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

Percentage Accurate: 88.3% → 99.3%

Time: 10.2s

Alternatives: 6

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: 88.3% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-306} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -2e-306) (not (<= t_0 0.0)))
     t_0
     (* (pow x 4.0) (* 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 <= -2e-306) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = pow(x, 4.0) * (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 <= (-2d-306)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (x ** 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 <= -2e-306) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x, 4.0) * (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 <= -2e-306) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = math.pow(x, 4.0) * (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 <= -2e-306) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64((x ^ 4.0) * 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 <= -2e-306) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (x ^ 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[Or[LessEqual[t$95$0, -2e-306], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -2.00000000000000006e-306 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 94.6%

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

   if -2.00000000000000006e-306 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 85.5%

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

   2. Taylor expanded in x around inf 99.9%

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

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

   4. Simplified 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-306} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]

Alternative 2: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-57} \lor \neg \left(x \leq 4.4 \cdot 10^{-61}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.2e-57) (not (<= x 4.4e-61)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.2e-57) || !(x <= 4.4e-61)) {
		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 <= (-1.2d-57)) .or. (.not. (x <= 4.4d-61))) 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 <= -1.2e-57) || !(x <= 4.4e-61)) {
		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 <= -1.2e-57) or not (x <= 4.4e-61):
		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 <= -1.2e-57) || !(x <= 4.4e-61))
		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 <= -1.2e-57) || ~((x <= 4.4e-61)))
		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, -1.2e-57], N[Not[LessEqual[x, 4.4e-61]], $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 < -1.20000000000000003e-57 or 4.40000000000000017e-61 < x

   1. Initial program 44.8%

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

   2. Taylor expanded in x around inf 93.2%

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

   4. Simplified 93.2%

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

   5. Taylor expanded in eps around 0 90.9%

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

   if -1.20000000000000003e-57 < x < 4.40000000000000017e-61

   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 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-57} \lor \neg \left(x \leq 4.4 \cdot 10^{-61}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 3: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-57} \lor \neg \left(x \leq 4.5 \cdot 10^{-61}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.22e-57) (not (<= x 4.5e-61)))
   (* (pow x 4.0) (* eps 5.0))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.22e-57) || !(x <= 4.5e-61)) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} 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 <= (-1.22d-57)) .or. (.not. (x <= 4.5d-61))) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -1.22e-57) || !(x <= 4.5e-61)) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.22e-57) or not (x <= 4.5e-61):
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.22e-57) || !(x <= 4.5e-61))
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.22e-57) || ~((x <= 4.5e-61)))
		tmp = (x ^ 4.0) * (eps * 5.0);
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.22e-57], N[Not[LessEqual[x, 4.5e-61]], $MachinePrecision]], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2200000000000001e-57 or 4.5e-61 < x

   1. Initial program 44.8%

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

   2. Taylor expanded in x around inf 91.0%

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

      1. distribute-rgt1-in 91.0%

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

      2. metadata-eval 91.0%

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

   4. Simplified 91.0%

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

   if -1.2200000000000001e-57 < x < 4.5e-61

   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 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-57} \lor \neg \left(x \leq 4.5 \cdot 10^{-61}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-57}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3e-57)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 4.5e-61) (pow eps 5.0) (* eps (* 5.0 (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3e-57) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 4.5e-61) {
		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 <= (-3d-57)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 4.5d-61) 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 <= -3e-57) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 4.5e-61) {
		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 <= -3e-57:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 4.5e-61:
		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 <= -3e-57)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 4.5e-61)
		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 <= -3e-57)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 4.5e-61)
		tmp = eps ^ 5.0;
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3e-57], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-61], 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 < -3.00000000000000001e-57

   1. Initial program 40.3%

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

   2. Taylor expanded in x around inf 93.4%

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

   4. Simplified 93.4%

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

   5. Taylor expanded in eps around 0 90.0%

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

   if -3.00000000000000001e-57 < x < 4.5e-61

   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 4.5e-61 < x

   1. Initial program 49.2%

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

   2. Taylor expanded in x around inf 92.1%

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

      1. *-commutative 92.1%

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

      2. distribute-rgt1-in 92.1%

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

      3. metadata-eval 92.1%

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

      4. *-commutative 92.1%

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

      5. associate-*r* 92.0%

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

   4. Simplified 92.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-57}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

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 87.0%

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

2. Taylor expanded in x around 0 85.2%

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

3. Final simplification 85.2%

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

Alternative 6: 71.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 87.0%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Step-by-step derivation

   1. add-cube-cbrt 79.8%

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

   2. unpow-prod-down 79.6%

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

   3. add-cube-cbrt 80.2%

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

   4. unpow-prod-down 86.3%

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

   5. prod-diff 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x + \varepsilon} \cdot \sqrt[3]{x + \varepsilon}\right)}^{5}, {\left(\sqrt[3]{x + \varepsilon}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{5}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}, {\left(\sqrt[3]{x}\right)}^{5} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right)} \]

   6. pow2 86.3%

      \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(\sqrt[3]{x + \varepsilon}\right)}^{2}\right)}}^{5}, {\left(\sqrt[3]{x + \varepsilon}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{5}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}, {\left(\sqrt[3]{x}\right)}^{5} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) \]

   7. pow2 86.3%

      \[\leadsto \mathsf{fma}\left({\left({\left(\sqrt[3]{x + \varepsilon}\right)}^{2}\right)}^{5}, {\left(\sqrt[3]{x + \varepsilon}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\color{blue}{\left({\left(\sqrt[3]{x}\right)}^{2}\right)}}^{5}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{5}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}, {\left(\sqrt[3]{x}\right)}^{5} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{5}\right) \]

3. Applied egg-rr 86.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left({\left(\sqrt[3]{x + \varepsilon}\right)}^{2}\right)}^{5}, {\left(\sqrt[3]{x + \varepsilon}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{5}, {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}, {\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right)} \]
4. Step-by-step derivation

   1. +-commutative 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{5}, {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}, {\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right) + \mathsf{fma}\left({\left({\left(\sqrt[3]{x + \varepsilon}\right)}^{2}\right)}^{5}, {\left(\sqrt[3]{x + \varepsilon}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right)} \]

   2. fma-udef 79.6%

      \[\leadsto \color{blue}{\left(\left(-{\left(\sqrt[3]{x}\right)}^{5}\right) \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5} + {\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right)} + \mathsf{fma}\left({\left({\left(\sqrt[3]{x + \varepsilon}\right)}^{2}\right)}^{5}, {\left(\sqrt[3]{x + \varepsilon}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right) \]

   3. distribute-lft-neg-in 79.6%

      \[\leadsto \left(\color{blue}{\left(-{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right)} + {\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right) + \mathsf{fma}\left({\left({\left(\sqrt[3]{x + \varepsilon}\right)}^{2}\right)}^{5}, {\left(\sqrt[3]{x + \varepsilon}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right) \]

   4. +-commutative 79.6%

      \[\leadsto \color{blue}{\left({\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5} + \left(-{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right)\right)} + \mathsf{fma}\left({\left({\left(\sqrt[3]{x + \varepsilon}\right)}^{2}\right)}^{5}, {\left(\sqrt[3]{x + \varepsilon}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right) \]

   5. sub-neg 79.6%

      \[\leadsto \color{blue}{\left({\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5} - {\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right)} + \mathsf{fma}\left({\left({\left(\sqrt[3]{x + \varepsilon}\right)}^{2}\right)}^{5}, {\left(\sqrt[3]{x + \varepsilon}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right) \]

   6. +-inverses 79.6%

      \[\leadsto \color{blue}{0} + \mathsf{fma}\left({\left({\left(\sqrt[3]{x + \varepsilon}\right)}^{2}\right)}^{5}, {\left(\sqrt[3]{x + \varepsilon}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right) \]

   7. fma-neg 86.3%

      \[\leadsto 0 + \color{blue}{\left({\left({\left(\sqrt[3]{x + \varepsilon}\right)}^{2}\right)}^{5} \cdot {\left(\sqrt[3]{x + \varepsilon}\right)}^{5} - {\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right)} \]

5. Simplified 86.3%

   \[\leadsto \color{blue}{0 + \left({\left({\left(\sqrt[3]{\varepsilon + x}\right)}^{2}\right)}^{5} \cdot {\left(\sqrt[3]{\varepsilon + x}\right)}^{5} - {\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right)} \]
6. Step-by-step derivation

   1. fma-neg 79.6%

      \[\leadsto 0 + \color{blue}{\mathsf{fma}\left({\left({\left(\sqrt[3]{\varepsilon + x}\right)}^{2}\right)}^{5}, {\left(\sqrt[3]{\varepsilon + x}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right)} \]

   2. pow-pow 79.6%

      \[\leadsto 0 + \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\varepsilon + x}\right)}^{\left(2 \cdot 5\right)}}, {\left(\sqrt[3]{\varepsilon + x}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right) \]

   3. metadata-eval 79.6%

      \[\leadsto 0 + \mathsf{fma}\left({\left(\sqrt[3]{\varepsilon + x}\right)}^{\color{blue}{10}}, {\left(\sqrt[3]{\varepsilon + x}\right)}^{5}, -{\left(\sqrt[3]{x}\right)}^{5} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5}\right) \]

   4. *-commutative 79.6%

      \[\leadsto 0 + \mathsf{fma}\left({\left(\sqrt[3]{\varepsilon + x}\right)}^{10}, {\left(\sqrt[3]{\varepsilon + x}\right)}^{5}, -\color{blue}{{\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{5} \cdot {\left(\sqrt[3]{x}\right)}^{5}}\right) \]

   5. pow-prod-down 79.5%

      \[\leadsto 0 + \mathsf{fma}\left({\left(\sqrt[3]{\varepsilon + x}\right)}^{10}, {\left(\sqrt[3]{\varepsilon + x}\right)}^{5}, -\color{blue}{{\left({\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}\right)}^{5}}\right) \]

   6. unpow2 79.5%

      \[\leadsto 0 + \mathsf{fma}\left({\left(\sqrt[3]{\varepsilon + x}\right)}^{10}, {\left(\sqrt[3]{\varepsilon + x}\right)}^{5}, -{\left(\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}\right)}^{5}\right) \]

   7. add-cube-cbrt 79.4%

      \[\leadsto 0 + \mathsf{fma}\left({\left(\sqrt[3]{\varepsilon + x}\right)}^{10}, {\left(\sqrt[3]{\varepsilon + x}\right)}^{5}, -{\color{blue}{x}}^{5}\right) \]

7. Applied egg-rr 79.4%

   \[\leadsto 0 + \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\varepsilon + x}\right)}^{10}, {\left(\sqrt[3]{\varepsilon + x}\right)}^{5}, -{x}^{5}\right)} \]
8. Step-by-step derivation

   1. fma-neg 79.7%

      \[\leadsto 0 + \color{blue}{\left({\left(\sqrt[3]{\varepsilon + x}\right)}^{10} \cdot {\left(\sqrt[3]{\varepsilon + x}\right)}^{5} - {x}^{5}\right)} \]

   2. metadata-eval 79.7%

      \[\leadsto 0 + \left({\left(\sqrt[3]{\varepsilon + x}\right)}^{\color{blue}{\left(2 \cdot 5\right)}} \cdot {\left(\sqrt[3]{\varepsilon + x}\right)}^{5} - {x}^{5}\right) \]

   3. pow-sqr 79.4%

      \[\leadsto 0 + \left(\color{blue}{\left({\left(\sqrt[3]{\varepsilon + x}\right)}^{5} \cdot {\left(\sqrt[3]{\varepsilon + x}\right)}^{5}\right)} \cdot {\left(\sqrt[3]{\varepsilon + x}\right)}^{5} - {x}^{5}\right) \]

   4. unpow3 79.4%

      \[\leadsto 0 + \left(\color{blue}{{\left({\left(\sqrt[3]{\varepsilon + x}\right)}^{5}\right)}^{3}} - {x}^{5}\right) \]

9. Simplified 79.4%

   \[\leadsto 0 + \color{blue}{\left({\left({\left(\sqrt[3]{\varepsilon + x}\right)}^{5}\right)}^{3} - {x}^{5}\right)} \]

10. Taylor expanded in eps around 0 72.0%

    \[\leadsto 0 + \color{blue}{\left({1}^{0.3333333333333333} \cdot {x}^{5} - {x}^{5}\right)} \]
11. Step-by-step derivation

    1. pow-base-1 72.0%

       \[\leadsto 0 + \left(\color{blue}{1} \cdot {x}^{5} - {x}^{5}\right) \]

    2. *-lft-identity 72.0%

       \[\leadsto 0 + \left(\color{blue}{{x}^{5}} - {x}^{5}\right) \]

    3. +-inverses 72.0%

       \[\leadsto 0 + \color{blue}{0} \]

12. Simplified 72.0%

    \[\leadsto 0 + \color{blue}{0} \]

13. Final simplification 72.0%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023315 
(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)))