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

Average Error: 7.7 → 12.3

Time: 5.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t_0 - {x}^{5} \leq 0\\ \mathbf{if}\;t_1:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{{x}^{5}}\right)}^{2}, \sqrt[3]{-{x}^{5}}, t_0\right)\\ \mathbf{elif}\;t_1:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x + \varepsilon\right) \cdot 10\right)\right) + 5 \cdot \mathsf{fma}\left(\varepsilon, {x}^{3}, {\varepsilon}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left({\varepsilon}^{3} \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0)) (t_1 (<= (- t_0 (pow x 5.0)) 0.0)))
   (if t_1
     (fma (pow (cbrt (pow x 5.0)) 2.0) (cbrt (- (pow x 5.0))) t_0)
     (if t_1
       (*
        x
        (+
         (* x (* (* eps eps) (* (+ x eps) 10.0)))
         (* 5.0 (fma eps (pow x 3.0) (pow eps 4.0)))))
       (+ (pow eps 5.0) (* x (* (pow eps 3.0) (+ (* eps 5.0) (* x 10.0)))))))))

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);
	int t_1 = (t_0 - pow(x, 5.0)) <= 0.0;
	double tmp;
	if (t_1) {
		tmp = fma(pow(cbrt(pow(x, 5.0)), 2.0), cbrt(-pow(x, 5.0)), t_0);
	} else if (t_1) {
		tmp = x * ((x * ((eps * eps) * ((x + eps) * 10.0))) + (5.0 * fma(eps, pow(x, 3.0), pow(eps, 4.0))));
	} else {
		tmp = pow(eps, 5.0) + (x * (pow(eps, 3.0) * ((eps * 5.0) + (x * 10.0))));
	}
	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(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0)) <= 0.0
	tmp = 0.0
	if (t_1)
		tmp = fma((cbrt((x ^ 5.0)) ^ 2.0), cbrt(Float64(-(x ^ 5.0))), t_0);
	elseif (t_1)
		tmp = Float64(x * Float64(Float64(x * Float64(Float64(eps * eps) * Float64(Float64(x + eps) * 10.0))) + Float64(5.0 * fma(eps, (x ^ 3.0), (eps ^ 4.0)))));
	else
		tmp = Float64((eps ^ 5.0) + Float64(x * Float64((eps ^ 3.0) * Float64(Float64(eps * 5.0) + Float64(x * 10.0)))));
	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[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = LessEqual[N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], 0.0]}, If[t$95$1, N[(N[Power[N[Power[N[Power[x, 5.0], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[(-N[Power[x, 5.0], $MachinePrecision]), 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], If[t$95$1, N[(x * N[(N[(x * N[(N[(eps * eps), $MachinePrecision] * N[(N[(x + eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(eps * N[Power[x, 3.0], $MachinePrecision] + N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 5.0], $MachinePrecision] + N[(x * N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(eps * 5.0), $MachinePrecision] + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

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)) < 0.0

   1. Initial program 8.3

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

   2. Applied egg-rr 13.3

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

   3. Applied egg-rr 13.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{{x}^{5}}\right)}^{2}}, \sqrt[3]{-{x}^{5}}, {\left(x + \varepsilon\right)}^{5}\right) \]

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

   1. Initial program 7.7

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

   2. Applied egg-rr 12.2

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

   3. Taylor expanded in eps around 0 10.6

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

   4. Simplified 10.6

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

   5. Taylor expanded in eps around 0 10.6

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

   6. Simplified 10.6

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

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

   1. Initial program 1.4

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

   2. Applied egg-rr 1.4

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

   3. Taylor expanded in eps around inf 3.0

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

   4. Simplified 3.0

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

4. Final simplification 12.3

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

Reproduce

herbie shell --seed 2022146 
(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)))