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

Percentage Accurate: 88.6% → 99.9%

Time: 11.2s

Alternatives: 6

Speedup: 1.7×

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: 88.6% 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: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\varepsilon \cdot \left(5 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right)\right)\right)\right) + {\varepsilon}^{5} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (*
   x
   (*
    eps
    (+
     (* 5.0 (pow x 3.0))
     (* eps (+ (* 10.0 (* x x)) (* eps (+ (* eps 5.0) (* x 10.0))))))))
  (pow eps 5.0)))

C

double code(double x, double eps) {
	return (x * (eps * ((5.0 * pow(x, 3.0)) + (eps * ((10.0 * (x * x)) + (eps * ((eps * 5.0) + (x * 10.0)))))))) + pow(eps, 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 ** 3.0d0)) + (eps * ((10.0d0 * (x * x)) + (eps * ((eps * 5.0d0) + (x * 10.0d0)))))))) + (eps ** 5.0d0)
end function

Java

public static double code(double x, double eps) {
	return (x * (eps * ((5.0 * Math.pow(x, 3.0)) + (eps * ((10.0 * (x * x)) + (eps * ((eps * 5.0) + (x * 10.0)))))))) + Math.pow(eps, 5.0);
}

Python

def code(x, eps):
	return (x * (eps * ((5.0 * math.pow(x, 3.0)) + (eps * ((10.0 * (x * x)) + (eps * ((eps * 5.0) + (x * 10.0)))))))) + math.pow(eps, 5.0)

Julia

function code(x, eps)
	return Float64(Float64(x * Float64(eps * Float64(Float64(5.0 * (x ^ 3.0)) + Float64(eps * Float64(Float64(10.0 * Float64(x * x)) + Float64(eps * Float64(Float64(eps * 5.0) + Float64(x * 10.0)))))))) + (eps ^ 5.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = (x * (eps * ((5.0 * (x ^ 3.0)) + (eps * ((10.0 * (x * x)) + (eps * ((eps * 5.0) + (x * 10.0)))))))) + (eps ^ 5.0);
end

Wolfram

code[x_, eps_] := N[(N[(x * N[(eps * N[(N[(5.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(eps * 5.0), $MachinePrecision] + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in x around 0 99.9%

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

4. Taylor expanded in eps around 0 99.9%

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

   1. unpow2 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto x \cdot \left(\varepsilon \cdot \left(5 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot \left(x \cdot x\right) + \varepsilon \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right)\right)\right)\right) + {\varepsilon}^{5} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-313} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-313) (not (<= t_0 0.0)))
     t_0
     (* eps (* (pow x 4.0) (+ 5.0 (* 10.0 (/ eps x))))))))

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-313) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-5d-313)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    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 t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-313) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * (Math.pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	}
	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-313) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = eps * (math.pow(x, 4.0) * (5.0 + (10.0 * (eps / x))))
	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-313) || !(t_0 <= 0.0))
		tmp = t_0;
	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)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -5e-313) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = eps * ((x ^ 4.0) * (5.0 + (10.0 * (eps / x))));
	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-313], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, 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 (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -5.00000000002e-313 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 99.2%

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

   if -5.00000000002e-313 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.5%

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

   3. Taylor expanded in eps around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(4, {x}^{3}, \mathsf{fma}\left(\varepsilon, {x}^{2} \cdot 10, {x}^{3} \cdot 6\right)\right) + 5 \cdot {x}^{4}\right)} \]

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-313} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ {\varepsilon}^{5} + x \cdot \left(\varepsilon \cdot \left(5 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (pow eps 5.0)
  (* x (* eps (+ (* 5.0 (pow x 3.0)) (* eps (* 10.0 (* x (+ x eps)))))))))

C

double code(double x, double eps) {
	return pow(eps, 5.0) + (x * (eps * ((5.0 * pow(x, 3.0)) + (eps * (10.0 * (x * (x + eps)))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps ** 5.0d0) + (x * (eps * ((5.0d0 * (x ** 3.0d0)) + (eps * (10.0d0 * (x * (x + eps)))))))
end function

Java

public static double code(double x, double eps) {
	return Math.pow(eps, 5.0) + (x * (eps * ((5.0 * Math.pow(x, 3.0)) + (eps * (10.0 * (x * (x + eps)))))));
}

Python

def code(x, eps):
	return math.pow(eps, 5.0) + (x * (eps * ((5.0 * math.pow(x, 3.0)) + (eps * (10.0 * (x * (x + eps)))))))

Julia

function code(x, eps)
	return Float64((eps ^ 5.0) + Float64(x * Float64(eps * Float64(Float64(5.0 * (x ^ 3.0)) + Float64(eps * Float64(10.0 * Float64(x * Float64(x + eps))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps ^ 5.0) + (x * (eps * ((5.0 * (x ^ 3.0)) + (eps * (10.0 * (x * (x + eps)))))));
end

Wolfram

code[x_, eps_] := N[(N[Power[eps, 5.0], $MachinePrecision] + N[(x * N[(eps * N[(N[(5.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(10.0 * N[(x * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in x around 0 99.9%

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

4. Taylor expanded in eps around 0 99.9%

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

5. Taylor expanded in eps around 0 99.1%

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

   1. distribute-lft-out 99.1%

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

   2. unpow2 99.1%

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

   3. distribute-rgt-in 99.1%

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

   4. +-commutative 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

   \[\leadsto {\varepsilon}^{5} + x \cdot \left(\varepsilon \cdot \left(5 \cdot {x}^{3} + \varepsilon \cdot \left(10 \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 4: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-63} \lor \neg \left(\varepsilon \leq 6.4 \cdot 10^{-65}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.2e-63) (not (<= eps 6.4e-65)))
   (* (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 <= -3.2e-63) || !(eps <= 6.4e-65)) {
		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 <= (-3.2d-63)) .or. (.not. (eps <= 6.4d-65))) 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 <= -3.2e-63) || !(eps <= 6.4e-65)) {
		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 <= -3.2e-63) or not (eps <= 6.4e-65):
		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 <= -3.2e-63) || !(eps <= 6.4e-65))
		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 <= -3.2e-63) || ~((eps <= 6.4e-65)))
		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, -3.2e-63], N[Not[LessEqual[eps, 6.4e-65]], $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 < -3.19999999999999989e-63 or 6.3999999999999998e-65 < eps

   1. Initial program 96.1%

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

   3. Taylor expanded in eps around inf 92.4%

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

      1. distribute-lft1-in 92.4%

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

      2. metadata-eval 92.4%

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

   5. Simplified 92.4%

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

   if -3.19999999999999989e-63 < eps < 6.3999999999999998e-65

   1. Initial program 89.4%

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

   3. Taylor expanded in eps around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(4, {x}^{3}, \mathsf{fma}\left(\varepsilon, {x}^{2} \cdot 10, {x}^{3} \cdot 6\right)\right) + 5 \cdot {x}^{4}\right)} \]

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-63} \lor \neg \left(\varepsilon \leq 6.4 \cdot 10^{-65}\right):\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps)))))

C

double code(double x, double eps) {
	return pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
end function

Java

public static double code(double x, double eps) {
	return Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
}

Python

def code(x, eps):
	return math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))

Julia

function code(x, eps)
	return Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
end

Wolfram

code[x_, eps_] := N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in eps around inf 90.1%

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

   1. distribute-lft1-in 90.1%

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

   2. metadata-eval 90.1%

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

5. Simplified 90.1%

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

Alternative 6: 87.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (pow eps 4.0) (+ eps (* x 5.0))))

C

double code(double x, double eps) {
	return pow(eps, 4.0) * (eps + (x * 5.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps ** 4.0d0) * (eps + (x * 5.0d0))
end function

Java

public static double code(double x, double eps) {
	return Math.pow(eps, 4.0) * (eps + (x * 5.0));
}

Python

def code(x, eps):
	return math.pow(eps, 4.0) * (eps + (x * 5.0))

Julia

function code(x, eps)
	return Float64((eps ^ 4.0) * Float64(eps + Float64(x * 5.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps ^ 4.0) * (eps + (x * 5.0));
end

Wolfram

code[x_, eps_] := N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in eps around inf 90.1%

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

   1. distribute-lft1-in 90.1%

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

   2. metadata-eval 90.1%

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

5. Simplified 90.1%

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

6. Taylor expanded in eps around 0 90.0%

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

   1. *-commutative 90.0%

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

8. Simplified 90.0%

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

Reproduce

herbie shell --seed 2024179 
(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)))