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

Percentage Accurate: 88.8% → 99.2%

Time: 9.2s

Alternatives: 9

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: 88.8% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-308) (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 <= -5e-308) || !(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 <= (-5d-308)) .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 <= -5e-308) || !(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 <= -5e-308) 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 <= -5e-308) || !(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 <= -5e-308) || ~((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, -5e-308], 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))) < -4.99999999999999955e-308 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.6%

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

   if -4.99999999999999955e-308 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 87.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-308} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 2.9 \cdot 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(\frac{x \cdot 5 + \frac{\left(x \cdot x\right) \cdot \left(10 - -10 \cdot \frac{x}{\varepsilon}\right)}{\varepsilon}}{\varepsilon} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.2e-64) (not (<= eps 2.9e-64)))
   (*
    (pow eps 5.0)
    (-
     (/ (+ (* x 5.0) (/ (* (* x x) (- 10.0 (* -10.0 (/ x eps)))) eps)) eps)
     -1.0))
   (* (pow x 4.0) (* eps 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.2e-64) || !(eps <= 2.9e-64)) {
		tmp = pow(eps, 5.0) * ((((x * 5.0) + (((x * x) * (10.0 - (-10.0 * (x / eps)))) / eps)) / eps) - -1.0);
	} 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) :: tmp
    if ((eps <= (-4.2d-64)) .or. (.not. (eps <= 2.9d-64))) then
        tmp = (eps ** 5.0d0) * ((((x * 5.0d0) + (((x * x) * (10.0d0 - ((-10.0d0) * (x / eps)))) / eps)) / eps) - (-1.0d0))
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.2e-64) || !(eps <= 2.9e-64)) {
		tmp = Math.pow(eps, 5.0) * ((((x * 5.0) + (((x * x) * (10.0 - (-10.0 * (x / eps)))) / eps)) / eps) - -1.0);
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.2e-64) or not (eps <= 2.9e-64):
		tmp = math.pow(eps, 5.0) * ((((x * 5.0) + (((x * x) * (10.0 - (-10.0 * (x / eps)))) / eps)) / eps) - -1.0)
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.2e-64) || !(eps <= 2.9e-64))
		tmp = Float64((eps ^ 5.0) * Float64(Float64(Float64(Float64(x * 5.0) + Float64(Float64(Float64(x * x) * Float64(10.0 - Float64(-10.0 * Float64(x / eps)))) / eps)) / eps) - -1.0));
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.2e-64) || ~((eps <= 2.9e-64)))
		tmp = (eps ^ 5.0) * ((((x * 5.0) + (((x * x) * (10.0 - (-10.0 * (x / eps)))) / eps)) / eps) - -1.0);
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.2e-64], N[Not[LessEqual[eps, 2.9e-64]], $MachinePrecision]], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(N[(N[(N[(x * 5.0), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * N[(10.0 - N[(-10.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.20000000000000023e-64 or 2.8999999999999999e-64 < eps

   1. Initial program 95.5%

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

   3. Taylor expanded in eps around -inf 93.9%

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + \left(-1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right) + -1 \cdot \frac{4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]

   5. Simplified 93.9%

      \[\leadsto \color{blue}{\left(-1 - \frac{5 \cdot x - \frac{{x}^{2} \cdot -10 - \frac{{x}^{3} \cdot 10}{\varepsilon}}{\varepsilon}}{\varepsilon}\right) \cdot \left(-{\varepsilon}^{5}\right)} \]

   6. Taylor expanded in x around 0 93.9%

      \[\leadsto \left(-1 - \frac{5 \cdot x - \frac{\color{blue}{{x}^{2} \cdot \left(-10 \cdot \frac{x}{\varepsilon} - 10\right)}}{\varepsilon}}{\varepsilon}\right) \cdot \left(-{\varepsilon}^{5}\right) \]
   7. Step-by-step derivation

      1. unpow2 93.9%

         \[\leadsto \left(-1 - \frac{5 \cdot x - \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(-10 \cdot \frac{x}{\varepsilon} - 10\right)}{\varepsilon}}{\varepsilon}\right) \cdot \left(-{\varepsilon}^{5}\right) \]

   8. Applied egg-rr 93.9%

      \[\leadsto \left(-1 - \frac{5 \cdot x - \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(-10 \cdot \frac{x}{\varepsilon} - 10\right)}{\varepsilon}}{\varepsilon}\right) \cdot \left(-{\varepsilon}^{5}\right) \]

   if -4.20000000000000023e-64 < eps < 2.8999999999999999e-64

   1. Initial program 88.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 2.9 \cdot 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(\frac{x \cdot 5 + \frac{\left(x \cdot x\right) \cdot \left(10 - -10 \cdot \frac{x}{\varepsilon}\right)}{\varepsilon}}{\varepsilon} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 1.16 \cdot 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(\frac{x \cdot \left(5 + \frac{x}{\varepsilon} \cdot 10\right)}{\varepsilon} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.7e-64) (not (<= eps 1.16e-64)))
   (* (pow eps 5.0) (- (/ (* x (+ 5.0 (* (/ x eps) 10.0))) eps) -1.0))
   (* (pow x 4.0) (* eps 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.7e-64) || !(eps <= 1.16e-64)) {
		tmp = pow(eps, 5.0) * (((x * (5.0 + ((x / eps) * 10.0))) / eps) - -1.0);
	} 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) :: tmp
    if ((eps <= (-1.7d-64)) .or. (.not. (eps <= 1.16d-64))) then
        tmp = (eps ** 5.0d0) * (((x * (5.0d0 + ((x / eps) * 10.0d0))) / eps) - (-1.0d0))
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.7e-64) || !(eps <= 1.16e-64)) {
		tmp = Math.pow(eps, 5.0) * (((x * (5.0 + ((x / eps) * 10.0))) / eps) - -1.0);
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.7e-64) or not (eps <= 1.16e-64):
		tmp = math.pow(eps, 5.0) * (((x * (5.0 + ((x / eps) * 10.0))) / eps) - -1.0)
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.7e-64) || !(eps <= 1.16e-64))
		tmp = Float64((eps ^ 5.0) * Float64(Float64(Float64(x * Float64(5.0 + Float64(Float64(x / eps) * 10.0))) / eps) - -1.0));
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.7e-64) || ~((eps <= 1.16e-64)))
		tmp = (eps ^ 5.0) * (((x * (5.0 + ((x / eps) * 10.0))) / eps) - -1.0);
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.7e-64], N[Not[LessEqual[eps, 1.16e-64]], $MachinePrecision]], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(N[(N[(x * N[(5.0 + N[(N[(x / eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.70000000000000006e-64 or 1.15999999999999992e-64 < eps

   1. Initial program 95.5%

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

   3. Taylor expanded in eps around -inf 93.9%

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + \left(-1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right) + -1 \cdot \frac{4 \cdot {x}^{3} + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]

   5. Simplified 93.9%

      \[\leadsto \color{blue}{\left(-1 - \frac{5 \cdot x - \frac{{x}^{2} \cdot -10 - \frac{{x}^{3} \cdot 10}{\varepsilon}}{\varepsilon}}{\varepsilon}\right) \cdot \left(-{\varepsilon}^{5}\right)} \]

   6. Taylor expanded in x around 0 93.9%

      \[\leadsto \left(-1 - \frac{5 \cdot x - \frac{\color{blue}{{x}^{2} \cdot \left(-10 \cdot \frac{x}{\varepsilon} - 10\right)}}{\varepsilon}}{\varepsilon}\right) \cdot \left(-{\varepsilon}^{5}\right) \]

   7. Taylor expanded in x around 0 93.3%

      \[\leadsto \left(-1 - \frac{\color{blue}{x \cdot \left(5 + 10 \cdot \frac{x}{\varepsilon}\right)}}{\varepsilon}\right) \cdot \left(-{\varepsilon}^{5}\right) \]
   8. Step-by-step derivation

      1. *-commutative 93.3%

         \[\leadsto \left(-1 - \frac{x \cdot \left(5 + \color{blue}{\frac{x}{\varepsilon} \cdot 10}\right)}{\varepsilon}\right) \cdot \left(-{\varepsilon}^{5}\right) \]

   9. Simplified 93.3%

      \[\leadsto \left(-1 - \frac{\color{blue}{x \cdot \left(5 + \frac{x}{\varepsilon} \cdot 10\right)}}{\varepsilon}\right) \cdot \left(-{\varepsilon}^{5}\right) \]

   if -1.70000000000000006e-64 < eps < 1.15999999999999992e-64

   1. Initial program 88.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 1.16 \cdot 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(\frac{x \cdot \left(5 + \frac{x}{\varepsilon} \cdot 10\right)}{\varepsilon} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 3.7 \cdot 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -5e-64) (not (<= eps 3.7e-64)))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
   (* (pow x 4.0) (* eps 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -5e-64) || !(eps <= 3.7e-64)) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} 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) :: tmp
    if ((eps <= (-5d-64)) .or. (.not. (eps <= 3.7d-64))) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -5e-64) || !(eps <= 3.7e-64)) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -5e-64) or not (eps <= 3.7e-64):
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -5e-64) || !(eps <= 3.7e-64))
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -5e-64) || ~((eps <= 3.7e-64)))
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -5e-64], N[Not[LessEqual[eps, 3.7e-64]], $MachinePrecision]], 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 * 5.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.00000000000000033e-64 or 3.69999999999999999e-64 < eps

   1. Initial program 95.5%

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

   3. Taylor expanded in eps around inf 92.1%

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

      1. distribute-lft1-in 92.1%

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

      2. metadata-eval 92.1%

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

   5. Simplified 92.1%

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

   if -5.00000000000000033e-64 < eps < 3.69999999999999999e-64

   1. Initial program 88.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 3.7 \cdot 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-65} \lor \neg \left(\varepsilon \leq 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -5e-65) (not (<= eps 1e-64)))
   (* (pow eps 4.0) (+ eps (* x 5.0)))
   (* (pow x 4.0) (* eps 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -5e-65) || !(eps <= 1e-64)) {
		tmp = pow(eps, 4.0) * (eps + (x * 5.0));
	} 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) :: tmp
    if ((eps <= (-5d-65)) .or. (.not. (eps <= 1d-64))) then
        tmp = (eps ** 4.0d0) * (eps + (x * 5.0d0))
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -5e-65) || !(eps <= 1e-64)) {
		tmp = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -5e-65) or not (eps <= 1e-64):
		tmp = math.pow(eps, 4.0) * (eps + (x * 5.0))
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -5e-65) || !(eps <= 1e-64))
		tmp = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)));
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -5e-65) || ~((eps <= 1e-64)))
		tmp = (eps ^ 4.0) * (eps + (x * 5.0));
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -5e-65], N[Not[LessEqual[eps, 1e-64]], $MachinePrecision]], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.99999999999999983e-65 or 9.99999999999999965e-65 < eps

   1. Initial program 95.5%

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

   3. Taylor expanded in eps around inf 92.1%

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

      1. distribute-lft1-in 92.1%

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

      2. metadata-eval 92.1%

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

   5. Simplified 92.1%

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

   6. Taylor expanded in eps around 0 91.7%

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

   if -4.99999999999999983e-65 < eps < 9.99999999999999965e-65

   1. Initial program 88.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-65} \lor \neg \left(\varepsilon \leq 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-50} \lor \neg \left(x \leq 2.4 \cdot 10^{-51}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5.5e-50) (not (<= x 2.4e-51)))
   (* eps (* 5.0 (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5.5e-50) || !(x <= 2.4e-51)) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-5.5d-50)) .or. (.not. (x <= 2.4d-51))) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -5.5e-50) || !(x <= 2.4e-51)) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -5.5e-50) or not (x <= 2.4e-51):
		tmp = eps * (5.0 * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5.5e-50) || !(x <= 2.4e-51))
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -5.5e-50) || ~((x <= 2.4e-51)))
		tmp = eps * (5.0 * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5.5e-50], N[Not[LessEqual[x, 2.4e-51]], $MachinePrecision]], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.49999999999999975e-50 or 2.4e-51 < x

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf 92.6%

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

      1. *-commutative 92.6%

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

      2. distribute-rgt1-in 92.6%

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

      3. metadata-eval 92.6%

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

      4. *-commutative 92.6%

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

      5. associate-*r* 92.6%

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

   5. Simplified 92.6%

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

   if -5.49999999999999975e-50 < x < 2.4e-51

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-50} \lor \neg \left(x \leq 2.4 \cdot 10^{-51}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-53} \lor \neg \left(x \leq 9.2 \cdot 10^{-51}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5.5e-53) (not (<= x 9.2e-51)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5.5e-53) || !(x <= 9.2e-51)) {
		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 <= (-5.5d-53)) .or. (.not. (x <= 9.2d-51))) 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 <= -5.5e-53) || !(x <= 9.2e-51)) {
		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 <= -5.5e-53) or not (x <= 9.2e-51):
		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 <= -5.5e-53) || !(x <= 9.2e-51))
		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 <= -5.5e-53) || ~((x <= 9.2e-51)))
		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, -5.5e-53], N[Not[LessEqual[x, 9.2e-51]], $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 < -5.50000000000000023e-53 or 9.20000000000000007e-51 < x

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf 92.6%

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

      1. *-commutative 92.6%

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

      2. distribute-rgt1-in 92.6%

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

      3. metadata-eval 92.6%

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

      4. *-commutative 92.6%

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

      5. associate-*r* 92.6%

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

   5. Simplified 92.6%

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

   6. Taylor expanded in eps around 0 92.3%

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

   if -5.50000000000000023e-53 < x < 9.20000000000000007e-51

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-53} \lor \neg \left(x \leq 9.2 \cdot 10^{-51}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-51}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-48}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.8e-51)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 1.45e-48) (pow eps 5.0) (* (pow x 4.0) (* eps 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.8e-51) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 1.45e-48) {
		tmp = pow(eps, 5.0);
	} 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) :: tmp
    if (x <= (-1.8d-51)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 1.45d-48) then
        tmp = eps ** 5.0d0
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.8e-51) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 1.45e-48) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.8e-51:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 1.45e-48:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.8e-51)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 1.45e-48)
		tmp = eps ^ 5.0;
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.8e-51)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 1.45e-48)
		tmp = eps ^ 5.0;
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.8e-51], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-48], N[Power[eps, 5.0], $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8e-51

   1. Initial program 37.0%

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

   3. Taylor expanded in x around inf 89.5%

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

      1. *-commutative 89.5%

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

      2. distribute-rgt1-in 89.5%

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

      3. metadata-eval 89.5%

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

      4. *-commutative 89.5%

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

      5. associate-*r* 89.6%

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

   5. Simplified 89.6%

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

   if -1.8e-51 < x < 1.4500000000000001e-48

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   if 1.4500000000000001e-48 < x

   1. Initial program 51.5%

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

   3. Taylor expanded in x around inf 95.9%

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

      1. distribute-rgt1-in 95.9%

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

      2. metadata-eval 95.9%

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

   5. Simplified 95.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-51}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-48}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.6% 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 89.5%

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

3. Taylor expanded in x around 0 88.8%

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

Reproduce

herbie shell --seed 2024146 
(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)))