2nthrt (problem 3.4.6)

Average Error: 32.3 → 1.4

Time: 21.5s

Precision: binary64

Cost: 65988

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\\ t_1 := \sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}\\ \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-t_1, t_1, t_0\right) + \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t_0\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\log x}{n}}\\ \end{array} \]

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 (cbrt (pow x (/ 3.0 n)))) (t_1 (cbrt (pow x (/ 1.5 n)))))
   (if (<= (/ 1.0 n) -0.2)
     (+ (* 2.0 (fma (- t_1) t_1 t_0)) (- (pow (+ 1.0 x) (/ 1.0 n)) t_0))
     (if (<= (/ 1.0 n) 2e-8)
       (/ (log1p (/ 1.0 x)) n)
       (- (exp (/ (log1p x) n)) (exp (/ (log x) 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 t_0 = cbrt(pow(x, (3.0 / n)));
	double t_1 = cbrt(pow(x, (1.5 / n)));
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = (2.0 * fma(-t_1, t_1, t_0)) + (pow((1.0 + x), (1.0 / n)) - t_0);
	} else if ((1.0 / n) <= 2e-8) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - exp((log(x) / 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)
	t_0 = cbrt((x ^ Float64(3.0 / n)))
	t_1 = cbrt((x ^ Float64(1.5 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.2)
		tmp = Float64(Float64(2.0 * fma(Float64(-t_1), t_1, t_0)) + Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0));
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - exp(Float64(log(x) / 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_] := Block[{t$95$0 = N[Power[N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[x, N[(1.5 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.2], N[(N[(2.0 * N[((-t$95$1) * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -0.20000000000000001

   1. Initial program 0.2

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

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right) + \left(\mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right) + \mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right)\right)} \]

   4. Simplified 0.4

      \[\leadsto \color{blue}{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right) + 2 \cdot \mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right)} \]
      Proof

      [Start] 0.4	\[ \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right) + \left(\mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right) + \mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right)\right) \]
      +-commutative [=>] 0.4	\[ \left({\color{blue}{\left(1 + x\right)}}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right) + \left(\mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right) + \mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right)\right) \]
      count-2 [=>] 0.4	\[ \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right) + \color{blue}{2 \cdot \mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right)} \]

   if -0.20000000000000001 < (/.f64 1 n) < 2e-8

   1. Initial program 44.5

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

   2. Taylor expanded in n around inf 15.3

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

   3. Simplified 15.3

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

      [Start] 15.3	\[ \frac{\log \left(1 + x\right) - \log x}{n} \]
      log1p-def [=>] 15.3	\[ \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Applied egg-rr 15.1

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

   5. Applied egg-rr 15.1

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

   6. Simplified 1.3

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

      [Start] 15.1	\[ \frac{\mathsf{log1p}\left(\frac{x + 1}{x} - 1\right)}{n} \]
      *-lft-identity [<=] 15.1	\[ \frac{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \left(x + 1\right)}}{x} - 1\right)}{n} \]
      associate-*l/ [<=] 17.4	\[ \frac{\mathsf{log1p}\left(\color{blue}{\frac{1}{x} \cdot \left(x + 1\right)} - 1\right)}{n} \]
      distribute-rgt-in [=>] 17.4	\[ \frac{\mathsf{log1p}\left(\color{blue}{\left(x \cdot \frac{1}{x} + 1 \cdot \frac{1}{x}\right)} - 1\right)}{n} \]
      +-commutative [=>] 17.4	\[ \frac{\mathsf{log1p}\left(\color{blue}{\left(1 \cdot \frac{1}{x} + x \cdot \frac{1}{x}\right)} - 1\right)}{n} \]
      *-lft-identity [=>] 17.4	\[ \frac{\mathsf{log1p}\left(\left(\color{blue}{\frac{1}{x}} + x \cdot \frac{1}{x}\right) - 1\right)}{n} \]
      rgt-mult-inverse [=>] 15.1	\[ \frac{\mathsf{log1p}\left(\left(\frac{1}{x} + \color{blue}{1}\right) - 1\right)}{n} \]
      associate--l+ [=>] 1.3	\[ \frac{\mathsf{log1p}\left(\color{blue}{\frac{1}{x} + \left(1 - 1\right)}\right)}{n} \]
      metadata-eval [=>] 1.3	\[ \frac{\mathsf{log1p}\left(\frac{1}{x} + \color{blue}{0}\right)}{n} \]

   7. Taylor expanded in n around 0 15.1

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

   8. Simplified 1.3

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

      [Start] 15.1	\[ \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]
      log1p-def [=>] 1.3	\[ \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

   if 2e-8 < (/.f64 1 n)

   1. Initial program 6.5

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

   2. Taylor expanded in n around 0 6.5

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

   3. Simplified 3.8

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

      [Start] 6.5	\[ e^{\frac{\log \left(1 + x\right)}{n}} - e^{\frac{\log x}{n}} \]
      log1p-def [=>] 3.8	\[ e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - e^{\frac{\log x}{n}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right) + \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\log x}{n}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	33028

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;{\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}\right)}^{3}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\log x}{n}}\\ \end{array} \]

Alternative 2
Error	1.4
Cost	26628

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{{x}^{\left(\frac{1.5}{n}\right)}}\right)}^{2}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\log x}{n}}\\ \end{array} \]

Alternative 3
Error	1.4
Cost	26568

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\log x}{n}}\\ \end{array} \]

Alternative 4
Error	1.4
Cost	26568

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt{\sqrt[3]{{x}^{\left(\frac{6}{n}\right)}}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\log x}{n}}\\ \end{array} \]

Alternative 5
Error	1.6
Cost	20164

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}\\ \end{array} \]

Alternative 6
Error	1.5
Cost	13896

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

Alternative 7
Error	1.5
Cost	13896

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

Alternative 8
Error	1.6
Cost	13640

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

Alternative 9
Error	7.1
Cost	7300

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

Alternative 10
Error	7.1
Cost	7172

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

Alternative 11
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 12
Error	16.3
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \frac{-0.5}{n \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 13
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 14
Error	28.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;n \leq -3.8 \lor \neg \left(n \leq 4.1 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 15
Error	40.4
Cost	320

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

Alternative 16
Error	40.0
Cost	320

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

Alternative 17
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))))