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

Average Error: 7.9 → 0.4

Time: 4.0s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-319}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left(x \cdot \left(x \cdot {\varepsilon}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 -4e-319)
     t_0
     (if (<= t_0 0.0)
       (fma (* eps 5.0) (pow x 4.0) (* 10.0 (* x (* x (pow eps 3.0)))))
       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 <= -4e-319) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = fma((eps * 5.0), pow(x, 4.0), (10.0 * (x * (x * pow(eps, 3.0)))));
	} 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 <= -4e-319)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = fma(Float64(eps * 5.0), (x ^ 4.0), Float64(10.0 * Float64(x * Float64(x * (eps ^ 3.0)))));
	else
		tmp = t_0;
	end
	return 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, -4e-319], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + N[(10.0 * N[(x * N[(x * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -4.0000049e-319 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 1.8

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

   if -4.0000049e-319 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 9.3

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

   2. Taylor expanded in x around inf 0.1

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

   3. Simplified 0.1

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

   4. Taylor expanded in eps around inf 0.1

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

   5. Simplified 0.1

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-319}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left(x \cdot \left(x \cdot {\varepsilon}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(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)))