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

Percentage Accurate: 87.8% → 87.8%

Time: 1.6s

Alternatives: 5

Speedup: 1.0×

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: 87.8% 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: 87.8% 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]

Derivation

1. Initial program 86.8%

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

Alternative 2: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ \mathbf{if}\;\varepsilon \leq -8.8 \cdot 10^{-94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\varepsilon \leq 1.25 \cdot 10^{-81}:\\ \;\;\;\;{t\_0}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0)))
   (if (<= eps -8.8e-94) t_0 (if (<= eps 1.25e-81) (pow t_0 5.0) t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double tmp;
	if (eps <= -8.8e-94) {
		tmp = t_0;
	} else if (eps <= 1.25e-81) {
		tmp = pow(t_0, 5.0);
	} else {
		tmp = t_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
    if (eps <= (-8.8d-94)) then
        tmp = t_0
    else if (eps <= 1.25d-81) then
        tmp = t_0 ** 5.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double tmp;
	if (eps <= -8.8e-94) {
		tmp = t_0;
	} else if (eps <= 1.25e-81) {
		tmp = Math.pow(t_0, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	tmp = 0
	if eps <= -8.8e-94:
		tmp = t_0
	elif eps <= 1.25e-81:
		tmp = math.pow(t_0, 5.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	tmp = 0.0
	if (eps <= -8.8e-94)
		tmp = t_0;
	elseif (eps <= 1.25e-81)
		tmp = t_0 ^ 5.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	tmp = 0.0;
	if (eps <= -8.8e-94)
		tmp = t_0;
	elseif (eps <= 1.25e-81)
		tmp = t_0 ^ 5.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, If[LessEqual[eps, -8.8e-94], t$95$0, If[LessEqual[eps, 1.25e-81], N[Power[t$95$0, 5.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if eps < -8.80000000000000004e-94 or 1.24999999999999995e-81 < eps

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 82.5%

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

   if -8.80000000000000004e-94 < eps < 1.24999999999999995e-81

   1. Initial program 87.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 87.0%

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

Alternative 3: 80.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 77.4%

   \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{5}} \]
6. Add Preprocessing

Alternative 4: 64.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{5} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return x ^ 5.0
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Power[x, 5.0], $MachinePrecision]

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 62.2%

   \[\leadsto \color{blue}{{x}^{5}} \]
6. Add Preprocessing

Alternative 5: 4.7% accurate, 52.3× speedup

Math

\[\begin{array}{l} \\ x + \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ x eps))

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x + eps
end function

Java

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

Python

def code(x, eps):
	return x + eps

Julia

function code(x, eps)
	return Float64(x + eps)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.6%

   \[\leadsto \color{blue}{x + \varepsilon} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(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)))