2nthrt (problem 3.4.6)

Average Error: 32.3 → 1.3

Time: 19.5s

Precision: binary64

Cost: 19977

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \leq -11500 \lor \neg \left(n \leq 22500000\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (if (or (<= n -11500.0) (not (<= n 22500000.0)))
   (/ (log1p (/ 1.0 x)) n)
   (- (exp (/ x n)) (cbrt (pow x (/ 3.0 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 tmp;
	if ((n <= -11500.0) || !(n <= 22500000.0)) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = exp((x / n)) - cbrt(pow(x, (3.0 / n)));
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double n) {
	double tmp;
	if ((n <= -11500.0) || !(n <= 22500000.0)) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - Math.cbrt(Math.pow(x, (3.0 / 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)
	tmp = 0.0
	if ((n <= -11500.0) || !(n <= 22500000.0))
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - cbrt((x ^ Float64(3.0 / 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_] := If[Or[LessEqual[n, -11500.0], N[Not[LessEqual[n, 22500000.0]], $MachinePrecision]], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -11500 or 2.25e7 < n

   1. Initial program 44.6

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

   2. Taylor expanded in n around inf 15.1

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

   3. Simplified 15.1

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

      [Start] 15.1	\[ \frac{\log \left(1 + x\right) - \log x}{n} \]
      log1p-def [=>] 15.1	\[ \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Applied egg-rr 14.9

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

   5. Applied egg-rr 14.9

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

   6. Simplified 1.1

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x} + 0\right)}}{n} \]
      Proof

      [Start] 14.9	\[ \frac{\mathsf{log1p}\left(\frac{x + 1}{x} - 1\right)}{n} \]
      *-lft-identity [<=] 14.9	\[ \frac{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \left(x + 1\right)}}{x} - 1\right)}{n} \]
      associate-*l/ [<=] 17.3	\[ \frac{\mathsf{log1p}\left(\color{blue}{\frac{1}{x} \cdot \left(x + 1\right)} - 1\right)}{n} \]
      distribute-rgt-in [=>] 17.3	\[ \frac{\mathsf{log1p}\left(\color{blue}{\left(x \cdot \frac{1}{x} + 1 \cdot \frac{1}{x}\right)} - 1\right)}{n} \]
      +-commutative [=>] 17.3	\[ \frac{\mathsf{log1p}\left(\color{blue}{\left(1 \cdot \frac{1}{x} + x \cdot \frac{1}{x}\right)} - 1\right)}{n} \]
      *-lft-identity [=>] 17.3	\[ \frac{\mathsf{log1p}\left(\left(\color{blue}{\frac{1}{x}} + x \cdot \frac{1}{x}\right) - 1\right)}{n} \]
      rgt-mult-inverse [=>] 15.0	\[ \frac{\mathsf{log1p}\left(\left(\frac{1}{x} + \color{blue}{1}\right) - 1\right)}{n} \]
      associate--l+ [=>] 1.1	\[ \frac{\mathsf{log1p}\left(\color{blue}{\frac{1}{x} + \left(1 - 1\right)}\right)}{n} \]
      metadata-eval [=>] 1.1	\[ \frac{\mathsf{log1p}\left(\frac{1}{x} + \color{blue}{0}\right)}{n} \]

   7. Taylor expanded in n around 0 15.0

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

   8. Simplified 1.1

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

      [Start] 15.0	\[ \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]
      log1p-def [=>] 1.1	\[ \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

   if -11500 < n < 2.25e7

   1. Initial program 2.8

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

   2. Applied egg-rr 1.9

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

   3. Simplified 1.9

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

      [Start] 1.9	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + \left(-{x}^{\left({n}^{-1}\right)}\right) \]
      sub-neg [<=] 1.9	\[ \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}} \]
      unpow-1 [=>] 1.9	\[ e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   4. Taylor expanded in x around 0 2.0

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

   5. Applied egg-rr 2.0

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -11500 \lor \neg \left(n \leq 22500000\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	13833

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2 \lor \neg \left(\frac{1}{n} \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \end{array} \]

Alternative 2
Error	1.5
Cost	7560

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \end{array} \]

Alternative 3
Error	2.4
Cost	7304

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4
Error	1.6
Cost	7304

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \]

Alternative 5
Error	1.6
Cost	7304

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \]

Alternative 6
Error	7.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \end{array} \]

Alternative 7
Error	16.3
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\frac{-0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 8
Error	35.5
Cost	836

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\left(1 + \frac{1}{n \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 9
Error	28.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;n \leq -3.8 \lor \neg \left(n \leq 1.7 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 10
Error	40.4
Cost	320

\[\frac{1}{n \cdot x} \]

Alternative 11
Error	40.0
Cost	320

\[\frac{\frac{1}{n}}{x} \]

Alternative 12
Error	61.1
Cost	192

\[\frac{x}{n} \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))