2nthrt (problem 3.4.6)

Average Error: 32.7 → 12.1

Time: 15.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -1.4696895820278125 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5.809244467416722 \cdot 10^{-171}:\\ \;\;\;\;-\frac{\log \left(\frac{-x}{-1 - x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.6142285024081105 \cdot 10^{-120}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8.796742391996114 \cdot 10^{-68}:\\ \;\;\;\;-\frac{\log \left(x \cdot \frac{1}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 9.870267292048051 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \frac{0.5}{x \cdot x}\right) - \frac{0.25}{{x}^{4}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.334499698959325 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{n} \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 0.26333508773832487:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \]

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 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -1.4696895820278125e-41)
     t_1
     (if (<= (/ 1.0 n) 5.809244467416722e-171)
       (- (/ (log (/ (- x) (- -1.0 x))) n))
       (if (<= (/ 1.0 n) 1.6142285024081105e-120)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 8.796742391996114e-68)
           (- (/ (log (* x (/ 1.0 (+ 1.0 x)))) n))
           (if (<= (/ 1.0 n) 9.870267292048051e-35)
             (/
              (-
               (-
                (+ (/ 1.0 x) (/ 0.3333333333333333 (pow x 3.0)))
                (/ 0.5 (* x x)))
               (/ 0.25 (pow x 4.0)))
              n)
             (if (<= (/ 1.0 n) 1.334499698959325e-25)
               (* (/ 1.0 n) (- (log1p x) (log x)))
               (if (<= (/ 1.0 n) 0.26333508773832487)
                 t_1
                 (- (exp (/ (log1p x) n)) t_0))))))))))

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 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1.4696895820278125e-41) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5.809244467416722e-171) {
		tmp = -(log(-x / (-1.0 - x)) / n);
	} else if ((1.0 / n) <= 1.6142285024081105e-120) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 8.796742391996114e-68) {
		tmp = -(log(x * (1.0 / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 9.870267292048051e-35) {
		tmp = ((((1.0 / x) + (0.3333333333333333 / pow(x, 3.0))) - (0.5 / (x * x))) - (0.25 / pow(x, 4.0))) / n;
	} else if ((1.0 / n) <= 1.334499698959325e-25) {
		tmp = (1.0 / n) * (log1p(x) - log(x));
	} else if ((1.0 / n) <= 0.26333508773832487) {
		tmp = t_1;
	} else {
		tmp = exp(log1p(x) / n) - t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 7 regimes

2. if (/.f64 1 n) < -1.46968958202781248e-41 or 1.3344996989593251e-25 < (/.f64 1 n) < 0.263335087738324869

   1. Initial program 13.9

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

   2. Taylor expanded in x around inf 9.1

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

   3. Simplified 9.1

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

   if -1.46968958202781248e-41 < (/.f64 1 n) < 5.8092444674167221e-171

   1. Initial program 41.3

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

   2. Taylor expanded in n around -inf 10.5

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

   3. Simplified 10.5

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

   4. Applied log1p-udef_binary64 10.5

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

   5. Applied diff-log_binary64 10.3

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

   6. Applied frac-2neg_binary64 10.3

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

   7. Simplified 10.3

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

   if 5.8092444674167221e-171 < (/.f64 1 n) < 1.6142285024081105e-120

   1. Initial program 46.5

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

   2. Taylor expanded in n around -inf 16.7

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

   3. Simplified 16.7

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

   4. Taylor expanded in x around inf 29.5

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

   if 1.6142285024081105e-120 < (/.f64 1 n) < 8.7967423919961143e-68

   1. Initial program 52.2

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

   2. Taylor expanded in n around -inf 22.6

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

   3. Simplified 22.6

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

   4. Applied log1p-udef_binary64 22.6

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

   5. Applied diff-log_binary64 22.5

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

   6. Applied div-inv_binary64 24.0

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

   7. Simplified 24.0

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

   if 8.7967423919961143e-68 < (/.f64 1 n) < 9.87026729204805135e-35

   1. Initial program 56.7

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

   2. Taylor expanded in n around -inf 26.5

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

   3. Simplified 26.5

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

   4. Taylor expanded in x around inf 31.4

      \[\leadsto -\frac{\color{blue}{\left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}}{n} \]

   5. Simplified 31.4

      \[\leadsto -\frac{\color{blue}{\left(\frac{0.5}{x \cdot x} - \left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right)\right) + \frac{0.25}{{x}^{4}}}}{n} \]

   if 9.87026729204805135e-35 < (/.f64 1 n) < 1.3344996989593251e-25

   1. Initial program 59.4

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

   2. Taylor expanded in n around -inf 27.2

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

   3. Simplified 27.2

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

   4. Applied div-inv_binary64 27.2

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

   if 0.263335087738324869 < (/.f64 1 n)

   1. Initial program 4.4

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

   2. Taylor expanded in n around 0 4.4

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

   3. Simplified 0.5

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.4696895820278125 \cdot 10^{-41}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5.809244467416722 \cdot 10^{-171}:\\ \;\;\;\;-\frac{\log \left(\frac{-x}{-1 - x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.6142285024081105 \cdot 10^{-120}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8.796742391996114 \cdot 10^{-68}:\\ \;\;\;\;-\frac{\log \left(x \cdot \frac{1}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 9.870267292048051 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \frac{0.5}{x \cdot x}\right) - \frac{0.25}{{x}^{4}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.334499698959325 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{n} \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 0.26333508773832487:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021224 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))