Average Error: 29.8 → 8.6
Time: 18.9s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ \mathbf{if}\;n \leq -38195.04975221294 \lor \neg \left(n \leq 137840.97768649113\right):\\ \;\;\;\;\left({x}^{\left(\frac{0.5}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \left(\mathsf{fma}\left(0.125, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \mathsf{fma}\left(0.0026041666666666665, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{4}}{{n}^{4}}, \mathsf{fma}\left(0.5, t_0, 0.020833333333333332 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3}}{{n}^{3}}\right)\right)\right) - \mathsf{fma}\left(0.020833333333333332, \frac{{\log x}^{3}}{{n}^{3}}, \mathsf{fma}\left(0.125, \frac{{\log x}^{2}}{n \cdot n}, \mathsf{fma}\left(0.0026041666666666665, \frac{{\log x}^{4}}{{n}^{4}}, \frac{\log \left(\sqrt{x}\right)}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{t_0} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
{\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}\\
\mathbf{if}\;n \leq -38195.04975221294 \lor \neg \left(n \leq 137840.97768649113\right):\\
\;\;\;\;\left({x}^{\left(\frac{0.5}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \left(\mathsf{fma}\left(0.125, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \mathsf{fma}\left(0.0026041666666666665, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{4}}{{n}^{4}}, \mathsf{fma}\left(0.5, t_0, 0.020833333333333332 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3}}{{n}^{3}}\right)\right)\right) - \mathsf{fma}\left(0.020833333333333332, \frac{{\log x}^{3}}{{n}^{3}}, \mathsf{fma}\left(0.125, \frac{{\log x}^{2}}{n \cdot n}, \mathsf{fma}\left(0.0026041666666666665, \frac{{\log x}^{4}}{{n}^{4}}, \frac{\log \left(\sqrt{x}\right)}{n}\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;e^{t_0} - {x}^{\left(\frac{1}{n}\right)}\\


\end{array}

(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)))
   (if (or (<= n -38195.04975221294) (not (<= n 137840.97768649113)))
     (*
      (+ (pow x (/ 0.5 n)) (pow (+ x 1.0) (/ 0.5 n)))
      (-
       (fma
        0.125
        (/ (pow (log1p x) 2.0) (* n n))
        (fma
         0.0026041666666666665
         (/ (pow (log1p x) 4.0) (pow n 4.0))
         (fma
          0.5
          t_0
          (* 0.020833333333333332 (/ (pow (log1p x) 3.0) (pow n 3.0))))))
       (fma
        0.020833333333333332
        (/ (pow (log x) 3.0) (pow n 3.0))
        (fma
         0.125
         (/ (pow (log x) 2.0) (* n n))
         (fma
          0.0026041666666666665
          (/ (pow (log x) 4.0) (pow n 4.0))
          (/ (log (sqrt x)) n))))))
     (- (exp t_0) (pow x (/ 1.0 n))))))
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 tmp;
	if ((n <= -38195.04975221294) || !(n <= 137840.97768649113)) {
		tmp = (pow(x, (0.5 / n)) + pow((x + 1.0), (0.5 / n))) * (fma(0.125, (pow(log1p(x), 2.0) / (n * n)), fma(0.0026041666666666665, (pow(log1p(x), 4.0) / pow(n, 4.0)), fma(0.5, t_0, (0.020833333333333332 * (pow(log1p(x), 3.0) / pow(n, 3.0)))))) - fma(0.020833333333333332, (pow(log(x), 3.0) / pow(n, 3.0)), fma(0.125, (pow(log(x), 2.0) / (n * n)), fma(0.0026041666666666665, (pow(log(x), 4.0) / pow(n, 4.0)), (log(sqrt(x)) / n)))));
	} else {
		tmp = exp(t_0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if n < -38195.0497522129372 or 137840.977686491125 < n

    1. Initial program 44.9

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

    2. Applied add-sqr-sqrt_binary6445.0

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

    3. Applied add-sqr-sqrt_binary6445.0

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

    4. Applied difference-of-squares_binary6445.0

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

    5. Simplified45.0

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

    6. Simplified45.0

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

    7. Taylor expanded in n around inf 14.4

      \[\leadsto \left({x}^{\left(\frac{0.5}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \color{blue}{\left(\left(0.125 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(0.0026041666666666665 \cdot \frac{{\log \left(1 + x\right)}^{4}}{{n}^{4}} + \left(0.020833333333333332 \cdot \frac{{\log \left(1 + x\right)}^{3}}{{n}^{3}} + 0.5 \cdot \frac{\log \left(1 + x\right)}{n}\right)\right)\right) - \left(0.020833333333333332 \cdot \frac{{\log x}^{3}}{{n}^{3}} + \left(0.125 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \left(0.0026041666666666665 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.5 \cdot \frac{\log x}{n}\right)\right)\right)\right)} \]

    8. Simplified14.4

      \[\leadsto \left({x}^{\left(\frac{0.5}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(0.125, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \mathsf{fma}\left(0.0026041666666666665, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{4}}{{n}^{4}}, \mathsf{fma}\left(0.5, \frac{\mathsf{log1p}\left(x\right)}{n}, 0.020833333333333332 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3}}{{n}^{3}}\right)\right)\right) - \mathsf{fma}\left(0.020833333333333332, \frac{{\log x}^{3}}{{n}^{3}}, \mathsf{fma}\left(0.125, \frac{{\log x}^{2}}{n \cdot n}, \mathsf{fma}\left(0.0026041666666666665, \frac{{\log x}^{4}}{{n}^{4}}, \frac{\log \left(\sqrt{x}\right)}{n}\right)\right)\right)\right)} \]

    if -38195.0497522129372 < n < 137840.977686491125

    1. Initial program 10.0

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

    2. Taylor expanded in n around 0 10.0

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

    3. Simplified1.1

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

    4. Applied pow1_binary641.1

      \[\leadsto \color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{1}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -38195.04975221294 \lor \neg \left(n \leq 137840.97768649113\right):\\ \;\;\;\;\left({x}^{\left(\frac{0.5}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \left(\mathsf{fma}\left(0.125, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \mathsf{fma}\left(0.0026041666666666665, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{4}}{{n}^{4}}, \mathsf{fma}\left(0.5, \frac{\mathsf{log1p}\left(x\right)}{n}, 0.020833333333333332 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3}}{{n}^{3}}\right)\right)\right) - \mathsf{fma}\left(0.020833333333333332, \frac{{\log x}^{3}}{{n}^{3}}, \mathsf{fma}\left(0.125, \frac{{\log x}^{2}}{n \cdot n}, \mathsf{fma}\left(0.0026041666666666665, \frac{{\log x}^{4}}{{n}^{4}}, \frac{\log \left(\sqrt{x}\right)}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))