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

Percentage Accurate: 88.3% → 99.3%

Time: 8.0s

Alternatives: 5

Speedup: 2.0×

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: 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-313} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -2e-313) (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 <= -2e-313) || !(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 <= (-2d-313)) .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 <= -2e-313) || !(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 <= -2e-313) 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 <= -2e-313) || !(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 <= -2e-313) || ~((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, -2e-313], 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)) < -1.99999999998e-313 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 99.0%

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

   if -1.99999999998e-313 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 86.1%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

      3. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in eps around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-313} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 2: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-55} \lor \neg \left(x \leq 7 \cdot 10^{-44}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.9e-55) (not (<= x 7e-44)))
   (* eps (* 5.0 (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.9e-55) || !(x <= 7e-44)) {
		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 <= (-1.9d-55)) .or. (.not. (x <= 7d-44))) 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 <= -1.9e-55) || !(x <= 7e-44)) {
		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 <= -1.9e-55) or not (x <= 7e-44):
		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 <= -1.9e-55) || !(x <= 7e-44))
		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 <= -1.9e-55) || ~((x <= 7e-44)))
		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, -1.9e-55], N[Not[LessEqual[x, 7e-44]], $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 < -1.8999999999999998e-55 or 6.9999999999999995e-44 < x

   1. Initial program 46.0%

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

   2. Taylor expanded in x around inf 92.2%

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

      1. distribute-lft1-in 92.2%

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

      2. metadata-eval 92.2%

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

      3. *-commutative 92.2%

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

   4. Simplified 92.2%

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

   5. Taylor expanded in eps around 0 92.1%

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

      1. associate-*r* 92.2%

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

      2. *-commutative 92.2%

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

      3. associate-*r* 92.2%

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

   7. Simplified 92.2%

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

   if -1.8999999999999998e-55 < x < 6.9999999999999995e-44

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 99.1%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-55} \lor \neg \left(x \leq 7 \cdot 10^{-44}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 3: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-55}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-44}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(5 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8.2e-55)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 1.85e-44) (pow eps 5.0) (* (* x x) (* 5.0 (* eps (* x x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8.2e-55) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 1.85e-44) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (x * x) * (5.0 * (eps * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-8.2d-55)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 1.85d-44) then
        tmp = eps ** 5.0d0
    else
        tmp = (x * x) * (5.0d0 * (eps * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -8.2e-55) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 1.85e-44) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = (x * x) * (5.0 * (eps * (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -8.2e-55:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 1.85e-44:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = (x * x) * (5.0 * (eps * (x * x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8.2e-55)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 1.85e-44)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(x * x) * Float64(5.0 * Float64(eps * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -8.2e-55)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 1.85e-44)
		tmp = eps ^ 5.0;
	else
		tmp = (x * x) * (5.0 * (eps * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8.2e-55], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.85e-44], N[Power[eps, 5.0], $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(5.0 * N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.1999999999999996e-55

   1. Initial program 40.9%

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

   2. Taylor expanded in x around inf 93.0%

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

      1. distribute-lft1-in 93.0%

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

      2. metadata-eval 93.0%

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

      3. associate-*l* 93.0%

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

   4. Simplified 93.0%

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

   if -8.1999999999999996e-55 < x < 1.85e-44

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 99.1%

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

   if 1.85e-44 < x

   1. Initial program 53.3%

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

      1. add-cube-cbrt 53.4%

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

      2. pow3 53.4%

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

   3. Applied egg-rr 53.4%

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

   4. Taylor expanded in x around inf 90.3%

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

      1. distribute-lft1-in 91.1%

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

      2. metadata-eval 91.1%

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

      3. *-commutative 91.1%

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

   6. Simplified 90.3%

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

      1. rem-cube-cbrt 91.1%

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

      2. metadata-eval 91.1%

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

      3. metadata-eval 91.1%

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

      4. metadata-eval 91.1%

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

      5. pow-prod-up 90.9%

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

      6. pow-prod-down 90.9%

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

      7. metadata-eval 90.9%

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

      8. pow2 90.9%

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

      9. associate-*r* 90.8%

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

   8. Applied egg-rr 90.8%

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

   9. Taylor expanded in eps around 0 91.0%

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

       1. unpow2 91.0%

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

   11. Simplified 91.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-55}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-44}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(5 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]

Alternative 4: 97.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-54} \lor \neg \left(x \leq 9.5 \cdot 10^{-45}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(5 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.6e-54) (not (<= x 9.5e-45)))
   (* (* x x) (* 5.0 (* eps (* x x))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.6e-54) || !(x <= 9.5e-45)) {
		tmp = (x * x) * (5.0 * (eps * (x * x)));
	} 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 <= (-3.6d-54)) .or. (.not. (x <= 9.5d-45))) then
        tmp = (x * x) * (5.0d0 * (eps * (x * x)))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -3.6e-54) || !(x <= 9.5e-45)) {
		tmp = (x * x) * (5.0 * (eps * (x * x)));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.6e-54) or not (x <= 9.5e-45):
		tmp = (x * x) * (5.0 * (eps * (x * x)))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.6e-54) || !(x <= 9.5e-45))
		tmp = Float64(Float64(x * x) * Float64(5.0 * Float64(eps * Float64(x * x))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -3.6e-54) || ~((x <= 9.5e-45)))
		tmp = (x * x) * (5.0 * (eps * (x * x)));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.6e-54], N[Not[LessEqual[x, 9.5e-45]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] * N[(5.0 * N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.59999999999999976e-54 or 9.5000000000000002e-45 < x

   1. Initial program 46.0%

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

      1. add-cube-cbrt 46.0%

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

      2. pow3 46.0%

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

   3. Applied egg-rr 46.0%

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

   4. Taylor expanded in x around inf 91.1%

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

      1. distribute-lft1-in 92.2%

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

      2. metadata-eval 92.2%

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

      3. *-commutative 92.2%

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

   6. Simplified 91.1%

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

      1. rem-cube-cbrt 92.2%

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

      2. metadata-eval 92.2%

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

      3. metadata-eval 92.2%

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

      4. metadata-eval 92.2%

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

      5. pow-prod-up 92.0%

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

      6. pow-prod-down 92.0%

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

      7. metadata-eval 92.0%

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

      8. pow2 92.0%

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

      9. associate-*r* 92.0%

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

   8. Applied egg-rr 92.0%

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

   9. Taylor expanded in eps around 0 92.1%

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

       1. unpow2 92.1%

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

   11. Simplified 92.1%

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

   if -3.59999999999999976e-54 < x < 9.5000000000000002e-45

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 99.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-54} \lor \neg \left(x \leq 9.5 \cdot 10^{-45}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(5 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5: 82.9% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return (x * x) * (5.0 * (eps * (x * x)));
}

Python

def code(x, eps):
	return (x * x) * (5.0 * (eps * (x * x)))

Julia

function code(x, eps)
	return Float64(Float64(x * x) * Float64(5.0 * Float64(eps * Float64(x * x))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. add-cube-cbrt 88.2%

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

   2. pow3 88.2%

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

3. Applied egg-rr 88.2%

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

4. Taylor expanded in x around inf 82.9%

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

   1. distribute-lft1-in 83.1%

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

   2. metadata-eval 83.1%

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

   3. *-commutative 83.1%

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

6. Simplified 82.9%

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

   1. rem-cube-cbrt 83.1%

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

   2. metadata-eval 83.1%

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

   3. metadata-eval 83.1%

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

   4. metadata-eval 83.1%

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

   5. pow-prod-up 83.1%

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

   6. pow-prod-down 83.1%

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

   7. metadata-eval 83.1%

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

   8. pow2 83.1%

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

   9. associate-*r* 83.1%

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

8. Applied egg-rr 83.1%

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

9. Taylor expanded in eps around 0 83.1%

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

    1. unpow2 83.1%

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

11. Simplified 83.1%

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

12. Final simplification 83.1%

    \[\leadsto \left(x \cdot x\right) \cdot \left(5 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right) \]

Reproduce

herbie shell --seed 2023270 
(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)))