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

Percentage Accurate: 88.5% → 99.3%

Time: 11.0s

Alternatives: 8

Speedup: 2.0×

Specification

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-308} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right) + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-308) (not (<= t_0 0.0)))
     t_0
     (+ (* (pow x 4.0) (* eps 5.0)) (* (* (* eps eps) 10.0) (pow x 3.0))))))

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-308) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (pow(x, 4.0) * (eps * 5.0)) + (((eps * eps) * 10.0) * pow(x, 3.0));
	}
	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-308)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = ((x ** 4.0d0) * (eps * 5.0d0)) + (((eps * eps) * 10.0d0) * (x ** 3.0d0))
    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-308) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (Math.pow(x, 4.0) * (eps * 5.0)) + (((eps * eps) * 10.0) * Math.pow(x, 3.0));
	}
	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-308) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (math.pow(x, 4.0) * (eps * 5.0)) + (((eps * eps) * 10.0) * math.pow(x, 3.0))
	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-308) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64((x ^ 4.0) * Float64(eps * 5.0)) + Float64(Float64(Float64(eps * eps) * 10.0) * (x ^ 3.0)));
	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-308) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = ((x ^ 4.0) * (eps * 5.0)) + (((eps * eps) * 10.0) * (x ^ 3.0));
	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-308], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.9999999999999991e-309 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.0%

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

   if -9.9999999999999991e-309 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 86.8%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. fma-def 99.9%

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

      2. distribute-rgt-out 99.9%

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

      3. unpow2 99.9%

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

      4. metadata-eval 99.9%

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

      5. distribute-rgt1-in 99.9%

         \[\leadsto \mathsf{fma}\left({x}^{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot 10, {x}^{4} \cdot \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)}\right) \]

      6. metadata-eval 99.9%

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

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot 10, {x}^{4} \cdot \left(5 \cdot \varepsilon\right)\right)} \]
   5. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-308} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right) + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot {x}^{3}\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-308} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-308) (not (<= t_0 0.0)))
     t_0
     (* eps (+ (* 5.0 (pow x 4.0)) (* eps (* 10.0 (pow x 3.0))))))))

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-308) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * ((5.0 * pow(x, 4.0)) + (eps * (10.0 * pow(x, 3.0))));
	}
	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-308)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = eps * ((5.0d0 * (x ** 4.0d0)) + (eps * (10.0d0 * (x ** 3.0d0))))
    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-308) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * ((5.0 * Math.pow(x, 4.0)) + (eps * (10.0 * Math.pow(x, 3.0))));
	}
	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-308) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = eps * ((5.0 * math.pow(x, 4.0)) + (eps * (10.0 * math.pow(x, 3.0))))
	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-308) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(eps * Float64(10.0 * (x ^ 3.0)))));
	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-308) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = eps * ((5.0 * (x ^ 4.0)) + (eps * (10.0 * (x ^ 3.0))));
	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-308], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(10.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.9999999999999991e-309 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.0%

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

   if -9.9999999999999991e-309 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 86.8%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. fma-def 99.9%

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

      2. distribute-rgt-out 99.9%

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

      3. unpow2 99.9%

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

      4. metadata-eval 99.9%

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

      5. distribute-rgt1-in 99.9%

         \[\leadsto \mathsf{fma}\left({x}^{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot 10, {x}^{4} \cdot \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)}\right) \]

      6. metadata-eval 99.9%

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

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot 10, {x}^{4} \cdot \left(5 \cdot \varepsilon\right)\right)} \]
   5. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. metadata-eval 99.9%

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

      8. pow-sqr 99.9%

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

      9. unpow2 99.9%

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

      10. unpow2 99.9%

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

      11. associate-*r* 99.9%

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

      12. *-commutative 99.9%

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

      13. *-commutative 99.9%

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

      14. associate-*l* 99.9%

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

      15. distribute-lft-out 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-308} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \end{array} \]

Alternative 3: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-308} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-308) (not (<= t_0 0.0)))
     t_0
     (* (pow x 4.0) (* eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -1e-308) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-1d-308)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -1e-308) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -1e-308) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -1e-308) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -1e-308) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (x ^ 4.0) * (eps * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-308], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.9999999999999991e-309 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.0%

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

   if -9.9999999999999991e-309 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 86.8%

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

   2. Taylor expanded in x around inf 99.9%

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

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

   4. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-308} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-50} \lor \neg \left(x \leq 5.2 \cdot 10^{-60}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.6e-50) (not (<= x 5.2e-60)))
   (* 5.0 (* eps (pow x 4.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.6e-50) || !(x <= 5.2e-60)) {
		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 <= (-2.6d-50)) .or. (.not. (x <= 5.2d-60))) 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 <= -2.6e-50) || !(x <= 5.2e-60)) {
		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 <= -2.6e-50) or not (x <= 5.2e-60):
		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 <= -2.6e-50) || !(x <= 5.2e-60))
		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 <= -2.6e-50) || ~((x <= 5.2e-60)))
		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, -2.6e-50], N[Not[LessEqual[x, 5.2e-60]], $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 < -2.6000000000000001e-50 or 5.1999999999999995e-60 < x

   1. Initial program 46.9%

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

   2. Taylor expanded in x around inf 89.6%

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

      1. *-commutative 89.6%

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

      2. distribute-rgt1-in 89.6%

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

      3. metadata-eval 89.6%

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

      4. *-commutative 89.6%

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

      5. associate-*r* 89.6%

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

   4. Simplified 89.6%

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

   5. Taylor expanded in eps around 0 89.6%

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

   if -2.6000000000000001e-50 < x < 5.1999999999999995e-60

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-50} \lor \neg \left(x \leq 5.2 \cdot 10^{-60}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-49}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.8e-49)
   (* eps (* 5.0 (pow x 4.0)))
   (if (<= x 7.8e-60) (pow eps 5.0) (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.8e-49) {
		tmp = eps * (5.0 * pow(x, 4.0));
	} else if (x <= 7.8e-60) {
		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 <= (-4.8d-49)) then
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    else if (x <= 7.8d-60) 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 <= -4.8e-49) {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	} else if (x <= 7.8e-60) {
		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 <= -4.8e-49:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	elif x <= 7.8e-60:
		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 <= -4.8e-49)
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	elseif (x <= 7.8e-60)
		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 <= -4.8e-49)
		tmp = eps * (5.0 * (x ^ 4.0));
	elseif (x <= 7.8e-60)
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.8e-49], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.8e-60], 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 < -4.79999999999999985e-49

   1. Initial program 33.0%

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

   2. Taylor expanded in x around inf 90.9%

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

      1. *-commutative 90.9%

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

      2. distribute-rgt1-in 90.9%

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

      3. metadata-eval 90.9%

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

      4. *-commutative 90.9%

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

      5. associate-*r* 90.8%

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

   4. Simplified 90.8%

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

   if -4.79999999999999985e-49 < x < 7.8000000000000004e-60

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.9%

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

   if 7.8000000000000004e-60 < x

   1. Initial program 60.3%

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

   2. Taylor expanded in x around inf 88.3%

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

      1. *-commutative 88.3%

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

      2. distribute-rgt1-in 88.3%

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

      3. metadata-eval 88.3%

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

      4. *-commutative 88.3%

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

      5. associate-*r* 88.3%

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

   4. Simplified 88.3%

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

   5. Taylor expanded in eps around 0 88.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-49}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 6: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-50}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.3e-50)
   (* (pow x 4.0) (* eps 5.0))
   (if (<= x 7.5e-60) (pow eps 5.0) (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.3e-50) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 7.5e-60) {
		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 <= (-2.3d-50)) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    else if (x <= 7.5d-60) 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 <= -2.3e-50) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 7.5e-60) {
		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 <= -2.3e-50:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	elif x <= 7.5e-60:
		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 <= -2.3e-50)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	elseif (x <= 7.5e-60)
		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 <= -2.3e-50)
		tmp = (x ^ 4.0) * (eps * 5.0);
	elseif (x <= 7.5e-60)
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.3e-50], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e-60], 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 < -2.3000000000000002e-50

   1. Initial program 33.0%

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

   2. Taylor expanded in x around inf 90.9%

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

      1. distribute-rgt1-in 90.9%

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

      2. metadata-eval 90.9%

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

   4. Simplified 90.9%

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

   if -2.3000000000000002e-50 < x < 7.5000000000000002e-60

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.9%

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

   if 7.5000000000000002e-60 < x

   1. Initial program 60.3%

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

   2. Taylor expanded in x around inf 88.3%

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

      1. *-commutative 88.3%

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

      2. distribute-rgt1-in 88.3%

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

      3. metadata-eval 88.3%

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

      4. *-commutative 88.3%

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

      5. associate-*r* 88.3%

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

   4. Simplified 88.3%

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

   5. Taylor expanded in eps around 0 88.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-50}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 7: 97.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-50} \lor \neg \left(x \leq 7.8 \cdot 10^{-60}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -8e-50) (not (<= x 7.8e-60)))
   (* eps (* (* x x) (* 5.0 (* x x))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -8e-50) || !(x <= 7.8e-60)) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} 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 <= (-8d-50)) .or. (.not. (x <= 7.8d-60))) then
        tmp = eps * ((x * x) * (5.0d0 * (x * x)))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -8e-50) || !(x <= 7.8e-60)) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -8e-50) or not (x <= 7.8e-60):
		tmp = eps * ((x * x) * (5.0 * (x * x)))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -8e-50) || !(x <= 7.8e-60))
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(5.0 * Float64(x * x))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -8e-50) || ~((x <= 7.8e-60)))
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -8e-50], N[Not[LessEqual[x, 7.8e-60]], $MachinePrecision]], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.00000000000000006e-50 or 7.8000000000000004e-60 < x

   1. Initial program 46.9%

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

   2. Taylor expanded in x around inf 89.6%

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

      1. *-commutative 89.6%

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

      2. distribute-rgt1-in 89.6%

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

      3. metadata-eval 89.6%

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

      4. *-commutative 89.6%

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

      5. associate-*r* 89.6%

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

   4. Simplified 89.6%

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

      1. add-sqr-sqrt 89.5%

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

      2. pow2 89.5%

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

      3. *-commutative 89.5%

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

      4. sqrt-prod 89.3%

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

      5. sqrt-pow1 89.3%

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

      6. metadata-eval 89.3%

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

      7. pow2 89.3%

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

   6. Applied egg-rr 89.3%

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

      1. associate-*l* 89.3%

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

   8. Simplified 89.3%

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

      1. unpow2 89.3%

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

      2. swap-sqr 89.3%

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

      3. *-commutative 89.3%

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

      4. *-commutative 89.3%

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

      5. swap-sqr 89.1%

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

      6. add-sqr-sqrt 89.4%

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

   10. Applied egg-rr 89.4%

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

   if -8.00000000000000006e-50 < x < 7.8000000000000004e-60

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-50} \lor \neg \left(x \leq 7.8 \cdot 10^{-60}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 8: 83.0% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return eps * ((x * x) * (5.0 * (x * x)));
}

Python

def code(x, eps):
	return eps * ((x * x) * (5.0 * (x * x)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.0%

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

2. Taylor expanded in x around inf 82.4%

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

   1. *-commutative 82.4%

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

   2. distribute-rgt1-in 82.4%

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

   3. metadata-eval 82.4%

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

   4. *-commutative 82.4%

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

   5. associate-*r* 82.4%

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

4. Simplified 82.4%

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

   1. add-sqr-sqrt 82.3%

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

   2. pow2 82.3%

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

   3. *-commutative 82.3%

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

   4. sqrt-prod 82.3%

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

   5. sqrt-pow1 82.3%

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

   6. metadata-eval 82.3%

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

   7. pow2 82.3%

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

6. Applied egg-rr 82.3%

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

   1. associate-*l* 82.3%

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

8. Simplified 82.3%

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

   1. unpow2 82.3%

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

   2. swap-sqr 82.3%

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

   3. *-commutative 82.3%

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

   4. *-commutative 82.3%

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

   5. swap-sqr 82.3%

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

   6. add-sqr-sqrt 82.3%

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

10. Applied egg-rr 82.3%

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

11. Final simplification 82.3%

    \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right) \]

Reproduce

herbie shell --seed 2023279 
(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)))