Average Error: 32.8 → 24.5
Time: 12.4s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2.839659274028174 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) + \log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 1.6431039643536093 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{n \cdot x} + \left(\frac{1}{n \cdot n} \cdot \left(\frac{\log 1}{x} + \frac{\log x}{x}\right) - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} - {x}^{\left(\frac{1}{n} \cdot 2\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -2.839659274028174 \cdot 10^{-16}:\\
\;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) + \log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\

\mathbf{elif}\;\frac{1}{n} \leq 1.6431039643536093 \cdot 10^{-27}:\\
\;\;\;\;\frac{1}{n \cdot x} + \left(\frac{1}{n \cdot n} \cdot \left(\frac{\log 1}{x} + \frac{\log x}{x}\right) - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{\frac{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} - {x}^{\left(\frac{1}{n} \cdot 2\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\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
 (if (<= (/ 1.0 n) -2.839659274028174e-16)
   (+
    (*
     2.0
     (* 0.3333333333333333 (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))))
    (log (cbrt (exp (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))))))
   (if (<= (/ 1.0 n) 1.6431039643536093e-27)
     (+
      (/ 1.0 (* n x))
      (-
       (* (/ 1.0 (* n n)) (+ (/ (log 1.0) x) (/ (log x) x)))
       (/ 0.5 (* x (* n x)))))
     (log
      (exp
       (/
        (- (pow (+ 1.0 x) (* (/ 1.0 n) 2.0)) (pow x (* (/ 1.0 n) 2.0)))
        (+ (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))))))))
double code(double x, double n) {
	return ((double) (((double) pow(((double) (x + 1.0)), (1.0 / n))) - ((double) pow(x, (1.0 / n)))));
}
↓
double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -2.839659274028174e-16)) {
		tmp = ((double) (((double) (2.0 * ((double) (0.3333333333333333 * ((double) (((double) pow(((double) (1.0 + x)), (1.0 / n))) - ((double) pow(x, (1.0 / n))))))))) + ((double) log(((double) cbrt(((double) exp(((double) (((double) pow(((double) (1.0 + x)), (1.0 / n))) - ((double) pow(x, (1.0 / n)))))))))))));
	} else {
		double tmp_1;
		if (((1.0 / n) <= 1.6431039643536093e-27)) {
			tmp_1 = ((double) ((1.0 / ((double) (n * x))) + ((double) (((double) ((1.0 / ((double) (n * n))) * ((double) ((((double) log(1.0)) / x) + (((double) log(x)) / x))))) - (0.5 / ((double) (x * ((double) (n * x)))))))));
		} else {
			tmp_1 = ((double) log(((double) exp((((double) (((double) pow(((double) (1.0 + x)), ((double) ((1.0 / n) * 2.0)))) - ((double) pow(x, ((double) ((1.0 / n) * 2.0)))))) / ((double) (((double) pow(((double) (1.0 + x)), (1.0 / n))) + ((double) pow(x, (1.0 / n))))))))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ 1.0 n) < -2.83965927402817416e-16

    1. Initial program Error: 3.6 bits

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    2. Using strategy rm

    3. Applied add-log-expError: 4.1 bits

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

    4. Applied add-log-expError: 4.0 bits

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

    5. Applied diff-logError: 4.0 bits

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

    6. SimplifiedError: 4.0 bits

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

    8. Applied add-cube-cbrtError: 4.1 bits

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

    9. Applied log-prodError: 4.1 bits

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

    10. SimplifiedError: 4.1 bits

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)} + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\]
    11. Using strategy rm

    12. Applied pow1/3Error: 4.1 bits

      \[\leadsto 2 \cdot \log \color{blue}{\left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{0.3333333333333333}\right)} + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\]

    13. Applied log-powError: 4.1 bits

      \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right)} + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\]

    14. SimplifiedError: 4.0 bits

      \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\]

    if -2.83965927402817416e-16 < (/ 1.0 n) < 1.64310396435360927e-27

    1. Initial program Error: 44.9 bits

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

    2. Taylor expanded around inf Error: 32.5 bits

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

    3. SimplifiedError: 32.4 bits

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

    if 1.64310396435360927e-27 < (/ 1.0 n)

    1. Initial program Error: 15.0 bits

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    2. Using strategy rm

    3. Applied add-log-expError: 15.1 bits

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

    4. Applied add-log-expError: 15.1 bits

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

    5. Applied diff-logError: 15.2 bits

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

    6. SimplifiedError: 15.1 bits

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

    8. Applied flip--Error: 15.1 bits

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

    9. SimplifiedError: 15.1 bits

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

  4. Final simplificationError: 24.5 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2.839659274028174 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) + \log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 1.6431039643536093 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{n \cdot x} + \left(\frac{1}{n \cdot n} \cdot \left(\frac{\log 1}{x} + \frac{\log x}{x}\right) - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} - {x}^{\left(\frac{1}{n} \cdot 2\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))