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

Percentage Accurate: 88.3% → 98.4%

Time: 11.3s

Alternatives: 6

Speedup: 1.8×

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: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{2} \cdot \left(10 \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-46}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6e-48)
   (* eps (+ (* 5.0 (pow x 4.0)) (* eps (* (pow x 2.0) (* 10.0 (+ x eps))))))
   (if (<= x 6.6e-46)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6e-48) {
		tmp = eps * ((5.0 * pow(x, 4.0)) + (eps * (pow(x, 2.0) * (10.0 * (x + eps)))));
	} else if (x <= 6.6e-46) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / 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 <= (-6d-48)) then
        tmp = eps * ((5.0d0 * (x ** 4.0d0)) + (eps * ((x ** 2.0d0) * (10.0d0 * (x + eps)))))
    else if (x <= 6.6d-46) then
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    else
        tmp = (x ** 4.0d0) * (eps * (5.0d0 + (10.0d0 * (eps / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6e-48) {
		tmp = eps * ((5.0 * Math.pow(x, 4.0)) + (eps * (Math.pow(x, 2.0) * (10.0 * (x + eps)))));
	} else if (x <= 6.6e-46) {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	} else {
		tmp = Math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6e-48:
		tmp = eps * ((5.0 * math.pow(x, 4.0)) + (eps * (math.pow(x, 2.0) * (10.0 * (x + eps)))))
	elif x <= 6.6e-46:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	else:
		tmp = math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6e-48)
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(eps * Float64((x ^ 2.0) * Float64(10.0 * Float64(x + eps))))));
	elseif (x <= 6.6e-46)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6e-48)
		tmp = eps * ((5.0 * (x ^ 4.0)) + (eps * ((x ^ 2.0) * (10.0 * (x + eps)))));
	elseif (x <= 6.6e-46)
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	else
		tmp = (x ^ 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6e-48], N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[Power[x, 2.0], $MachinePrecision] * N[(10.0 * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e-46], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.9999999999999998e-48

   1. Initial program 30.0%

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

   3. Taylor expanded in eps around 0 94.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 94.9%

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

      2. associate-+r+ 94.9%

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

      3. distribute-lft1-in 94.9%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(4 + 1\right) \cdot {x}^{4}} + \varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right)\right) \]

      4. metadata-eval 94.9%

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

      5. +-commutative 94.9%

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

   5. Simplified 94.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10 + \varepsilon \cdot \left({x}^{2} \cdot 10\right)\right)\right)} \]

   6. Taylor expanded in eps around 0 94.9%

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \color{blue}{\varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*r* 94.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{\left(10 \cdot \varepsilon\right) \cdot {x}^{2}} + 10 \cdot {x}^{3}\right)\right) \]

      2. *-commutative 94.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot 10\right)} \cdot {x}^{2} + 10 \cdot {x}^{3}\right)\right) \]

      3. associate-*r* 94.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \left(10 \cdot {x}^{2}\right)} + 10 \cdot {x}^{3}\right)\right) \]

      4. fma-undefine 94.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, 10 \cdot {x}^{2}, 10 \cdot {x}^{3}\right)}\right) \]

      5. fma-undefine 94.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(10 \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)}\right) \]

      6. associate-*r* 94.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot 10\right) \cdot {x}^{2}} + 10 \cdot {x}^{3}\right)\right) \]

      7. *-commutative 94.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\color{blue}{\left(10 \cdot \varepsilon\right)} \cdot {x}^{2} + 10 \cdot {x}^{3}\right)\right) \]

      8. cube-mult 94.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(10 \cdot \varepsilon\right) \cdot {x}^{2} + 10 \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \]

      9. unpow2 94.9%

         \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(10 \cdot \varepsilon\right) \cdot {x}^{2} + 10 \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

      10. associate-*r* 94.9%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(\left(10 \cdot \varepsilon\right) \cdot {x}^{2} + \color{blue}{\left(10 \cdot x\right) \cdot {x}^{2}}\right)\right) \]

      11. distribute-rgt-in 94.9%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \color{blue}{\left({x}^{2} \cdot \left(10 \cdot \varepsilon + 10 \cdot x\right)\right)}\right) \]

      12. distribute-lft-out 94.9%

          \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{2} \cdot \color{blue}{\left(10 \cdot \left(\varepsilon + x\right)\right)}\right)\right) \]

   8. Simplified 94.9%

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \color{blue}{\varepsilon \cdot \left({x}^{2} \cdot \left(10 \cdot \left(\varepsilon + x\right)\right)\right)}\right) \]

   if -5.9999999999999998e-48 < x < 6.60000000000000027e-46

   1. Initial program 100.0%

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

   if 6.60000000000000027e-46 < x

   1. Initial program 57.0%

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

   3. Taylor expanded in x around inf 99.9%

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

   4. Taylor expanded in eps around 0 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{2} \cdot \left(10 \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-46}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-48} \lor \neg \left(x \leq 1.28 \cdot 10^{-45}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -9e-48) (not (<= x 1.28e-45)))
   (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x)))))
   (- (pow (+ x eps) 5.0) (pow x 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -9e-48) || !(x <= 1.28e-45)) {
		tmp = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	} else {
		tmp = pow((x + eps), 5.0) - pow(x, 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-48)) .or. (.not. (x <= 1.28d-45))) then
        tmp = (x ** 4.0d0) * (eps * (5.0d0 + (10.0d0 * (eps / x))))
    else
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -9e-48) || !(x <= 1.28e-45)) {
		tmp = Math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	} else {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -9e-48) or not (x <= 1.28e-45):
		tmp = math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))))
	else:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -9e-48) || !(x <= 1.28e-45))
		tmp = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	else
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -9e-48) || ~((x <= 1.28e-45)))
		tmp = (x ^ 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	else
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -9e-48], N[Not[LessEqual[x, 1.28e-45]], $MachinePrecision]], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.99999999999999977e-48 or 1.28e-45 < x

   1. Initial program 39.0%

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

   3. Taylor expanded in x around inf 95.4%

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

   4. Taylor expanded in eps around 0 95.3%

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

   if -8.99999999999999977e-48 < x < 1.28e-45

   1. Initial program 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-48} \lor \neg \left(x \leq 1.28 \cdot 10^{-45}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-46} \lor \neg \left(x \leq 3.8 \cdot 10^{-46}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.95e-46) (not (<= x 3.8e-46)))
   (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x)))))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.95e-46) || !(x <= 3.8e-46)) {
		tmp = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	} else {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-1.95d-46)) .or. (.not. (x <= 3.8d-46))) then
        tmp = (x ** 4.0d0) * (eps * (5.0d0 + (10.0d0 * (eps / x))))
    else
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -1.95e-46) || !(x <= 3.8e-46)) {
		tmp = Math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	} else {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.95e-46) or not (x <= 3.8e-46):
		tmp = math.pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))))
	else:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.95e-46) || !(x <= 3.8e-46))
		tmp = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	else
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.95e-46) || ~((x <= 3.8e-46)))
		tmp = (x ^ 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	else
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.95e-46], N[Not[LessEqual[x, 3.8e-46]], $MachinePrecision]], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9500000000000001e-46 or 3.7999999999999997e-46 < x

   1. Initial program 40.5%

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

   3. Taylor expanded in x around inf 93.4%

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

   4. Taylor expanded in eps around 0 93.4%

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

   if -1.9500000000000001e-46 < x < 3.7999999999999997e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 99.8%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft1-in 99.8%

         \[\leadsto {\varepsilon}^{5} \cdot \left(1 + \color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}}\right) \]

      2. metadata-eval 99.8%

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-46} \lor \neg \left(x \leq 3.8 \cdot 10^{-46}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-48} \lor \neg \left(x \leq 9 \cdot 10^{-46}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.3e-48) (not (<= x 9e-46)))
   (* 5.0 (* eps (pow x 4.0)))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.3e-48) || !(x <= 9e-46)) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-3.3d-48)) .or. (.not. (x <= 9d-46))) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -3.3e-48) || !(x <= 9e-46)) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.3e-48) or not (x <= 9e-46):
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.3e-48) || !(x <= 9e-46))
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -3.3e-48) || ~((x <= 9e-46)))
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.3e-48], N[Not[LessEqual[x, 9e-46]], $MachinePrecision]], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.3e-48 or 9.00000000000000001e-46 < x

   1. Initial program 39.0%

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

      1. sqr-pow 16.5%

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

      2. metadata-eval 16.5%

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{2.5}} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)} - {x}^{5} \]

      3. metadata-eval 16.5%

         \[\leadsto {\left(x + \varepsilon\right)}^{2.5} \cdot {\left(x + \varepsilon\right)}^{\color{blue}{2.5}} - {x}^{5} \]

   4. Applied egg-rr 16.5%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{2.5} \cdot {\left(x + \varepsilon\right)}^{2.5}} - {x}^{5} \]

   5. Taylor expanded in x around inf 92.7%

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

      1. *-commutative 92.7%

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

      2. distribute-rgt1-in 92.7%

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

      3. metadata-eval 92.7%

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

      4. associate-*r* 92.7%

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

   7. Simplified 92.7%

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

   if -3.3e-48 < x < 9.00000000000000001e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 99.4%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft1-in 99.4%

         \[\leadsto {\varepsilon}^{5} \cdot \left(1 + \color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}}\right) \]

      2. metadata-eval 99.4%

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

   5. Simplified 99.4%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-48} \lor \neg \left(x \leq 9 \cdot 10^{-46}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-47} \lor \neg \left(x \leq 3.8 \cdot 10^{-46}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.6e-47) (not (<= x 3.8e-46)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.6e-47) || !(x <= 3.8e-46)) {
		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 <= (-3.6d-47)) .or. (.not. (x <= 3.8d-46))) 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 <= -3.6e-47) || !(x <= 3.8e-46)) {
		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 <= -3.6e-47) or not (x <= 3.8e-46):
		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 <= -3.6e-47) || !(x <= 3.8e-46))
		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 <= -3.6e-47) || ~((x <= 3.8e-46)))
		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, -3.6e-47], N[Not[LessEqual[x, 3.8e-46]], $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 < -3.59999999999999991e-47 or 3.7999999999999997e-46 < x

   1. Initial program 40.5%

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

      1. sqr-pow 18.6%

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

      2. metadata-eval 18.6%

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{2.5}} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)} - {x}^{5} \]

      3. metadata-eval 18.6%

         \[\leadsto {\left(x + \varepsilon\right)}^{2.5} \cdot {\left(x + \varepsilon\right)}^{\color{blue}{2.5}} - {x}^{5} \]

   4. Applied egg-rr 18.6%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{2.5} \cdot {\left(x + \varepsilon\right)}^{2.5}} - {x}^{5} \]

   5. Taylor expanded in x around inf 90.7%

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

      1. *-commutative 90.7%

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

      2. distribute-rgt1-in 90.7%

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

      3. metadata-eval 90.7%

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

      4. associate-*r* 90.7%

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

   7. Simplified 90.7%

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

   if -3.59999999999999991e-47 < x < 3.7999999999999997e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-47} \lor \neg \left(x \leq 3.8 \cdot 10^{-46}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 90.7%

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

3. Taylor expanded in x around 0 88.9%

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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)))