2nthrt (problem 3.4.6)

Average Error: 33.2 → 24.1

Time: 12.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -916.7070234872914:\\ \;\;\;\;{\left({\left(e^{\sqrt[3]{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\right)}^{\left(\sqrt[3]{\log \left(\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5.079153628185159 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{\sqrt[3]{\log \left(\frac{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} - {x}^{\left(\frac{1}{n} \cdot 2\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\right)}^{\left(\sqrt[3]{\log \left(\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right)}\right)}\\ \end{array}\]

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -916.7070234872914)
   (pow
    (pow
     (exp (cbrt (log (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n))))))
     (cbrt (log (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n))))))
    (cbrt (log (log (exp (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n))))))))
   (if (<= (/ 1.0 n) 5.079153628185159e-13)
     (/ 1.0 (* n x))
     (pow
      (pow
       (exp
        (cbrt
         (log
          (/
           (- (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)))))))
       (cbrt (log (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n))))))
      (cbrt
       (log (log (exp (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))))))))))

C

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) <= -916.7070234872914)) {
		tmp = ((double) pow(((double) pow(((double) exp(((double) cbrt(((double) log(((double) (((double) pow(((double) (1.0 + x)), (1.0 / n))) - ((double) pow(x, (1.0 / n))))))))))), ((double) cbrt(((double) log(((double) (((double) pow(((double) (1.0 + x)), (1.0 / n))) - ((double) pow(x, (1.0 / n))))))))))), ((double) cbrt(((double) log(((double) log(((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) <= 5.079153628185159e-13)) {
			tmp_1 = (1.0 / ((double) (n * x)));
		} else {
			tmp_1 = ((double) pow(((double) pow(((double) exp(((double) cbrt(((double) log((((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)))))))))))), ((double) cbrt(((double) log(((double) (((double) pow(((double) (1.0 + x)), (1.0 / n))) - ((double) pow(x, (1.0 / n))))))))))), ((double) cbrt(((double) log(((double) log(((double) exp(((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

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -916.707023487291394

   1. Initial program Error: 0 bits

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

   3. Applied add-exp-log Error: 0 bits

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

   5. Applied add-cube-cbrt Error: 0 bits

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

   6. Applied exp-prod Error: 0 bits

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

   7. Simplified Error: 0 bits

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

   9. Applied add-log-exp Error: 0 bits

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

   10. Applied add-log-exp Error: 0 bits

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

   11. Applied diff-log Error: 0 bits

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

   12. Simplified Error: 0 bits

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

   if -916.707023487291394 < (/ 1.0 n) < 5.079153628185159e-13

   1. Initial program Error: 44.3 bits

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

   2. Taylor expanded around -inf Error: 64.0 bits

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

   3. Simplified Error: 31.7 bits

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}}\]

   if 5.079153628185159e-13 < (/ 1.0 n)

   1. Initial program Error: 9.0 bits

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

   3. Applied add-exp-log Error: 9.0 bits

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

   5. Applied add-cube-cbrt Error: 9.0 bits

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

   6. Applied exp-prod Error: 9.0 bits

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

   7. Simplified Error: 9.0 bits

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

   9. Applied add-log-exp Error: 9.1 bits

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

   10. Applied add-log-exp Error: 9.2 bits

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

   11. Applied diff-log Error: 9.2 bits

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

   12. Simplified Error: 9.1 bits

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

   14. Applied flip-- Error: 9.2 bits

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

   15. Simplified Error: 9.1 bits

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

4. Final simplification Error: 24.1 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -916.7070234872914:\\ \;\;\;\;{\left({\left(e^{\sqrt[3]{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\right)}^{\left(\sqrt[3]{\log \left(\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5.079153628185159 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{\sqrt[3]{\log \left(\frac{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} - {x}^{\left(\frac{1}{n} \cdot 2\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\right)}^{\left(\sqrt[3]{\log \left(\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))