2nthrt (problem 3.4.6)

Average Error: 33.0 → 7.5

Time: 14.3s

Precision: binary64

Math

\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

↓

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_1 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 1.3156010340719408 \cdot 10^{-288}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \mathsf{fma}\left(0.16666666666666666, {t_0}^{3}, t_0\right)\right) - \mathsf{fma}\left(0.16666666666666666, {t_1}^{3}, \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, t_1\right)\right)\\ \mathbf{elif}\;x \leq 1.2988131937612738 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(-1, {x}^{\left(\frac{1}{n}\right)}, e^{t_0}\right)\\ \mathbf{elif}\;x \leq 9705.140304668308:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t_1}}{x \cdot n}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))

↓

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log1p x) n)) (t_1 (/ (log x) n)))
   (if (<= x 1.3156010340719408e-288)
     (-
      (fma
       0.5
       (/ (pow (log1p x) 2.0) (* n n))
       (fma 0.16666666666666666 (pow t_0 3.0) t_0))
      (fma
       0.16666666666666666
       (pow t_1 3.0)
       (fma 0.5 (/ (pow (log x) 2.0) (* n n)) t_1)))
     (if (<= x 1.2988131937612738e-262)
       (fma -1.0 (pow x (/ 1.0 n)) (exp t_0))
       (if (<= x 9705.140304668308)
         (/ (- (log1p x) (log x)) n)
         (/ (exp t_1) (* x n)))))))

C

double code(double x, double n) {
	return pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
}

↓

double code(double x, double n) {
	double t_0 = log1p(x) / n;
	double t_1 = log(x) / n;
	double tmp;
	if (x <= 1.3156010340719408e-288) {
		tmp = fma(0.5, (pow(log1p(x), 2.0) / (n * n)), fma(0.16666666666666666, pow(t_0, 3.0), t_0)) - fma(0.16666666666666666, pow(t_1, 3.0), fma(0.5, (pow(log(x), 2.0) / (n * n)), t_1));
	} else if (x <= 1.2988131937612738e-262) {
		tmp = fma(-1.0, pow(x, (1.0 / n)), exp(t_0));
	} else if (x <= 9705.140304668308) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp(t_1) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

↓

function code(x, n)
	t_0 = Float64(log1p(x) / n)
	t_1 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 1.3156010340719408e-288)
		tmp = Float64(fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), fma(0.16666666666666666, (t_0 ^ 3.0), t_0)) - fma(0.16666666666666666, (t_1 ^ 3.0), fma(0.5, Float64((log(x) ^ 2.0) / Float64(n * n)), t_1)));
	elseif (x <= 1.2988131937612738e-262)
		tmp = fma(-1.0, (x ^ Float64(1.0 / n)), exp(t_0));
	elseif (x <= 9705.140304668308)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(t_1) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, n_] := Block[{t$95$0 = N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 1.3156010340719408e-288], N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 * N[Power[t$95$1, 3.0], $MachinePrecision] + N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2988131937612738e-262], N[(-1.0 * N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] + N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9705.140304668308], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[t$95$1], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 4 regimes

2. if x < 1.3156010340719408e-288

   1. Initial program 40.6

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 20.0

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(0.16666666666666666 \cdot \frac{{\log \left(1 + x\right)}^{3}}{{n}^{3}} + \frac{\log \left(1 + x\right)}{n}\right)\right) - \left(0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)\right)} \]

   3. Simplified 20.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)}^{3}, \frac{\mathsf{log1p}\left(x\right)}{n}\right)\right) - \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\log x}{n}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)\right)} \]

   if 1.3156010340719408e-288 < x < 1.29881319376127382e-262

   1. Initial program 37.7

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Applied egg-rr 37.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, {x}^{\left(\frac{1}{n}\right)}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)} \]

   if 1.29881319376127382e-262 < x < 9705.1403046683081

   1. Initial program 49.2

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 11.8

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   3. Simplified 11.8

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   4. Applied egg-rr 11.8

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]

   5. Applied egg-rr 11.8

      \[\leadsto \color{blue}{0 + \frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 9705.1403046683081 < x

   1. Initial program 21.3

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 1.5

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]

   3. Simplified 1.5

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3156010340719408 \cdot 10^{-288}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)}^{3}, \frac{\mathsf{log1p}\left(x\right)}{n}\right)\right) - \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\log x}{n}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)\right)\\ \mathbf{elif}\;x \leq 1.2988131937612738 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(-1, {x}^{\left(\frac{1}{n}\right)}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \mathbf{elif}\;x \leq 9705.140304668308:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))