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

Percentage Accurate: 88.5% → 99.4%

Time: 10.4s

Alternatives: 11

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: 88.5% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-322} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -2e-322) (not (<= t_0 0.0)))
     t_0
     (* (pow x 4.0) (* eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -2e-322) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-2d-322)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -2e-322) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -2e-322) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -2e-322) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -2e-322) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-322], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.97626e-322 or -0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.3%

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

   if -1.97626e-322 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 89.7%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. distribute-rgt1-in 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-322} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{{x}^{8}} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-58}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-34)
   (* (sqrt (pow x 8.0)) (* eps 5.0))
   (if (<= x 8.2e-58)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-34) {
		tmp = sqrt(pow(x, 8.0)) * (eps * 5.0);
	} else if (x <= 8.2e-58) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} 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 <= (-2.7d-34)) then
        tmp = sqrt((x ** 8.0d0)) * (eps * 5.0d0)
    else if (x <= 8.2d-58) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    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 <= -2.7e-34) {
		tmp = Math.sqrt(Math.pow(x, 8.0)) * (eps * 5.0);
	} else if (x <= 8.2e-58) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} 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 <= -2.7e-34:
		tmp = math.sqrt(math.pow(x, 8.0)) * (eps * 5.0)
	elif x <= 8.2e-58:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	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 <= -2.7e-34)
		tmp = Float64(sqrt((x ^ 8.0)) * Float64(eps * 5.0));
	elseif (x <= 8.2e-58)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	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 <= -2.7e-34)
		tmp = sqrt((x ^ 8.0)) * (eps * 5.0);
	elseif (x <= 8.2e-58)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = (x ^ 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-34], N[(N[Sqrt[N[Power[x, 8.0], $MachinePrecision]], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e-58], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $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 < -2.70000000000000017e-34

   1. Initial program 11.2%

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

   3. Taylor expanded in x around inf 11.2%

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

   4. Taylor expanded in x around inf 99.5%

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

      1. remove-double-neg 99.5%

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

      2. metadata-eval 99.5%

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

      3. associate-*r* 99.5%

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

      4. mul-1-neg 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. +-commutative 99.5%

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

      7. neg-mul-1 99.5%

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

      8. distribute-rgt-out 99.5%

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

      9. metadata-eval 99.5%

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

      10. distribute-rgt-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

   6. Simplified 99.5%

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

      1. add-sqr-sqrt 98.9%

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

      2. sqrt-unprod 99.5%

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

      3. pow-prod-up 99.7%

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

      4. metadata-eval 99.7%

         \[\leadsto \sqrt{{x}^{\color{blue}{8}}} \cdot \left(\varepsilon \cdot 5\right) \]

   8. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\sqrt{{x}^{8}}} \cdot \left(\varepsilon \cdot 5\right) \]

   if -2.70000000000000017e-34 < x < 8.20000000000000056e-58

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   if 8.20000000000000056e-58 < x

   1. Initial program 55.9%

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

   3. Taylor expanded in x around -inf 87.3%

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

      1. +-commutative 87.3%

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

      2. associate-+r+ 87.3%

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

      3. mul-1-neg 87.3%

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

      4. unsub-neg 87.3%

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

      5. distribute-rgt1-in 87.3%

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

      6. metadata-eval 87.3%

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

      7. *-commutative 87.3%

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

   5. Simplified 87.3%

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

   6. Taylor expanded in eps around 0 87.3%

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

Alternative 3: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 3.95 \cdot 10^{-57}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-34)
   (* (pow x 4.0) (* eps 5.0))
   (if (<= x 3.95e-57)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-34) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 3.95e-57) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} 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 <= (-2.7d-34)) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    else if (x <= 3.95d-57) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    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 <= -2.7e-34) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 3.95e-57) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} 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 <= -2.7e-34:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	elif x <= 3.95e-57:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	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 <= -2.7e-34)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	elseif (x <= 3.95e-57)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	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 <= -2.7e-34)
		tmp = (x ^ 4.0) * (eps * 5.0);
	elseif (x <= 3.95e-57)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = (x ^ 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-34], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.95e-57], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $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 < -2.70000000000000017e-34

   1. Initial program 11.2%

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

   3. Taylor expanded in x around inf 99.5%

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

      1. distribute-rgt1-in 99.5%

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

      2. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   if -2.70000000000000017e-34 < x < 3.9499999999999999e-57

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   if 3.9499999999999999e-57 < x

   1. Initial program 55.9%

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

   3. Taylor expanded in x around -inf 87.3%

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

      1. +-commutative 87.3%

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

      2. associate-+r+ 87.3%

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

      3. mul-1-neg 87.3%

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

      4. unsub-neg 87.3%

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

      5. distribute-rgt1-in 87.3%

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

      6. metadata-eval 87.3%

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

      7. *-commutative 87.3%

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

   5. Simplified 87.3%

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

   6. Taylor expanded in eps around 0 87.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 3.95 \cdot 10^{-57}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-57}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-34)
   (* (pow x 4.0) (* eps 5.0))
   (if (<= x 7.4e-57)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* (pow x 3.0) (* eps (+ (* x 5.0) (* eps 10.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-34) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 7.4e-57) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = pow(x, 3.0) * (eps * ((x * 5.0) + (eps * 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.7d-34)) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    else if (x <= 7.4d-57) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = (x ** 3.0d0) * (eps * ((x * 5.0d0) + (eps * 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-34) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 7.4e-57) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = Math.pow(x, 3.0) * (eps * ((x * 5.0) + (eps * 10.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.7e-34:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	elif x <= 7.4e-57:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = math.pow(x, 3.0) * (eps * ((x * 5.0) + (eps * 10.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-34)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	elseif (x <= 7.4e-57)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64((x ^ 3.0) * Float64(eps * Float64(Float64(x * 5.0) + Float64(eps * 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.7e-34)
		tmp = (x ^ 4.0) * (eps * 5.0);
	elseif (x <= 7.4e-57)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = (x ^ 3.0) * (eps * ((x * 5.0) + (eps * 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-34], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.4e-57], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 3.0], $MachinePrecision] * N[(eps * N[(N[(x * 5.0), $MachinePrecision] + N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.70000000000000017e-34

   1. Initial program 11.2%

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

   3. Taylor expanded in x around inf 99.5%

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

      1. distribute-rgt1-in 99.5%

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

      2. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   if -2.70000000000000017e-34 < x < 7.4e-57

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   if 7.4e-57 < x

   1. Initial program 55.9%

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

   3. Taylor expanded in x around -inf 87.3%

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

      1. +-commutative 87.3%

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

      2. associate-+r+ 87.3%

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

      3. mul-1-neg 87.3%

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

      4. unsub-neg 87.3%

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

      5. distribute-rgt1-in 87.3%

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

      6. metadata-eval 87.3%

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

      7. *-commutative 87.3%

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

   5. Simplified 87.3%

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

   6. Taylor expanded in x around 0 87.1%

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

   7. Taylor expanded in eps around 0 87.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-57}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34} \lor \neg \left(x \leq 2 \cdot 10^{-36}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.7e-34) (not (<= x 2e-36)))
   (* (pow x 4.0) (* eps 5.0))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-34) || !(x <= 2e-36)) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} 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 <= (-2.7d-34)) .or. (.not. (x <= 2d-36))) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    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 <= -2.7e-34) || !(x <= 2e-36)) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.7e-34) or not (x <= 2e-36):
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	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 <= -2.7e-34) || !(x <= 2e-36))
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	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 <= -2.7e-34) || ~((x <= 2e-36)))
		tmp = (x ^ 4.0) * (eps * 5.0);
	else
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.7e-34], N[Not[LessEqual[x, 2e-36]], $MachinePrecision]], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $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 < -2.70000000000000017e-34 or 1.9999999999999999e-36 < x

   1. Initial program 28.3%

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

   3. Taylor expanded in x around inf 95.3%

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

      1. distribute-rgt1-in 95.3%

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

      2. metadata-eval 95.3%

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

   5. Simplified 95.3%

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

   if -2.70000000000000017e-34 < x < 1.9999999999999999e-36

   1. Initial program 99.1%

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

   3. Taylor expanded in eps around inf 98.2%

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

      1. distribute-lft1-in 98.2%

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

      2. metadata-eval 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34} \lor \neg \left(x \leq 2 \cdot 10^{-36}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34} \lor \neg \left(x \leq 1.45 \cdot 10^{-36}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.7e-34) (not (<= x 1.45e-36)))
   (* (pow x 4.0) (* eps 5.0))
   (* (pow eps 4.0) (+ eps (* x 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-34) || !(x <= 1.45e-36)) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} 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 <= (-2.7d-34)) .or. (.not. (x <= 1.45d-36))) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    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 <= -2.7e-34) || !(x <= 1.45e-36)) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.7e-34) or not (x <= 1.45e-36):
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	else:
		tmp = math.pow(eps, 4.0) * (eps + (x * 5.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.7e-34) || !(x <= 1.45e-36))
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	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 <= -2.7e-34) || ~((x <= 1.45e-36)))
		tmp = (x ^ 4.0) * (eps * 5.0);
	else
		tmp = (eps ^ 4.0) * (eps + (x * 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.7e-34], N[Not[LessEqual[x, 1.45e-36]], $MachinePrecision]], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $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 < -2.70000000000000017e-34 or 1.45000000000000006e-36 < x

   1. Initial program 28.3%

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

   3. Taylor expanded in x around inf 95.3%

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

      1. distribute-rgt1-in 95.3%

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

      2. metadata-eval 95.3%

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

   5. Simplified 95.3%

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

   if -2.70000000000000017e-34 < x < 1.45000000000000006e-36

   1. Initial program 99.1%

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

   3. Taylor expanded in eps around inf 98.2%

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

      1. distribute-lft1-in 98.2%

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

      2. metadata-eval 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in eps around 0 98.2%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34} \lor \neg \left(x \leq 1.45 \cdot 10^{-36}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34} \lor \neg \left(x \leq 8.4 \cdot 10^{-57}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.7e-34) (not (<= x 8.4e-57)))
   (* (pow x 4.0) (* eps 5.0))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-34) || !(x <= 8.4e-57)) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.7d-34)) .or. (.not. (x <= 8.4d-57))) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-34) || !(x <= 8.4e-57)) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.7e-34) or not (x <= 8.4e-57):
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.7e-34) || !(x <= 8.4e-57))
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.7e-34) || ~((x <= 8.4e-57)))
		tmp = (x ^ 4.0) * (eps * 5.0);
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.7e-34], N[Not[LessEqual[x, 8.4e-57]], $MachinePrecision]], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.70000000000000017e-34 or 8.3999999999999998e-57 < x

   1. Initial program 44.8%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. distribute-rgt1-in 88.5%

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

      2. metadata-eval 88.5%

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

   5. Simplified 88.5%

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

   if -2.70000000000000017e-34 < x < 8.3999999999999998e-57

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34} \lor \neg \left(x \leq 8.4 \cdot 10^{-57}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34} \lor \neg \left(x \leq 1.15 \cdot 10^{-56}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.7e-34) (not (<= x 1.15e-56)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-34) || !(x <= 1.15e-56)) {
		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 <= (-2.7d-34)) .or. (.not. (x <= 1.15d-56))) 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 <= -2.7e-34) || !(x <= 1.15e-56)) {
		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 <= -2.7e-34) or not (x <= 1.15e-56):
		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 <= -2.7e-34) || !(x <= 1.15e-56))
		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 <= -2.7e-34) || ~((x <= 1.15e-56)))
		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, -2.7e-34], N[Not[LessEqual[x, 1.15e-56]], $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 < -2.70000000000000017e-34 or 1.15000000000000001e-56 < x

   1. Initial program 44.8%

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

   3. Taylor expanded in x around inf 44.7%

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

   4. Taylor expanded in x around inf 88.5%

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

      1. *-commutative 88.5%

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

      2. distribute-rgt1-in 88.5%

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

      3. metadata-eval 88.5%

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

      4. associate-*r* 88.1%

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

   6. Simplified 88.1%

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

   if -2.70000000000000017e-34 < x < 1.15000000000000001e-56

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34} \lor \neg \left(x \leq 1.15 \cdot 10^{-56}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-34}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-56}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-34)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 1.4e-56) (pow eps 5.0) (* eps (* 5.0 (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-34) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 1.4e-56) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * (5.0 * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.7d-34)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 1.4d-56) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-34) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 1.4e-56) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.7e-34:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 1.4e-56:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-34)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 1.4e-56)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.7e-34)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 1.4e-56)
		tmp = eps ^ 5.0;
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-34], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-56], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.70000000000000017e-34

   1. Initial program 11.2%

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

   3. Taylor expanded in x around inf 11.2%

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

   4. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-rgt1-in 99.5%

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

      3. metadata-eval 99.5%

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

      4. associate-*r* 99.4%

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

   6. Simplified 99.4%

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

   if -2.70000000000000017e-34 < x < 1.39999999999999997e-56

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.3%

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

   if 1.39999999999999997e-56 < x

   1. Initial program 55.9%

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

   3. Taylor expanded in x around inf 84.8%

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

      1. *-commutative 84.8%

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

      2. distribute-rgt1-in 84.8%

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

      3. metadata-eval 84.8%

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

      4. *-commutative 84.8%

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

      5. associate-*r* 84.6%

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

   5. Simplified 84.6%

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

Alternative 10: 87.5% 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 91.3%

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

3. Taylor expanded in x around 0 89.6%

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

Alternative 11: 70.6% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 91.3%

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

   1. sub-neg 91.3%

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

   2. +-commutative 91.3%

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

   3. sqr-pow 47.2%

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

   4. distribute-rgt-neg-in 47.2%

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

   5. fma-define 44.6%

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

   6. metadata-eval 44.6%

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

   7. metadata-eval 44.6%

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

4. Applied egg-rr 44.6%

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

5. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{-1 \cdot \left({x}^{5} \cdot \left(-1 \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto -1 \cdot \color{blue}{\left(\left(-1 \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot {x}^{5}\right)} \]

   2. *-commutative 0.0%

      \[\leadsto -1 \cdot \left(\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot -1} - 1\right) \cdot {x}^{5}\right) \]

   3. unpow2 0.0%

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

   4. rem-square-sqrt 74.5%

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

   5. metadata-eval 74.5%

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

   6. metadata-eval 74.5%

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

   7. mul0-lft 74.5%

      \[\leadsto -1 \cdot \color{blue}{0} \]

   8. metadata-eval 74.5%

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

7. Simplified 74.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(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)))