2nthrt (problem 3.4.6)

Average Error: 32.6 → 1.2

Time: 20.0s

Precision: binary64

Cost: 59332

Math

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

↓

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\\ t_2 := {x}^{\left(\frac{0.5}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{fma}\left(-t_2, t_2, t_0\right) + \left(t_1 - t_0\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(t_1\right)\right) - t_0\\ \end{array} \]

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 (exp (/ (log1p x) n)))
        (t_2 (pow x (/ 0.5 n))))
   (if (<= (/ 1.0 n) -5e-5)
     (pow (cbrt (+ (fma (- t_2) t_2 t_0) (- t_1 t_0))) 3.0)
     (if (<= (/ 1.0 n) 0.0002)
       (/ (log1p (/ 1.0 x)) n)
       (- (expm1 (log1p t_1)) 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 = exp((log1p(x) / n));
	double t_2 = pow(x, (0.5 / n));
	double tmp;
	if ((1.0 / n) <= -5e-5) {
		tmp = pow(cbrt((fma(-t_2, t_2, t_0) + (t_1 - t_0))), 3.0);
	} else if ((1.0 / n) <= 0.0002) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = expm1(log1p(t_1)) - t_0;
	}
	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 = x ^ Float64(1.0 / n)
	t_1 = exp(Float64(log1p(x) / n))
	t_2 = x ^ Float64(0.5 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-5)
		tmp = cbrt(Float64(fma(Float64(-t_2), t_2, t_0) + Float64(t_1 - t_0))) ^ 3.0;
	elseif (Float64(1.0 / n) <= 0.0002)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(expm1(log1p(t_1)) - t_0);
	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[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[x, N[(0.5 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-5], N[Power[N[Power[N[(N[((-t$95$2) * t$95$2 + t$95$0), $MachinePrecision] + N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0002], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(Exp[N[Log[1 + t$95$1], $MachinePrecision]] - 1), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5.00000000000000024e-5

   1. Initial program 0.8

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

   2. Applied egg-rr 0.9

      \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}}\right)}^{3}} \]

   3. Applied egg-rr 0.9

      \[\leadsto {\left(\sqrt[3]{\color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + \left(\left(-{x}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-{x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{1}{n}\right)}\right)\right)}}\right)}^{3} \]

   4. Simplified 0.9

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(-{x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{1}{n}\right)}\right) + \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{3} \]
      Proof

      [Start] 0.9	\[ {\left(\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + \left(\left(-{x}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-{x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{1}{n}\right)}\right)\right)}\right)}^{3} \]
      associate-+r+ [=>] 0.9	\[ {\left(\sqrt[3]{\color{blue}{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + \left(-{x}^{\left(\frac{1}{n}\right)}\right)\right) + \mathsf{fma}\left(-{x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{3} \]
      sub-neg [<=] 0.9	\[ {\left(\sqrt[3]{\color{blue}{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)} + \mathsf{fma}\left(-{x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{1}{n}\right)}\right)}\right)}^{3} \]
      +-commutative [=>] 0.9	\[ {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(-{x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{1}{n}\right)}\right) + \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{3} \]

   if -5.00000000000000024e-5 < (/.f64 1 n) < 2.0000000000000001e-4

   1. Initial program 44.9

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

   2. Taylor expanded in n around inf 15.5

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

   3. Simplified 15.5

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

      [Start] 15.5	\[ \frac{\log \left(1 + x\right) - \log x}{n} \]
      log1p-def [=>] 15.5	\[ \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Applied egg-rr 15.4

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

   5. Applied egg-rr 15.4

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

   6. Simplified 1.3

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

      [Start] 15.4	\[ \frac{\mathsf{log1p}\left(\frac{x + 1}{x} - 1\right)}{n} \]
      *-lft-identity [<=] 15.4	\[ \frac{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \left(x + 1\right)}}{x} - 1\right)}{n} \]
      associate-*l/ [<=] 17.5	\[ \frac{\mathsf{log1p}\left(\color{blue}{\frac{1}{x} \cdot \left(x + 1\right)} - 1\right)}{n} \]
      distribute-rgt-in [=>] 17.5	\[ \frac{\mathsf{log1p}\left(\color{blue}{\left(x \cdot \frac{1}{x} + 1 \cdot \frac{1}{x}\right)} - 1\right)}{n} \]
      +-commutative [=>] 17.5	\[ \frac{\mathsf{log1p}\left(\color{blue}{\left(1 \cdot \frac{1}{x} + x \cdot \frac{1}{x}\right)} - 1\right)}{n} \]
      *-lft-identity [=>] 17.5	\[ \frac{\mathsf{log1p}\left(\left(\color{blue}{\frac{1}{x}} + x \cdot \frac{1}{x}\right) - 1\right)}{n} \]
      rgt-mult-inverse [=>] 15.4	\[ \frac{\mathsf{log1p}\left(\left(\frac{1}{x} + \color{blue}{1}\right) - 1\right)}{n} \]
      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 15.4

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

   8. Simplified 1.3

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

      [Start] 15.4	\[ \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]
      log1p-def [=>] 1.3	\[ \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

   if 2.0000000000000001e-4 < (/.f64 1 n)

   1. Initial program 4.5

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

   2. Applied egg-rr 1.3

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{fma}\left(-{x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{0.5}{n}\right)}, {x}^{\left(\frac{1}{n}\right)}\right) + \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	33032

\[\begin{array}{l} t_0 := {\left(1 + x\right)}^{\left(\frac{0.5}{n}\right)}\\ t_1 := {x}^{\left(\frac{0.5}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\left(t_0 - t_1\right) \cdot \left(t_1 + t_0\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2
Error	1.2
Cost	27332

\[\begin{array}{l} t_0 := {\left(1 + x\right)}^{\left(\frac{0.5}{n}\right)}\\ t_1 := {x}^{\left(\frac{0.5}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\left(t_0 - t_1\right) \cdot \left(t_1 + t_0\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3
Error	1.2
Cost	20232

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \]

Alternative 4
Error	1.5
Cost	13769

\[\begin{array}{l} \mathbf{if}\;n \leq -550000 \lor \neg \left(n \leq 5600\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5
Error	1.6
Cost	7436

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -8.4 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.5676004395515114 \cdot 10^{-285}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;n \leq 4900:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	2.3
Cost	7304

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 7
Error	1.7
Cost	7180

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -8.4 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.5676004395515114 \cdot 10^{-285}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;n \leq 5500:\\ \;\;\;\;1 - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	7.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \end{array} \]

Alternative 9
Error	16.9
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\frac{-0.5}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+167}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 10
Error	35.8
Cost	840

\[\begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ \mathbf{if}\;n \leq -12500:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.25 \cdot 10^{-193}:\\ \;\;\;\;\left(1 + t_0\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 11
Error	29.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;n \leq -5.8:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-70}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 12
Error	40.4
Cost	320

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

Alternative 13
Error	40.0
Cost	320

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

Alternative 14
Error	61.0
Cost	192

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

Reproduce

herbie shell --seed 2023045 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))