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

Percentage Accurate: 88.8% → 98.1%

Time: 8.1s

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: 88.8% 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: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-250} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-250) (not (<= t_0 0.0)))
     t_0
     (* eps (+ (* 5.0 (pow x 4.0)) (* eps (* (pow x 3.0) 10.0)))))))

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-250) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * ((5.0 * pow(x, 4.0)) + (eps * (pow(x, 3.0) * 10.0)));
	}
	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-250)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = eps * ((5.0d0 * (x ** 4.0d0)) + (eps * ((x ** 3.0d0) * 10.0d0)))
    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-250) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * ((5.0 * Math.pow(x, 4.0)) + (eps * (Math.pow(x, 3.0) * 10.0)));
	}
	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-250) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = eps * ((5.0 * math.pow(x, 4.0)) + (eps * (math.pow(x, 3.0) * 10.0)))
	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-250) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(eps * Float64((x ^ 3.0) * 10.0))));
	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-250) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = eps * ((5.0 * (x ^ 4.0)) + (eps * ((x ^ 3.0) * 10.0)));
	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-250], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[Power[x, 3.0], $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -5.00000000000000027e-250 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.3%

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

   if -5.00000000000000027e-250 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 86.2%

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

   2. Taylor expanded in eps around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*l* 99.9%

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

      4. distribute-lft-out 99.9%

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

      5. distribute-lft1-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. *-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. distribute-rgt-out 99.9%

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

      10. associate-*r* 99.9%

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

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-250} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)\\ \end{array} \]

Alternative 2: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-250} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-250) (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-250) || !(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-250)) .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-250) || !(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-250) 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-250) || !(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-250) || ~((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-250], 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)) < -5.00000000000000027e-250 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.3%

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

   if -5.00000000000000027e-250 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 86.2%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. distribute-lft1-in 99.8%

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

      2. metadata-eval 99.8%

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

      3. associate-*l* 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-250} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 3: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-41} \lor \neg \left(x \leq 2.1 \cdot 10^{-53}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.45e-41) (not (<= x 2.1e-53)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.45e-41) || !(x <= 2.1e-53)) {
		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 <= (-1.45d-41)) .or. (.not. (x <= 2.1d-53))) 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 <= -1.45e-41) || !(x <= 2.1e-53)) {
		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 <= -1.45e-41) or not (x <= 2.1e-53):
		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 <= -1.45e-41) || !(x <= 2.1e-53))
		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 <= -1.45e-41) || ~((x <= 2.1e-53)))
		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, -1.45e-41], N[Not[LessEqual[x, 2.1e-53]], $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 < -1.44999999999999989e-41 or 2.09999999999999977e-53 < x

   1. Initial program 28.3%

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

   2. Taylor expanded in x around inf 91.8%

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

      1. distribute-lft1-in 91.8%

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

      2. metadata-eval 91.8%

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

      3. associate-*l* 91.9%

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

   4. Simplified 91.9%

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

   if -1.44999999999999989e-41 < x < 2.09999999999999977e-53

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-41} \lor \neg \left(x \leq 2.1 \cdot 10^{-53}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-42}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-53}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.2e-42)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 2.1e-53) (pow eps 5.0) (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-42) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 2.1e-53) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.2d-42)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 2.1d-53) then
        tmp = eps ** 5.0d0
    else
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-42) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 2.1e-53) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.2e-42:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 2.1e-53:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.2e-42)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 2.1e-53)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.2e-42)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 2.1e-53)
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.2e-42], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.1e-53], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.2e-42

   1. Initial program 14.9%

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

   2. Taylor expanded in x around inf 95.4%

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

      1. distribute-lft1-in 95.4%

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

      2. metadata-eval 95.4%

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

      3. *-commutative 95.4%

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

   4. Simplified 95.4%

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

   5. Taylor expanded in eps around 0 95.3%

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

      1. associate-*r* 95.4%

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

      2. *-commutative 95.4%

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

      3. associate-*r* 95.5%

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

   7. Simplified 95.5%

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

   if -5.2e-42 < x < 2.09999999999999977e-53

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

   if 2.09999999999999977e-53 < x

   1. Initial program 47.3%

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

   2. Taylor expanded in x around inf 86.8%

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

      1. distribute-lft1-in 86.8%

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

      2. metadata-eval 86.8%

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

      3. associate-*l* 87.0%

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

   4. Simplified 87.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-42}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-53}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 5: 87.8% 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 88.1%

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

2. Taylor expanded in x around 0 87.3%

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

3. Final simplification 87.3%

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

Reproduce

herbie shell --seed 2023174 
(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)))