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

Average Error: 7.4 → 0.8

Time: 4.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1.9715732985220083 \cdot 10^{-294}:\\ \;\;\;\;10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + \left({\varepsilon}^{5} + \left(5 \cdot \left(x \cdot {\varepsilon}^{4}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(5 \cdot \left(x \cdot \left({\varepsilon}^{3} + {x}^{3}\right)\right), \varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \left(10 \cdot \left(\left(x + \varepsilon\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 -1.9715732985220083e-294)
     (+
      (* 10.0 (* (pow eps 3.0) (pow x 2.0)))
      (+
       (pow eps 5.0)
       (+ (* 5.0 (* x (pow eps 4.0))) (* 10.0 (* (pow eps 2.0) (pow x 3.0))))))
     (if (<= t_0 0.0)
       (fma
        (* 5.0 (* x (+ (pow eps 3.0) (pow x 3.0))))
        eps
        (* eps (* eps (* 10.0 (* (+ x eps) (* x x))))))
       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 <= -1.9715732985220083e-294) {
		tmp = (10.0 * (pow(eps, 3.0) * pow(x, 2.0))) + (pow(eps, 5.0) + ((5.0 * (x * pow(eps, 4.0))) + (10.0 * (pow(eps, 2.0) * pow(x, 3.0)))));
	} else if (t_0 <= 0.0) {
		tmp = fma((5.0 * (x * (pow(eps, 3.0) + pow(x, 3.0)))), eps, (eps * (eps * (10.0 * ((x + eps) * (x * x))))));
	} 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 <= -1.9715732985220083e-294)
		tmp = Float64(Float64(10.0 * Float64((eps ^ 3.0) * (x ^ 2.0))) + Float64((eps ^ 5.0) + Float64(Float64(5.0 * Float64(x * (eps ^ 4.0))) + Float64(10.0 * Float64((eps ^ 2.0) * (x ^ 3.0))))));
	elseif (t_0 <= 0.0)
		tmp = fma(Float64(5.0 * Float64(x * Float64((eps ^ 3.0) + (x ^ 3.0)))), eps, Float64(eps * Float64(eps * Float64(10.0 * Float64(Float64(x + eps) * Float64(x * x))))));
	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, -1.9715732985220083e-294], N[(N[(10.0 * N[(N[Power[eps, 3.0], $MachinePrecision] * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 5.0], $MachinePrecision] + N[(N[(5.0 * N[(x * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(10.0 * N[(N[Power[eps, 2.0], $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * N[(x * N[(N[Power[eps, 3.0], $MachinePrecision] + N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[(eps * N[(eps * N[(10.0 * N[(N[(x + eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -1.97157329852200827e-294

   1. Initial program 1.4

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

   2. Taylor expanded in x around 0 3.2

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

   if -1.97157329852200827e-294 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 8.7

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

   2. Taylor expanded in x around inf 0.5

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

   3. Simplified 0.5

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

   4. Applied egg-rr 0.4

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

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

   1. Initial program 1.7

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1.9715732985220083 \cdot 10^{-294}:\\ \;\;\;\;10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + \left({\varepsilon}^{5} + \left(5 \cdot \left(x \cdot {\varepsilon}^{4}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(5 \cdot \left(x \cdot \left({\varepsilon}^{3} + {x}^{3}\right)\right), \varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \left(10 \cdot \left(\left(x + \varepsilon\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))