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

Percentage Accurate: 87.5% → 99.0%

Time: 9.4s

Alternatives: 10

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: 87.5% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-295} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-295) (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 <= -1e-295) || !(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 <= (-1d-295)) .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 <= -1e-295) || !(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 <= -1e-295) 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 <= -1e-295) || !(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 <= -1e-295) || ~((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, -1e-295], 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))) < -1.00000000000000006e-295 or -0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 98.4%

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

   if -1.00000000000000006e-295 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 88.0%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. distribute-rgt1-in 99.9%

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

      2. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-295} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 2.8 \cdot 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \left(x \cdot x\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.65e-62) (not (<= eps 2.8e-64)))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
   (* eps (+ (* 5.0 (pow x 4.0)) (* (* x x) (* 10.0 (* eps (+ x eps))))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.65e-62) || !(eps <= 2.8e-64)) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * ((5.0 * pow(x, 4.0)) + ((x * x) * (10.0 * (eps * (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 ((eps <= (-3.65d-62)) .or. (.not. (eps <= 2.8d-64))) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = eps * ((5.0d0 * (x ** 4.0d0)) + ((x * x) * (10.0d0 * (eps * (x + eps)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.65e-62) || !(eps <= 2.8e-64)) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * ((5.0 * Math.pow(x, 4.0)) + ((x * x) * (10.0 * (eps * (x + eps)))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.65e-62) or not (eps <= 2.8e-64):
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = eps * ((5.0 * math.pow(x, 4.0)) + ((x * x) * (10.0 * (eps * (x + eps)))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.65e-62) || !(eps <= 2.8e-64))
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(Float64(x * x) * Float64(10.0 * Float64(eps * Float64(x + eps))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.65e-62) || ~((eps <= 2.8e-64)))
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = eps * ((5.0 * (x ^ 4.0)) + ((x * x) * (10.0 * (eps * (x + eps)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.65e-62], N[Not[LessEqual[eps, 2.8e-64]], $MachinePrecision]], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(10.0 * N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.6499999999999999e-62 or 2.80000000000000004e-64 < eps

   1. Initial program 95.4%

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

   3. Taylor expanded in eps around inf 94.5%

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

      1. distribute-lft1-in 94.5%

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

      2. metadata-eval 94.5%

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

   5. Simplified 94.5%

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

   if -3.6499999999999999e-62 < eps < 2.80000000000000004e-64

   1. Initial program 88.7%

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

   3. Taylor expanded in eps around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r+ 99.9%

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

      3. distribute-lft1-in 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. unpow2 99.9%

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

      3. distribute-lft-out 99.9%

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

      4. +-commutative 99.9%

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

   8. Simplified 99.9%

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

      1. unpow2 99.9%

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

   10. Applied egg-rr 99.9%

       \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \color{blue}{\left(x \cdot x\right)} \cdot \left(10 \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 2.8 \cdot 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \left(x \cdot x\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 8.2 \cdot 10^{-65}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.65e-62) (not (<= eps 8.2e-65)))
   (* (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 ((eps <= -3.65e-62) || !(eps <= 8.2e-65)) {
		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 ((eps <= (-3.65d-62)) .or. (.not. (eps <= 8.2d-65))) 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 ((eps <= -3.65e-62) || !(eps <= 8.2e-65)) {
		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 (eps <= -3.65e-62) or not (eps <= 8.2e-65):
		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 ((eps <= -3.65e-62) || !(eps <= 8.2e-65))
		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 ((eps <= -3.65e-62) || ~((eps <= 8.2e-65)))
		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[Or[LessEqual[eps, -3.65e-62], N[Not[LessEqual[eps, 8.2e-65]], $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 * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.6499999999999999e-62 or 8.19999999999999975e-65 < eps

   1. Initial program 95.4%

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

   3. Taylor expanded in eps around inf 94.5%

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

      1. distribute-lft1-in 94.5%

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

      2. metadata-eval 94.5%

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

   5. Simplified 94.5%

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

   if -3.6499999999999999e-62 < eps < 8.19999999999999975e-65

   1. Initial program 88.7%

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

   3. Taylor expanded in x around -inf 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r+ 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

      5. distribute-rgt1-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in eps around 0 99.9%

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

      1. *-commutative 99.9%

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

   8. Simplified 99.9%

      \[\leadsto {x}^{4} \cdot \color{blue}{\left(\varepsilon \cdot \left(5 + \frac{\varepsilon}{x} \cdot 10\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 8.2 \cdot 10^{-65}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 1.35 \cdot 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.65e-62) (not (<= eps 1.35e-64)))
   (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
   (* eps (* (pow x 4.0) (+ 5.0 (* 10.0 (/ eps x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.65e-62) || !(eps <= 1.35e-64)) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * (pow(x, 4.0) * (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 ((eps <= (-3.65d-62)) .or. (.not. (eps <= 1.35d-64))) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else
        tmp = eps * ((x ** 4.0d0) * (5.0d0 + (10.0d0 * (eps / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.65e-62) || !(eps <= 1.35e-64)) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = eps * (Math.pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.65e-62) or not (eps <= 1.35e-64):
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = eps * (math.pow(x, 4.0) * (5.0 + (10.0 * (eps / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.65e-62) || !(eps <= 1.35e-64))
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	else
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.65e-62) || ~((eps <= 1.35e-64)))
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = eps * ((x ^ 4.0) * (5.0 + (10.0 * (eps / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.65e-62], N[Not[LessEqual[eps, 1.35e-64]], $MachinePrecision]], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.6499999999999999e-62 or 1.34999999999999993e-64 < eps

   1. Initial program 95.4%

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

   3. Taylor expanded in eps around inf 94.5%

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

      1. distribute-lft1-in 94.5%

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

      2. metadata-eval 94.5%

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

   5. Simplified 94.5%

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

   if -3.6499999999999999e-62 < eps < 1.34999999999999993e-64

   1. Initial program 88.7%

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

   3. Taylor expanded in eps around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r+ 99.9%

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

      3. distribute-lft1-in 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 99.9%

      \[\leadsto \varepsilon \cdot \color{blue}{\left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 1.35 \cdot 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 6.5 \cdot 10^{-65}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.65e-62) (not (<= eps 6.5e-65)))
   (* (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 <= -3.65e-62) || !(eps <= 6.5e-65)) {
		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 <= (-3.65d-62)) .or. (.not. (eps <= 6.5d-65))) 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 <= -3.65e-62) || !(eps <= 6.5e-65)) {
		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 <= -3.65e-62) or not (eps <= 6.5e-65):
		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 <= -3.65e-62) || !(eps <= 6.5e-65))
		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 <= -3.65e-62) || ~((eps <= 6.5e-65)))
		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, -3.65e-62], N[Not[LessEqual[eps, 6.5e-65]], $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 < -3.6499999999999999e-62 or 6.5e-65 < eps

   1. Initial program 95.4%

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

   3. Taylor expanded in eps around inf 94.5%

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

      1. distribute-lft1-in 94.5%

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

      2. metadata-eval 94.5%

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

   5. Simplified 94.5%

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

   if -3.6499999999999999e-62 < eps < 6.5e-65

   1. Initial program 88.7%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. distribute-rgt1-in 99.7%

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

      2. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 6.5 \cdot 10^{-65}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 1.25 \cdot 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.65e-62) (not (<= eps 1.25e-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 <= -3.65e-62) || !(eps <= 1.25e-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 <= (-3.65d-62)) .or. (.not. (eps <= 1.25d-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 <= -3.65e-62) || !(eps <= 1.25e-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 <= -3.65e-62) or not (eps <= 1.25e-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 <= -3.65e-62) || !(eps <= 1.25e-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 <= -3.65e-62) || ~((eps <= 1.25e-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, -3.65e-62], N[Not[LessEqual[eps, 1.25e-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 < -3.6499999999999999e-62 or 1.25000000000000008e-64 < eps

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0 94.5%

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

      1. fma-define 94.5%

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

      2. distribute-lft1-in 94.5%

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

      3. metadata-eval 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in eps around 0 94.2%

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

   if -3.6499999999999999e-62 < eps < 1.25000000000000008e-64

   1. Initial program 88.7%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. distribute-rgt1-in 99.7%

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

      2. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 1.25 \cdot 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-65}\right):\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.65e-62) (not (<= eps 3.8e-65)))
   (pow eps 5.0)
   (* (pow x 4.0) (* eps 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.65e-62) || !(eps <= 3.8e-65)) {
		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 ((eps <= (-3.65d-62)) .or. (.not. (eps <= 3.8d-65))) 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 ((eps <= -3.65e-62) || !(eps <= 3.8e-65)) {
		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 (eps <= -3.65e-62) or not (eps <= 3.8e-65):
		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 ((eps <= -3.65e-62) || !(eps <= 3.8e-65))
		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 ((eps <= -3.65e-62) || ~((eps <= 3.8e-65)))
		tmp = eps ^ 5.0;
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.65e-62], N[Not[LessEqual[eps, 3.8e-65]], $MachinePrecision]], N[Power[eps, 5.0], $MachinePrecision], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.6499999999999999e-62 or 3.8000000000000002e-65 < eps

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0 92.5%

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

   if -3.6499999999999999e-62 < eps < 3.8000000000000002e-65

   1. Initial program 88.7%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. distribute-rgt1-in 99.7%

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

      2. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-65}\right):\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 7.5 \cdot 10^{-65}\right):\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.65e-62) (not (<= eps 7.5e-65)))
   (pow eps 5.0)
   (* eps (* 5.0 (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.65e-62) || !(eps <= 7.5e-65)) {
		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 ((eps <= (-3.65d-62)) .or. (.not. (eps <= 7.5d-65))) 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 ((eps <= -3.65e-62) || !(eps <= 7.5e-65)) {
		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 (eps <= -3.65e-62) or not (eps <= 7.5e-65):
		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 ((eps <= -3.65e-62) || !(eps <= 7.5e-65))
		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 ((eps <= -3.65e-62) || ~((eps <= 7.5e-65)))
		tmp = eps ^ 5.0;
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.65e-62], N[Not[LessEqual[eps, 7.5e-65]], $MachinePrecision]], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.6499999999999999e-62 or 7.5000000000000002e-65 < eps

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0 92.5%

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

   if -3.6499999999999999e-62 < eps < 7.5000000000000002e-65

   1. Initial program 88.7%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-rgt1-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 7.5 \cdot 10^{-65}\right):\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.65e-62) (not (<= eps 1e-64)))
   (pow eps 5.0)
   (* 5.0 (* eps (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.65e-62) || !(eps <= 1e-64)) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * 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 ((eps <= (-3.65d-62)) .or. (.not. (eps <= 1d-64))) then
        tmp = eps ** 5.0d0
    else
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.65e-62) || !(eps <= 1e-64)) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.65e-62) or not (eps <= 1e-64):
		tmp = math.pow(eps, 5.0)
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.65e-62) || !(eps <= 1e-64))
		tmp = eps ^ 5.0;
	else
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.65e-62) || ~((eps <= 1e-64)))
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.65e-62], N[Not[LessEqual[eps, 1e-64]], $MachinePrecision]], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.6499999999999999e-62 or 9.99999999999999965e-65 < eps

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0 92.5%

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

   if -3.6499999999999999e-62 < eps < 9.99999999999999965e-65

   1. Initial program 88.7%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. distribute-rgt1-in 99.7%

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

      2. metadata-eval 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 99.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 10^{-64}\right):\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.6% 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 90.1%

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

3. Taylor expanded in x around 0 89.3%

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

Reproduce

herbie shell --seed 2024133 
(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)))