 + (4.0d0 * (eps ** 2.0d0)))))) + (((x ** 3.0d0) * (t_1 + ((eps ** 2.0d0) * 8.0d0))) + ((x ** 4.0d0) * (eps + (eps * 4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double t_1 = 2.0 * Math.pow(eps, 2.0);
	double tmp;
	if ((t_0 <= -5e-280) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (Math.pow(x, 2.0) * ((4.0 * Math.pow(eps, 3.0)) + (eps * (t_1 + (4.0 * Math.pow(eps, 2.0)))))) + ((Math.pow(x, 3.0) * (t_1 + (Math.pow(eps, 2.0) * 8.0))) + (Math.pow(x, 4.0) * (eps + (eps * 4.0))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	t_1 = 2.0 * math.pow(eps, 2.0)
	tmp = 0
	if (t_0 <= -5e-280) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (math.pow(x, 2.0) * ((4.0 * math.pow(eps, 3.0)) + (eps * (t_1 + (4.0 * math.pow(eps, 2.0)))))) + ((math.pow(x, 3.0) * (t_1 + (math.pow(eps, 2.0) * 8.0))) + (math.pow(x, 4.0) * (eps + (eps * 4.0))))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(2.0 * (eps ^ 2.0))
	tmp = 0.0
	if ((t_0 <= -5e-280) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64((x ^ 2.0) * Float64(Float64(4.0 * (eps ^ 3.0)) + Float64(eps * Float64(t_1 + Float64(4.0 * (eps ^ 2.0)))))) + Float64(Float64((x ^ 3.0) * Float64(t_1 + Float64((eps ^ 2.0) * 8.0))) + Float64((x ^ 4.0) * Float64(eps + Float64(eps * 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	t_1 = 2.0 * (eps ^ 2.0);
	tmp = 0.0;
	if ((t_0 <= -5e-280) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = ((x ^ 2.0) * ((4.0 * (eps ^ 3.0)) + (eps * (t_1 + (4.0 * (eps ^ 2.0)))))) + (((x ^ 3.0) * (t_1 + ((eps ^ 2.0) * 8.0))) + ((x ^ 4.0) * (eps + (eps * 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-280], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(4.0 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(t$95$1 + N[(4.0 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 3.0], $MachinePrecision] * N[(t$95$1 + N[(N[Power[eps, 2.0], $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps + N[(eps * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -5.00000000000000028e-280 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.0%

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

   if -5.00000000000000028e-280 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 85.3%

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

   3. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(4 \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right) + \left({x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) + {x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-280} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot \left(4 \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)\right) + \left({x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + {\varepsilon}^{2} \cdot 8\right) + {x}^{4} \cdot \left(\varepsilon + \varepsilon \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-284} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-284) (not (<= t_0 0.0)))
     t_0
     (* eps (* 5.0 (pow x 4.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-284) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * (5.0 * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-5d-284)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-284) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -5e-284) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-284) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -5e-284) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-284], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -4.99999999999999973e-284 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 97.6%

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

   if -4.99999999999999973e-284 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 85.4%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-284} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-56} \lor \neg \left(x \leq 3.2 \cdot 10^{-72}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5.2e-56) (not (<= x 3.2e-72)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5.2e-56) || !(x <= 3.2e-72)) {
		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 <= (-5.2d-56)) .or. (.not. (x <= 3.2d-72))) 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 <= -5.2e-56) || !(x <= 3.2e-72)) {
		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 <= -5.2e-56) or not (x <= 3.2e-72):
		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 <= -5.2e-56) || !(x <= 3.2e-72))
		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 <= -5.2e-56) || ~((x <= 3.2e-72)))
		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, -5.2e-56], N[Not[LessEqual[x, 3.2e-72]], $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 < -5.19999999999999994e-56 or 3.19999999999999999e-72 < x

   1. Initial program 46.6%

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

   3. Taylor expanded in x around inf 93.0%

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

      1. distribute-rgt1-in 93.0%

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

      2. metadata-eval 93.0%

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

   5. Simplified 93.0%

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

   6. Taylor expanded in x around 0 92.9%

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

   if -5.19999999999999994e-56 < x < 3.19999999999999999e-72

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-56} \lor \neg \left(x \leq 3.2 \cdot 10^{-72}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-56} \lor \neg \left(x \leq 2 \cdot 10^{-72}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.7e-56) (not (<= x 2e-72)))
   (* eps (* 5.0 (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.7e-56) || !(x <= 2e-72)) {
		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 <= (-1.7d-56)) .or. (.not. (x <= 2d-72))) 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 <= -1.7e-56) || !(x <= 2e-72)) {
		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 <= -1.7e-56) or not (x <= 2e-72):
		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 <= -1.7e-56) || !(x <= 2e-72))
		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 <= -1.7e-56) || ~((x <= 2e-72)))
		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, -1.7e-56], N[Not[LessEqual[x, 2e-72]], $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 < -1.69999999999999991e-56 or 1.9999999999999999e-72 < x

   1. Initial program 46.6%

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

   3. Taylor expanded in x around inf 93.0%

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

      1. *-commutative 93.0%

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

      2. distribute-rgt1-in 93.0%

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

      3. metadata-eval 93.0%

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

      4. *-commutative 93.0%

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

      5. associate-*r* 93.1%

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

   5. Simplified 93.1%

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

   if -1.69999999999999991e-56 < x < 1.9999999999999999e-72

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-56} \lor \neg \left(x \leq 2 \cdot 10^{-72}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.4% 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 87.7%

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

3. Taylor expanded in x around 0 86.1%

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

4. Final simplification 86.1%

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

Reproduce

herbie shell --seed 2024046 
(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)))