2nthrt (problem 3.4.6)

Average Error: 32.9 → 11.1

Time: 12.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_2 := e^{t_1}\\ t_3 := {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ t_4 := \frac{\log x}{n}\\ t_5 := e^{t_4}\\ t_6 := t_3 \cdot t_0\\ \mathbf{if}\;\frac{1}{n} \leq -0.0002:\\ \;\;\;\;t_2 - t_5\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \mathsf{fma}\left(0.041666666666666664, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{4}}{{n}^{4}}, \mathsf{fma}\left(0.16666666666666666, {t_1}^{3}, t_1\right)\right)\right) - \mathsf{fma}\left(0.16666666666666666, {t_4}^{3}, \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \mathsf{fma}\left(0.041666666666666664, \frac{{\log x}^{4}}{{n}^{4}}, t_4\right)\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\log \left(\frac{-x}{-1 - x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{t_5}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t_2, -t_6\right) + \mathsf{fma}\left(-t_3, t_0, t_6\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow (pow (cbrt x) 2.0) (/ 1.0 n)))
        (t_1 (/ (log1p x) n))
        (t_2 (exp t_1))
        (t_3 (pow (cbrt x) (/ 1.0 n)))
        (t_4 (/ (log x) n))
        (t_5 (exp t_4))
        (t_6 (* t_3 t_0)))
   (if (<= (/ 1.0 n) -0.0002)
     (- t_2 t_5)
     (if (<= (/ 1.0 n) -2e-82)
       (-
        (fma
         0.5
         (/ (pow (log1p x) 2.0) (* n n))
         (fma
          0.041666666666666664
          (/ (pow (log1p x) 4.0) (pow n 4.0))
          (fma 0.16666666666666666 (pow t_1 3.0) t_1)))
        (fma
         0.16666666666666666
         (pow t_4 3.0)
         (fma
          0.5
          (/ (pow (log x) 2.0) (* n n))
          (fma 0.041666666666666664 (/ (pow (log x) 4.0) (pow n 4.0)) t_4))))
       (if (<= (/ 1.0 n) 2e-61)
         (/ (- (log (/ (- x) (- -1.0 x)))) n)
         (if (<= (/ 1.0 n) 0.02)
           (/ t_5 (* n x))
           (+ (fma 1.0 t_2 (- t_6)) (fma (- t_3) t_0 t_6))))))))

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(pow(cbrt(x), 2.0), (1.0 / n));
	double t_1 = log1p(x) / n;
	double t_2 = exp(t_1);
	double t_3 = pow(cbrt(x), (1.0 / n));
	double t_4 = log(x) / n;
	double t_5 = exp(t_4);
	double t_6 = t_3 * t_0;
	double tmp;
	if ((1.0 / n) <= -0.0002) {
		tmp = t_2 - t_5;
	} else if ((1.0 / n) <= -2e-82) {
		tmp = fma(0.5, (pow(log1p(x), 2.0) / (n * n)), fma(0.041666666666666664, (pow(log1p(x), 4.0) / pow(n, 4.0)), fma(0.16666666666666666, pow(t_1, 3.0), t_1))) - fma(0.16666666666666666, pow(t_4, 3.0), fma(0.5, (pow(log(x), 2.0) / (n * n)), fma(0.041666666666666664, (pow(log(x), 4.0) / pow(n, 4.0)), t_4)));
	} else if ((1.0 / n) <= 2e-61) {
		tmp = -log((-x / (-1.0 - x))) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = t_5 / (n * x);
	} else {
		tmp = fma(1.0, t_2, -t_6) + fma(-t_3, t_0, t_6);
	}
	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) ^ 2.0) ^ Float64(1.0 / n)
	t_1 = Float64(log1p(x) / n)
	t_2 = exp(t_1)
	t_3 = cbrt(x) ^ Float64(1.0 / n)
	t_4 = Float64(log(x) / n)
	t_5 = exp(t_4)
	t_6 = Float64(t_3 * t_0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.0002)
		tmp = Float64(t_2 - t_5);
	elseif (Float64(1.0 / n) <= -2e-82)
		tmp = Float64(fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), fma(0.041666666666666664, Float64((log1p(x) ^ 4.0) / (n ^ 4.0)), fma(0.16666666666666666, (t_1 ^ 3.0), t_1))) - fma(0.16666666666666666, (t_4 ^ 3.0), fma(0.5, Float64((log(x) ^ 2.0) / Float64(n * n)), fma(0.041666666666666664, Float64((log(x) ^ 4.0) / (n ^ 4.0)), t_4))));
	elseif (Float64(1.0 / n) <= 2e-61)
		tmp = Float64(Float64(-log(Float64(Float64(-x) / Float64(-1.0 - x)))) / n);
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(t_5 / Float64(n * x));
	else
		tmp = Float64(fma(1.0, t_2, Float64(-t_6)) + fma(Float64(-t_3), t_0, t_6));
	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[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$2 = N[Exp[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Power[x, 1/3], $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$5 = N[Exp[t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 * t$95$0), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.0002], N[(t$95$2 - t$95$5), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-82], N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[Power[t$95$1, 3.0], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 * N[Power[t$95$4, 3.0], $MachinePrecision] + N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[(N[Power[N[Log[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-61], N[((-N[Log[N[((-x) / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(t$95$5 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * t$95$2 + (-t$95$6)), $MachinePrecision] + N[((-t$95$3) * t$95$0 + t$95$6), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 5 regimes

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

   1. Initial program 0.5

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

   2. Taylor expanded in n around 0 0.5

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

   3. Simplified 0.5

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

   if -2.0000000000000001e-4 < (/.f64 1 n) < -2e-82

   1. Initial program 54.7

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

   2. Taylor expanded in n around inf 25.8

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log \left(1 + x\right)}^{4}}{{n}^{4}} + \left(0.16666666666666666 \cdot \frac{{\log \left(1 + x\right)}^{3}}{{n}^{3}} + \frac{\log \left(1 + x\right)}{n}\right)\right)\right) - \left(0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + \frac{\log x}{n}\right)\right)\right)} \]

   3. Simplified 25.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \mathsf{fma}\left(0.041666666666666664, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{4}}{{n}^{4}}, \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)}^{3}, \frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)\right) - \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\log x}{n}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \mathsf{fma}\left(0.041666666666666664, \frac{{\log x}^{4}}{{n}^{4}}, \frac{\log x}{n}\right)\right)\right)} \]

   if -2e-82 < (/.f64 1 n) < 2.0000000000000001e-61

   1. Initial program 41.6

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

   2. Taylor expanded in n around inf 11.0

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

   3. Simplified 11.0

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

   4. Applied egg-rr 10.9

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

   5. Applied egg-rr 10.9

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

   6. Applied egg-rr 10.9

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

   if 2.0000000000000001e-61 < (/.f64 1 n) < 0.0200000000000000004

   1. Initial program 51.8

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

   2. Taylor expanded in x around inf 32.6

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

   3. Simplified 32.6

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   if 0.0200000000000000004 < (/.f64 1 n)

   1. Initial program 4.4

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

   2. Applied egg-rr 0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{\left(\frac{1}{n}\right)}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0002:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\log x}{n}}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \mathsf{fma}\left(0.041666666666666664, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{4}}{{n}^{4}}, \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)}^{3}, \frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)\right) - \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\log x}{n}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \mathsf{fma}\left(0.041666666666666664, \frac{{\log x}^{4}}{{n}^{4}}, \frac{\log x}{n}\right)\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\log \left(\frac{-x}{-1 - x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))