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

Percentage Accurate: 92.8% → 99.3%

Time: 12.3s

Precision: binary64

Cost: 39880

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-319}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-319)
     (pow eps 5.0)
     (if (<= t_0 0.0) (* (* x x) (* eps (* x (* x 5.0)))) t_0))))

C

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

↓

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-319) {
		tmp = pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (x * x) * (eps * (x * (x * 5.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-5d-319)) then
        tmp = eps ** 5.0d0
    else if (t_0 <= 0.0d0) then
        tmp = (x * x) * (eps * (x * (x * 5.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-319) {
		tmp = Math.pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (x * x) * (eps * (x * (x * 5.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -5e-319:
		tmp = math.pow(eps, 5.0)
	elif t_0 <= 0.0:
		tmp = (x * x) * (eps * (x * (x * 5.0)))
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-319)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x * x) * Float64(eps * Float64(x * Float64(x * 5.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -5e-319)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = (x * x) * (eps * (x * (x * 5.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

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[LessEqual[t$95$0, -5e-319], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x * x), $MachinePrecision] * N[(eps * N[(x * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -4.9999937e-319

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if -4.9999937e-319 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 88.7%

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

   2. Taylor expanded in eps around 0 100.0%

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

   3. Applied egg-rr 99.7%

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

      [Start] 100.0%	\[ \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right) \]
      distribute-lft1-in [=>] 100.0%	\[ \varepsilon \cdot \color{blue}{\left(\left(4 + 1\right) \cdot {x}^{4}\right)} \]
      metadata-eval [=>] 100.0%	\[ \varepsilon \cdot \left(\color{blue}{5} \cdot {x}^{4}\right) \]
      associate-*l* [<=] 99.9%	\[ \color{blue}{\left(\varepsilon \cdot 5\right) \cdot {x}^{4}} \]
      add-cube-cbrt [=>] 99.7%	\[ \color{blue}{\left(\sqrt[3]{\left(\varepsilon \cdot 5\right) \cdot {x}^{4}} \cdot \sqrt[3]{\left(\varepsilon \cdot 5\right) \cdot {x}^{4}}\right) \cdot \sqrt[3]{\left(\varepsilon \cdot 5\right) \cdot {x}^{4}}} \]
      pow3 [=>] 99.7%	\[ \color{blue}{{\left(\sqrt[3]{\left(\varepsilon \cdot 5\right) \cdot {x}^{4}}\right)}^{3}} \]
      *-commutative [=>] 99.7%	\[ {\left(\sqrt[3]{\color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5\right)}}\right)}^{3} \]

   4. Applied egg-rr 100.0%

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

      [Start] 99.7%	\[ {\left(\sqrt[3]{{x}^{4} \cdot \left(\varepsilon \cdot 5\right)}\right)}^{3} \]
      rem-cube-cbrt [=>] 99.9%	\[ \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5\right)} \]
      *-commutative [=>] 99.9%	\[ \color{blue}{\left(\varepsilon \cdot 5\right) \cdot {x}^{4}} \]
      sqr-pow [=>] 99.8%	\[ \left(\varepsilon \cdot 5\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right)} \]
      metadata-eval [=>] 99.8%	\[ \left(\varepsilon \cdot 5\right) \cdot \left({x}^{\color{blue}{2}} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \]
      pow2 [<=] 99.8%	\[ \left(\varepsilon \cdot 5\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \]
      metadata-eval [=>] 99.8%	\[ \left(\varepsilon \cdot 5\right) \cdot \left(\left(x \cdot x\right) \cdot {x}^{\color{blue}{2}}\right) \]
      pow2 [<=] 99.8%	\[ \left(\varepsilon \cdot 5\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
      associate-*r* [=>] 99.9%	\[ \color{blue}{\left(\left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]
      associate-*r* [<=] 99.9%	\[ \color{blue}{\left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)} \cdot \left(x \cdot x\right) \]
      *-commutative [=>] 99.9%	\[ \color{blue}{\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)} \]
      *-commutative [=>] 99.9%	\[ \left(x \cdot x\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 5\right)}\right) \]
      associate-*l* [=>] 100.0%	\[ \left(x \cdot x\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(x \cdot \left(x \cdot 5\right)\right)}\right) \]

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

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-319}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	39880

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-319}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	98.4%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-51}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	98.2%
Cost	6793

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-47} \lor \neg \left(x \leq 2.75 \cdot 10^{-45}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 4
Accuracy	86.0%
Cost	704

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

Alternative 5
Accuracy	86.0%
Cost	704

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

Reproduce

herbie shell --seed 2023277 
(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)))