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

Average Error: 7.1 → 1.2

Time: 2.8s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\varepsilon \cdot \left(10 \cdot \varepsilon\right)\right) \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right) + \varepsilon \cdot \left(5 \cdot {x}^{3}\right)\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-47}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4} + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t_0 + {x}^{3} \cdot \left(\varepsilon \cdot 5\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* eps (* 10.0 eps)) (* x (+ x eps)))))
   (if (<= x -7.6e-54)
     (*
      x
      (+ (* 10.0 (* (pow eps 2.0) (pow x 2.0))) (* eps (* 5.0 (pow x 3.0)))))
     (if (<= x 4e-47)
       (+ (pow eps 5.0) (* x (+ (* 5.0 (pow eps 4.0)) t_0)))
       (* x (+ t_0 (* (pow x 3.0) (* eps 5.0))))))))

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 = (eps * (10.0 * eps)) * (x * (x + eps));
	double tmp;
	if (x <= -7.6e-54) {
		tmp = x * ((10.0 * (pow(eps, 2.0) * pow(x, 2.0))) + (eps * (5.0 * pow(x, 3.0))));
	} else if (x <= 4e-47) {
		tmp = pow(eps, 5.0) + (x * ((5.0 * pow(eps, 4.0)) + t_0));
	} else {
		tmp = x * (t_0 + (pow(x, 3.0) * (eps * 5.0)));
	}
	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 = (eps * (10.0d0 * eps)) * (x * (x + eps))
    if (x <= (-7.6d-54)) then
        tmp = x * ((10.0d0 * ((eps ** 2.0d0) * (x ** 2.0d0))) + (eps * (5.0d0 * (x ** 3.0d0))))
    else if (x <= 4d-47) then
        tmp = (eps ** 5.0d0) + (x * ((5.0d0 * (eps ** 4.0d0)) + t_0))
    else
        tmp = x * (t_0 + ((x ** 3.0d0) * (eps * 5.0d0)))
    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 = (eps * (10.0 * eps)) * (x * (x + eps));
	double tmp;
	if (x <= -7.6e-54) {
		tmp = x * ((10.0 * (Math.pow(eps, 2.0) * Math.pow(x, 2.0))) + (eps * (5.0 * Math.pow(x, 3.0))));
	} else if (x <= 4e-47) {
		tmp = Math.pow(eps, 5.0) + (x * ((5.0 * Math.pow(eps, 4.0)) + t_0));
	} else {
		tmp = x * (t_0 + (Math.pow(x, 3.0) * (eps * 5.0)));
	}
	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 = (eps * (10.0 * eps)) * (x * (x + eps))
	tmp = 0
	if x <= -7.6e-54:
		tmp = x * ((10.0 * (math.pow(eps, 2.0) * math.pow(x, 2.0))) + (eps * (5.0 * math.pow(x, 3.0))))
	elif x <= 4e-47:
		tmp = math.pow(eps, 5.0) + (x * ((5.0 * math.pow(eps, 4.0)) + t_0))
	else:
		tmp = x * (t_0 + (math.pow(x, 3.0) * (eps * 5.0)))
	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(eps * Float64(10.0 * eps)) * Float64(x * Float64(x + eps)))
	tmp = 0.0
	if (x <= -7.6e-54)
		tmp = Float64(x * Float64(Float64(10.0 * Float64((eps ^ 2.0) * (x ^ 2.0))) + Float64(eps * Float64(5.0 * (x ^ 3.0)))));
	elseif (x <= 4e-47)
		tmp = Float64((eps ^ 5.0) + Float64(x * Float64(Float64(5.0 * (eps ^ 4.0)) + t_0)));
	else
		tmp = Float64(x * Float64(t_0 + Float64((x ^ 3.0) * Float64(eps * 5.0))));
	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 = (eps * (10.0 * eps)) * (x * (x + eps));
	tmp = 0.0;
	if (x <= -7.6e-54)
		tmp = x * ((10.0 * ((eps ^ 2.0) * (x ^ 2.0))) + (eps * (5.0 * (x ^ 3.0))));
	elseif (x <= 4e-47)
		tmp = (eps ^ 5.0) + (x * ((5.0 * (eps ^ 4.0)) + t_0));
	else
		tmp = x * (t_0 + ((x ^ 3.0) * (eps * 5.0)));
	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[(eps * N[(10.0 * eps), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.6e-54], N[(x * N[(N[(10.0 * N[(N[Power[eps, 2.0], $MachinePrecision] * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(5.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e-47], N[(N[Power[eps, 5.0], $MachinePrecision] + N[(x * N[(N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$0 + N[(N[Power[x, 3.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if x < -7.6000000000000005e-54

   1. Initial program 37.3

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

   2. Taylor expanded in x around inf 4.5

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

   3. Simplified 4.4

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

   4. Taylor expanded in eps around 0 4.5

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

   5. Simplified 4.5

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

   6. Taylor expanded in eps around 0 4.9

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

   if -7.6000000000000005e-54 < x < 3.9999999999999999e-47

   1. Initial program 0.2

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

   2. Taylor expanded in x around 0 0.3

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

   3. Simplified 0.3

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

   if 3.9999999999999999e-47 < x

   1. Initial program 40.0

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

   2. Taylor expanded in x around inf 5.1

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

   3. Simplified 5.1

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

   4. Taylor expanded in eps around 0 5.1

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

   5. Simplified 5.2

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

   6. Taylor expanded in eps around 0 5.1

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

   7. Simplified 5.1

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \left(10 \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right) + \varepsilon \cdot \left(5 \cdot {x}^{3}\right)\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-47}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4} + \left(\varepsilon \cdot \left(10 \cdot \varepsilon\right)\right) \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\varepsilon \cdot \left(10 \cdot \varepsilon\right)\right) \cdot \left(x \cdot \left(x + \varepsilon\right)\right) + {x}^{3} \cdot \left(\varepsilon \cdot 5\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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)))