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

Percentage Accurate: 87.9% → 98.6%

Time: 9.3s

Alternatives: 5

Speedup: 1.9×

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: 87.9% 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: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-303)
     t_0
     (if (<= t_0 0.0)
       (* 5.0 (* eps (pow x 4.0)))
       (+ (* x (* 5.0 (pow eps 4.0))) (pow 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-303) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = (x * (5.0 * pow(eps, 4.0))) + 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) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-5d-303)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = (x * (5.0d0 * (eps ** 4.0d0))) + (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-303) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = (x * (5.0 * Math.pow(eps, 4.0))) + Math.pow(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-303:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = (x * (5.0 * math.pow(eps, 4.0))) + math.pow(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-303)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = Float64(Float64(x * Float64(5.0 * (eps ^ 4.0))) + (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-303)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = (x * (5.0 * (eps ^ 4.0))) + (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[LessEqual[t$95$0, -5e-303], t$95$0, If[LessEqual[t$95$0, 0.0], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -4.9999999999999998e-303

   1. Initial program 99.8%

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

   if -4.9999999999999998e-303 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 84.5%

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

   2. Taylor expanded in eps around 0 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in eps around 0 99.9%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in eps around inf 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-303}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(5 \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5}\\ \end{array} \]

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-303} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-303) (not (<= t_0 0.0)))
     t_0
     (* 5.0 (* eps (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-303) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = 5.0 * (eps * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-5d-303)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = 5.0d0 * (eps * (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-303) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = 5.0 * (eps * 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-303) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = 5.0 * (eps * 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-303) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(5.0 * Float64(eps * (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-303) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = 5.0 * (eps * (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-303], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(5.0 * N[(eps * 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)) < -4.9999999999999998e-303 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 99.9%

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

   if -4.9999999999999998e-303 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 84.5%

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

   2. Taylor expanded in eps around 0 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in eps around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-303} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 3: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-46} \lor \neg \left(x \leq 1.02 \cdot 10^{-72}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -9e-46) (not (<= x 1.02e-72)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -9e-46) || !(x <= 1.02e-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 <= (-9d-46)) .or. (.not. (x <= 1.02d-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 <= -9e-46) || !(x <= 1.02e-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 <= -9e-46) or not (x <= 1.02e-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 <= -9e-46) || !(x <= 1.02e-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 <= -9e-46) || ~((x <= 1.02e-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, -9e-46], N[Not[LessEqual[x, 1.02e-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 < -9.00000000000000001e-46 or 1.02e-72 < x

   1. Initial program 39.0%

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

   2. Taylor expanded in eps around 0 90.7%

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

      1. distribute-lft1-in 90.7%

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

      2. metadata-eval 90.7%

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

   4. Simplified 90.7%

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

   5. Taylor expanded in eps around 0 90.8%

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

   if -9.00000000000000001e-46 < x < 1.02e-72

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-46} \lor \neg \left(x \leq 1.02 \cdot 10^{-72}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 4: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-72}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.4e-45)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 1.02e-72) (pow eps 5.0) (* eps (* 5.0 (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.4e-45) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 1.02e-72) {
		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 <= (-1.4d-45)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 1.02d-72) 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 <= -1.4e-45) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 1.02e-72) {
		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 <= -1.4e-45:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 1.02e-72:
		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 <= -1.4e-45)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 1.02e-72)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.4e-45)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 1.02e-72)
		tmp = eps ^ 5.0;
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.4e-45], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.02e-72], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4000000000000001e-45

   1. Initial program 39.1%

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

   2. Taylor expanded in eps around 0 85.2%

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

      1. distribute-lft1-in 85.2%

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

      2. metadata-eval 85.2%

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

   4. Simplified 85.2%

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

   5. Taylor expanded in eps around 0 85.4%

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

   if -1.4000000000000001e-45 < x < 1.02e-72

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.8%

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

   if 1.02e-72 < x

   1. Initial program 38.9%

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

   2. Taylor expanded in eps around 0 96.4%

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

      1. distribute-lft1-in 96.4%

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

      2. metadata-eval 96.4%

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

   4. Simplified 96.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-72}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 5: 86.6% 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 87.8%

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

2. Taylor expanded in x around 0 87.2%

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

3. Final simplification 87.2%

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

Reproduce

herbie shell --seed 2023297 
(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)))