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

Percentage Accurate: 88.7% → 98.0%

Time: 9.8s

Alternatives: 7

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: 88.7% 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: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-44} \lor \neg \left(x \leq 2.6 \cdot 10^{-52}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 + \frac{\frac{{\varepsilon}^{3} \cdot 10}{x} - -10 \cdot {\varepsilon}^{2}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -7.8e-44) (not (<= x 2.6e-52)))
   (*
    (pow x 4.0)
    (+
     (* eps 5.0)
     (/ (- (/ (* (pow eps 3.0) 10.0) x) (* -10.0 (pow eps 2.0))) x)))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -7.8e-44) || !(x <= 2.6e-52)) {
		tmp = pow(x, 4.0) * ((eps * 5.0) + ((((pow(eps, 3.0) * 10.0) / x) - (-10.0 * pow(eps, 2.0))) / 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 <= (-7.8d-44)) .or. (.not. (x <= 2.6d-52))) then
        tmp = (x ** 4.0d0) * ((eps * 5.0d0) + (((((eps ** 3.0d0) * 10.0d0) / x) - ((-10.0d0) * (eps ** 2.0d0))) / 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 <= -7.8e-44) || !(x <= 2.6e-52)) {
		tmp = Math.pow(x, 4.0) * ((eps * 5.0) + ((((Math.pow(eps, 3.0) * 10.0) / x) - (-10.0 * Math.pow(eps, 2.0))) / 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 <= -7.8e-44) or not (x <= 2.6e-52):
		tmp = math.pow(x, 4.0) * ((eps * 5.0) + ((((math.pow(eps, 3.0) * 10.0) / x) - (-10.0 * math.pow(eps, 2.0))) / 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 <= -7.8e-44) || !(x <= 2.6e-52))
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) + Float64(Float64(Float64(Float64((eps ^ 3.0) * 10.0) / x) - Float64(-10.0 * (eps ^ 2.0))) / 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 <= -7.8e-44) || ~((x <= 2.6e-52)))
		tmp = (x ^ 4.0) * ((eps * 5.0) + (((((eps ^ 3.0) * 10.0) / x) - (-10.0 * (eps ^ 2.0))) / 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, -7.8e-44], N[Not[LessEqual[x, 2.6e-52]], $MachinePrecision]], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] + N[(N[(N[(N[(N[Power[eps, 3.0], $MachinePrecision] * 10.0), $MachinePrecision] / x), $MachinePrecision] - N[(-10.0 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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 < -7.8000000000000004e-44 or 2.5999999999999999e-52 < x

   1. Initial program 39.6%

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

   3. Taylor expanded in x around -inf 97.9%

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

   5. Simplified 98.0%

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

   if -7.8000000000000004e-44 < x < 2.5999999999999999e-52

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-44} \lor \neg \left(x \leq 2.6 \cdot 10^{-52}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 + \frac{\frac{{\varepsilon}^{3} \cdot 10}{x} - -10 \cdot {\varepsilon}^{2}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-44} \lor \neg \left(x \leq 2.4 \cdot 10^{-52}\right):\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot \left(5 + \varepsilon \cdot \frac{10}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.2e-44) (not (<= x 2.4e-52)))
   (* (* (pow x 4.0) eps) (+ 5.0 (* eps (/ 10.0 x))))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.2e-44) || !(x <= 2.4e-52)) {
		tmp = (pow(x, 4.0) * eps) * (5.0 + (eps * (10.0 / 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 <= (-3.2d-44)) .or. (.not. (x <= 2.4d-52))) then
        tmp = ((x ** 4.0d0) * eps) * (5.0d0 + (eps * (10.0d0 / 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 <= -3.2e-44) || !(x <= 2.4e-52)) {
		tmp = (Math.pow(x, 4.0) * eps) * (5.0 + (eps * (10.0 / 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 <= -3.2e-44) or not (x <= 2.4e-52):
		tmp = (math.pow(x, 4.0) * eps) * (5.0 + (eps * (10.0 / 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 <= -3.2e-44) || !(x <= 2.4e-52))
		tmp = Float64(Float64((x ^ 4.0) * eps) * Float64(5.0 + Float64(eps * Float64(10.0 / 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 <= -3.2e-44) || ~((x <= 2.4e-52)))
		tmp = ((x ^ 4.0) * eps) * (5.0 + (eps * (10.0 / 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, -3.2e-44], N[Not[LessEqual[x, 2.4e-52]], $MachinePrecision]], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * N[(5.0 + N[(eps * N[(10.0 / x), $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 < -3.19999999999999995e-44 or 2.4000000000000002e-52 < x

   1. Initial program 39.6%

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

   3. Taylor expanded in x around -inf 97.6%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 97.6%

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

      2. associate-+r+ 97.7%

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

      3. mul-1-neg 97.7%

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

      4. unsub-neg 97.7%

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

      5. distribute-rgt1-in 97.7%

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

      6. metadata-eval 97.7%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      7. *-commutative 97.7%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

   5. Simplified 97.7%

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

      1. associate-/l* 97.7%

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

      2. unpow2 97.7%

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

      3. associate-*l* 97.7%

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

   7. Applied egg-rr 97.7%

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

   8. Taylor expanded in x around inf 97.7%

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

   10. Simplified 97.7%

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

   if -3.19999999999999995e-44 < x < 2.4000000000000002e-52

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-44} \lor \neg \left(x \leq 2.4 \cdot 10^{-52}\right):\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot \left(5 + \varepsilon \cdot \frac{10}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-44}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \varepsilon \cdot \left(\varepsilon \cdot \frac{-10}{x}\right)\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot \left(5 + \varepsilon \cdot \frac{10}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.15e-44)
   (* (pow x 4.0) (- (* eps 5.0) (* eps (* eps (/ -10.0 x)))))
   (if (<= x 2.4e-52)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* (* (pow x 4.0) eps) (+ 5.0 (* eps (/ 10.0 x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.15e-44) {
		tmp = pow(x, 4.0) * ((eps * 5.0) - (eps * (eps * (-10.0 / x))));
	} else if (x <= 2.4e-52) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = (pow(x, 4.0) * eps) * (5.0 + (eps * (10.0 / 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 <= (-2.15d-44)) then
        tmp = (x ** 4.0d0) * ((eps * 5.0d0) - (eps * (eps * ((-10.0d0) / x))))
    else if (x <= 2.4d-52) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = ((x ** 4.0d0) * eps) * (5.0d0 + (eps * (10.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.15e-44) {
		tmp = Math.pow(x, 4.0) * ((eps * 5.0) - (eps * (eps * (-10.0 / x))));
	} else if (x <= 2.4e-52) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = (Math.pow(x, 4.0) * eps) * (5.0 + (eps * (10.0 / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.15e-44:
		tmp = math.pow(x, 4.0) * ((eps * 5.0) - (eps * (eps * (-10.0 / x))))
	elif x <= 2.4e-52:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = (math.pow(x, 4.0) * eps) * (5.0 + (eps * (10.0 / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.15e-44)
		tmp = Float64((x ^ 4.0) * Float64(Float64(eps * 5.0) - Float64(eps * Float64(eps * Float64(-10.0 / x)))));
	elseif (x <= 2.4e-52)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64(Float64((x ^ 4.0) * eps) * Float64(5.0 + Float64(eps * Float64(10.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.15e-44)
		tmp = (x ^ 4.0) * ((eps * 5.0) - (eps * (eps * (-10.0 / x))));
	elseif (x <= 2.4e-52)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = ((x ^ 4.0) * eps) * (5.0 + (eps * (10.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.15e-44], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] - N[(eps * N[(eps * N[(-10.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-52], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * N[(5.0 + N[(eps * N[(10.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.15000000000000007e-44

   1. Initial program 34.9%

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

   3. Taylor expanded in x around -inf 95.1%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 95.1%

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

      2. associate-+r+ 95.2%

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

      3. mul-1-neg 95.2%

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

      4. unsub-neg 95.2%

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

      5. distribute-rgt1-in 95.2%

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

      6. metadata-eval 95.2%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      7. *-commutative 95.2%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

   5. Simplified 95.2%

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

      1. associate-/l* 95.2%

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

      2. unpow2 95.2%

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

      3. associate-*l* 95.2%

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

   7. Applied egg-rr 95.2%

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

   if -2.15000000000000007e-44 < x < 2.4000000000000002e-52

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   if 2.4000000000000002e-52 < x

   1. Initial program 42.3%

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

   3. Taylor expanded in x around -inf 99.1%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.1%

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

      2. associate-+r+ 99.1%

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

      3. mul-1-neg 99.1%

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

      4. unsub-neg 99.1%

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

      5. distribute-rgt1-in 99.1%

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

      6. metadata-eval 99.1%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      7. *-commutative 99.1%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

   5. Simplified 99.1%

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

      1. associate-/l* 99.1%

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

      2. unpow2 99.1%

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

      3. associate-*l* 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in x around inf 99.1%

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

   10. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-44}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5 - \varepsilon \cdot \left(\varepsilon \cdot \frac{-10}{x}\right)\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-52}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot \left(5 + \varepsilon \cdot \frac{10}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-44} \lor \neg \left(x \leq 2.6 \cdot 10^{-52}\right):\\ \;\;\;\;5 \cdot \left({x}^{4} \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.15e-44) (not (<= x 2.6e-52)))
   (* 5.0 (* (pow x 4.0) eps))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.15e-44) || !(x <= 2.6e-52)) {
		tmp = 5.0 * (pow(x, 4.0) * eps);
	} 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 <= (-2.15d-44)) .or. (.not. (x <= 2.6d-52))) then
        tmp = 5.0d0 * ((x ** 4.0d0) * eps)
    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 <= -2.15e-44) || !(x <= 2.6e-52)) {
		tmp = 5.0 * (Math.pow(x, 4.0) * eps);
	} else {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.15e-44) or not (x <= 2.6e-52):
		tmp = 5.0 * (math.pow(x, 4.0) * eps)
	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 <= -2.15e-44) || !(x <= 2.6e-52))
		tmp = Float64(5.0 * Float64((x ^ 4.0) * eps));
	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 <= -2.15e-44) || ~((x <= 2.6e-52)))
		tmp = 5.0 * ((x ^ 4.0) * eps);
	else
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.15e-44], N[Not[LessEqual[x, 2.6e-52]], $MachinePrecision]], N[(5.0 * N[(N[Power[x, 4.0], $MachinePrecision] * eps), $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 < -2.15000000000000007e-44 or 2.5999999999999999e-52 < x

   1. Initial program 39.6%

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

   3. Taylor expanded in x around inf 96.2%

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

      1. *-commutative 96.2%

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

      2. distribute-rgt1-in 96.2%

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

      3. metadata-eval 96.2%

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

      4. *-commutative 96.2%

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

      5. associate-*r* 96.2%

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

   5. Simplified 96.2%

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

   6. Taylor expanded in eps around 0 96.3%

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

   if -2.15000000000000007e-44 < x < 2.5999999999999999e-52

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-44} \lor \neg \left(x \leq 2.6 \cdot 10^{-52}\right):\\ \;\;\;\;5 \cdot \left({x}^{4} \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-44} \lor \neg \left(x \leq 2.7 \cdot 10^{-52}\right):\\ \;\;\;\;5 \cdot \left({x}^{4} \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.2e-44) (not (<= x 2.7e-52)))
   (* 5.0 (* (pow x 4.0) eps))
   (* (pow eps 4.0) (+ eps (* x 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.2e-44) || !(x <= 2.7e-52)) {
		tmp = 5.0 * (pow(x, 4.0) * eps);
	} else {
		tmp = pow(eps, 4.0) * (eps + (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 <= (-2.2d-44)) .or. (.not. (x <= 2.7d-52))) then
        tmp = 5.0d0 * ((x ** 4.0d0) * eps)
    else
        tmp = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.2e-44) || !(x <= 2.7e-52)) {
		tmp = 5.0 * (Math.pow(x, 4.0) * eps);
	} else {
		tmp = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.2e-44) or not (x <= 2.7e-52):
		tmp = 5.0 * (math.pow(x, 4.0) * eps)
	else:
		tmp = math.pow(eps, 4.0) * (eps + (x * 5.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.2e-44) || !(x <= 2.7e-52))
		tmp = Float64(5.0 * Float64((x ^ 4.0) * eps));
	else
		tmp = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.2e-44) || ~((x <= 2.7e-52)))
		tmp = 5.0 * ((x ^ 4.0) * eps);
	else
		tmp = (eps ^ 4.0) * (eps + (x * 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.2e-44], N[Not[LessEqual[x, 2.7e-52]], $MachinePrecision]], N[(5.0 * N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.20000000000000012e-44 or 2.70000000000000009e-52 < x

   1. Initial program 39.6%

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

   3. Taylor expanded in x around inf 96.2%

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

      1. *-commutative 96.2%

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

      2. distribute-rgt1-in 96.2%

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

      3. metadata-eval 96.2%

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

      4. *-commutative 96.2%

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

      5. associate-*r* 96.2%

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

   5. Simplified 96.2%

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

   6. Taylor expanded in eps around 0 96.3%

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

   if -2.20000000000000012e-44 < x < 2.70000000000000009e-52

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in eps around 0 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-44} \lor \neg \left(x \leq 2.7 \cdot 10^{-52}\right):\\ \;\;\;\;5 \cdot \left({x}^{4} \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-44} \lor \neg \left(x \leq 2.7 \cdot 10^{-52}\right):\\ \;\;\;\;5 \cdot \left({x}^{4} \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.15e-44) (not (<= x 2.7e-52)))
   (* 5.0 (* (pow x 4.0) eps))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.15e-44) || !(x <= 2.7e-52)) {
		tmp = 5.0 * (pow(x, 4.0) * eps);
	} 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 <= (-2.15d-44)) .or. (.not. (x <= 2.7d-52))) then
        tmp = 5.0d0 * ((x ** 4.0d0) * eps)
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.15e-44) || !(x <= 2.7e-52)) {
		tmp = 5.0 * (Math.pow(x, 4.0) * eps);
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.15e-44) or not (x <= 2.7e-52):
		tmp = 5.0 * (math.pow(x, 4.0) * eps)
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.15e-44) || !(x <= 2.7e-52))
		tmp = Float64(5.0 * Float64((x ^ 4.0) * eps));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.15e-44) || ~((x <= 2.7e-52)))
		tmp = 5.0 * ((x ^ 4.0) * eps);
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.15e-44], N[Not[LessEqual[x, 2.7e-52]], $MachinePrecision]], N[(5.0 * N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.15000000000000007e-44 or 2.70000000000000009e-52 < x

   1. Initial program 39.6%

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

   3. Taylor expanded in x around inf 96.2%

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

      1. *-commutative 96.2%

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

      2. distribute-rgt1-in 96.2%

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

      3. metadata-eval 96.2%

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

      4. *-commutative 96.2%

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

      5. associate-*r* 96.2%

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

   5. Simplified 96.2%

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

   6. Taylor expanded in eps around 0 96.3%

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

   if -2.15000000000000007e-44 < x < 2.70000000000000009e-52

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-44} \lor \neg \left(x \leq 2.7 \cdot 10^{-52}\right):\\ \;\;\;\;5 \cdot \left({x}^{4} \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.7% 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 87.9%

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

3. Taylor expanded in x around 0 87.0%

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

4. Final simplification 87.0%

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

Reproduce

herbie shell --seed 2024089 
(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)))