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

Percentage Accurate: 88.0% → 99.9%

Time: 13.2s

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: 88.0% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 5 \cdot {\varepsilon}^{4} + x \cdot \mathsf{fma}\left(4, {\varepsilon}^{3}, \mathsf{fma}\left(x, \mathsf{fma}\left({\varepsilon}^{2}, 10, \varepsilon \cdot \left(x \cdot 5\right)\right), {\varepsilon}^{3} \cdot 6\right)\right), {\varepsilon}^{5}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  x
  (+
   (* 5.0 (pow eps 4.0))
   (*
    x
    (fma
     4.0
     (pow eps 3.0)
     (fma
      x
      (fma (pow eps 2.0) 10.0 (* eps (* x 5.0)))
      (* (pow eps 3.0) 6.0)))))
  (pow eps 5.0)))

C

double code(double x, double eps) {
	return fma(x, ((5.0 * pow(eps, 4.0)) + (x * fma(4.0, pow(eps, 3.0), fma(x, fma(pow(eps, 2.0), 10.0, (eps * (x * 5.0))), (pow(eps, 3.0) * 6.0))))), pow(eps, 5.0));
}

Julia

function code(x, eps)
	return fma(x, Float64(Float64(5.0 * (eps ^ 4.0)) + Float64(x * fma(4.0, (eps ^ 3.0), fma(x, fma((eps ^ 2.0), 10.0, Float64(eps * Float64(x * 5.0))), Float64((eps ^ 3.0) * 6.0))))), (eps ^ 5.0))
end

Wolfram

code[x_, eps_] := N[(x * N[(N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(4.0 * N[Power[eps, 3.0], $MachinePrecision] + N[(x * N[(N[Power[eps, 2.0], $MachinePrecision] * 10.0 + N[(eps * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in x around 0 99.9%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 5 \cdot {\varepsilon}^{4} + x \cdot \left(\varepsilon \cdot \left(5 \cdot {x}^{2} + \left(\varepsilon \cdot 10\right) \cdot \left(x + \varepsilon\right)\right)\right), {\varepsilon}^{5}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  x
  (+
   (* 5.0 (pow eps 4.0))
   (* x (* eps (+ (* 5.0 (pow x 2.0)) (* (* eps 10.0) (+ x eps))))))
  (pow eps 5.0)))

C

double code(double x, double eps) {
	return fma(x, ((5.0 * pow(eps, 4.0)) + (x * (eps * ((5.0 * pow(x, 2.0)) + ((eps * 10.0) * (x + eps)))))), pow(eps, 5.0));
}

Julia

function code(x, eps)
	return fma(x, Float64(Float64(5.0 * (eps ^ 4.0)) + Float64(x * Float64(eps * Float64(Float64(5.0 * (x ^ 2.0)) + Float64(Float64(eps * 10.0) * Float64(x + eps)))))), (eps ^ 5.0))
end

Wolfram

code[x_, eps_] := N[(x * N[(N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(eps * N[(N[(5.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(eps * 10.0), $MachinePrecision] * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in x around 0 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eps around 0 99.9%

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

7. Taylor expanded in x around 0 99.9%

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

   1. distribute-lft-out 99.9%

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

   2. unpow2 99.9%

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

   3. distribute-lft-out 99.9%

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

   4. +-commutative 99.9%

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

   5. associate-*l* 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

    \[\leadsto \mathsf{fma}\left(x, 5 \cdot {\varepsilon}^{4} + x \cdot \left(\varepsilon \cdot \left(5 \cdot {x}^{2} + \left(\varepsilon \cdot 10\right) \cdot \left(x + \varepsilon\right)\right)\right), {\varepsilon}^{5}\right) \]
11. Add Preprocessing

Alternative 3: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 5 \cdot {\varepsilon}^{4} + x \cdot \left(5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right), {\varepsilon}^{5}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  x
  (+ (* 5.0 (pow eps 4.0)) (* x (* 5.0 (* eps (pow x 2.0)))))
  (pow eps 5.0)))

C

double code(double x, double eps) {
	return fma(x, ((5.0 * pow(eps, 4.0)) + (x * (5.0 * (eps * pow(x, 2.0))))), pow(eps, 5.0));
}

Julia

function code(x, eps)
	return fma(x, Float64(Float64(5.0 * (eps ^ 4.0)) + Float64(x * Float64(5.0 * Float64(eps * (x ^ 2.0))))), (eps ^ 5.0))
end

Wolfram

code[x_, eps_] := N[(x * N[(N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(5.0 * N[(eps * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in x around 0 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eps around 0 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(x, 5 \cdot {\varepsilon}^{4} + x \cdot \left(5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right), {\varepsilon}^{5}\right) \]
8. Add Preprocessing

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.5e-38)
   (* (pow x 4.0) (* eps (+ 5.0 (/ (* eps 10.0) x))))
   (if (<= x 1.35e-66)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.5e-38) {
		tmp = pow(x, 4.0) * (eps * (5.0 + ((eps * 10.0) / x)));
	} else if (x <= 1.35e-66) {
		tmp = pow((x + eps), 5.0) - pow(x, 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 (x <= (-2.5d-38)) then
        tmp = (x ** 4.0d0) * (eps * (5.0d0 + ((eps * 10.0d0) / x)))
    else if (x <= 1.35d-66) then
        tmp = ((x + eps) ** 5.0d0) - (x ** 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 (x <= -2.5e-38) {
		tmp = Math.pow(x, 4.0) * (eps * (5.0 + ((eps * 10.0) / x)));
	} else if (x <= 1.35e-66) {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.5e-38:
		tmp = math.pow(x, 4.0) * (eps * (5.0 + ((eps * 10.0) / x)))
	elif x <= 1.35e-66:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.5e-38)
		tmp = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(Float64(eps * 10.0) / x))));
	elseif (x <= 1.35e-66)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 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 (x <= -2.5e-38)
		tmp = (x ^ 4.0) * (eps * (5.0 + ((eps * 10.0) / x)));
	elseif (x <= 1.35e-66)
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.5e-38], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(N[(eps * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e-66], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000017e-38

   1. Initial program 10.5%

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

   3. Taylor expanded in x around inf 99.6%

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

   4. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-*r/ 99.6%

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

   6. Simplified 99.6%

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

   if -2.50000000000000017e-38 < x < 1.34999999999999998e-66

   1. Initial program 100.0%

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

   if 1.34999999999999998e-66 < x

   1. Initial program 48.5%

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

   3. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-rgt1-in 99.5%

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

      3. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in eps around 0 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.5e-38)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 1.35e-66)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.5e-38) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 1.35e-66) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} 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 (x <= (-2.5d-38)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 1.35d-66) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    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 (x <= -2.5e-38) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 1.35e-66) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.5e-38:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 1.35e-66:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.5e-38)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 1.35e-66)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	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 (x <= -2.5e-38)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 1.35e-66)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.5e-38], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e-66], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000017e-38

   1. Initial program 10.5%

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

   3. Taylor expanded in x around inf 98.2%

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

      1. *-commutative 98.2%

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

      2. distribute-rgt1-in 98.2%

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

      3. metadata-eval 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-*r* 98.3%

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

   5. Simplified 98.3%

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

   if -2.50000000000000017e-38 < x < 1.34999999999999998e-66

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   if 1.34999999999999998e-66 < x

   1. Initial program 48.5%

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

   3. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-rgt1-in 99.5%

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

      3. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in eps around 0 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.5e-38)
   (* (pow x 4.0) (* eps (+ 5.0 (/ (* eps 10.0) x))))
   (if (<= x 1.35e-66)
     (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
     (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.5e-38) {
		tmp = pow(x, 4.0) * (eps * (5.0 + ((eps * 10.0) / x)));
	} else if (x <= 1.35e-66) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} 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 (x <= (-2.5d-38)) then
        tmp = (x ** 4.0d0) * (eps * (5.0d0 + ((eps * 10.0d0) / x)))
    else if (x <= 1.35d-66) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    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 (x <= -2.5e-38) {
		tmp = Math.pow(x, 4.0) * (eps * (5.0 + ((eps * 10.0) / x)));
	} else if (x <= 1.35e-66) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.5e-38:
		tmp = math.pow(x, 4.0) * (eps * (5.0 + ((eps * 10.0) / x)))
	elif x <= 1.35e-66:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.5e-38)
		tmp = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(Float64(eps * 10.0) / x))));
	elseif (x <= 1.35e-66)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	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 (x <= -2.5e-38)
		tmp = (x ^ 4.0) * (eps * (5.0 + ((eps * 10.0) / x)));
	elseif (x <= 1.35e-66)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.5e-38], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(N[(eps * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e-66], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000017e-38

   1. Initial program 10.5%

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

   3. Taylor expanded in x around inf 99.6%

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

   4. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-*r/ 99.6%

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

   6. Simplified 99.6%

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

   if -2.50000000000000017e-38 < x < 1.34999999999999998e-66

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   if 1.34999999999999998e-66 < x

   1. Initial program 48.5%

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

   3. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-rgt1-in 99.5%

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

      3. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in eps around 0 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot \left(5 + \frac{\varepsilon \cdot 10}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.5e-38)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 1.35e-66)
     (* (pow eps 4.0) (+ eps (* x 5.0)))
     (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.5e-38) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 1.35e-66) {
		tmp = pow(eps, 4.0) * (eps + (x * 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 (x <= (-2.5d-38)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 1.35d-66) then
        tmp = (eps ** 4.0d0) * (eps + (x * 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 (x <= -2.5e-38) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 1.35e-66) {
		tmp = Math.pow(eps, 4.0) * (eps + (x * 5.0));
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.5e-38:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 1.35e-66:
		tmp = math.pow(eps, 4.0) * (eps + (x * 5.0))
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.5e-38)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 1.35e-66)
		tmp = Float64((eps ^ 4.0) * Float64(eps + Float64(x * 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 (x <= -2.5e-38)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 1.35e-66)
		tmp = (eps ^ 4.0) * (eps + (x * 5.0));
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.5e-38], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e-66], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000017e-38

   1. Initial program 10.5%

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

   3. Taylor expanded in x around inf 98.2%

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

      1. *-commutative 98.2%

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

      2. distribute-rgt1-in 98.2%

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

      3. metadata-eval 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-*r* 98.3%

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

   5. Simplified 98.3%

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

   if -2.50000000000000017e-38 < x < 1.34999999999999998e-66

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf 99.5%

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

      1. distribute-lft1-in 99.5%

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

      2. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in eps around 0 99.4%

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

      1. *-commutative 99.4%

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

   8. Simplified 99.4%

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

   if 1.34999999999999998e-66 < x

   1. Initial program 48.5%

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

   3. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-rgt1-in 99.5%

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

      3. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in eps around 0 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-38} \lor \neg \left(x \leq 1.35 \cdot 10^{-66}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.9e-38) (not (<= x 1.35e-66)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.9e-38) || !(x <= 1.35e-66)) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.9d-38)) .or. (.not. (x <= 1.35d-66))) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.9e-38) || !(x <= 1.35e-66)) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.9e-38) or not (x <= 1.35e-66):
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.9e-38) || !(x <= 1.35e-66))
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.9e-38) || ~((x <= 1.35e-66)))
		tmp = 5.0 * (eps * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.9e-38], N[Not[LessEqual[x, 1.35e-66]], $MachinePrecision]], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.89999999999999994e-38 or 1.34999999999999998e-66 < x

   1. Initial program 34.1%

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

   3. Taylor expanded in x around inf 99.0%

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

      1. *-commutative 99.0%

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

      2. distribute-rgt1-in 99.0%

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

      3. metadata-eval 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in eps around 0 99.2%

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

   if -2.89999999999999994e-38 < x < 1.34999999999999998e-66

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-38} \lor \neg \left(x \leq 1.35 \cdot 10^{-66}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.5e-38)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 1.35e-66) (pow eps 5.0) (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.5e-38) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 1.35e-66) {
		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 (x <= (-2.5d-38)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 1.35d-66) 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 (x <= -2.5e-38) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 1.35e-66) {
		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 x <= -2.5e-38:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 1.35e-66:
		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 (x <= -2.5e-38)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 1.35e-66)
		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 (x <= -2.5e-38)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 1.35e-66)
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.5e-38], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e-66], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000017e-38

   1. Initial program 10.5%

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

   3. Taylor expanded in x around inf 98.2%

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

      1. *-commutative 98.2%

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

      2. distribute-rgt1-in 98.2%

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

      3. metadata-eval 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-*r* 98.3%

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

   5. Simplified 98.3%

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

   if -2.50000000000000017e-38 < x < 1.34999999999999998e-66

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.4%

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

   if 1.34999999999999998e-66 < x

   1. Initial program 48.5%

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

   3. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-rgt1-in 99.5%

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

      3. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in eps around 0 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\varepsilon}^{5} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (pow eps 5.0))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return eps ^ 5.0
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Power[eps, 5.0], $MachinePrecision]

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in x around 0 86.5%

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

4. Final simplification 86.5%

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

Reproduce

herbie shell --seed 2024055 
(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)))