2nthrt (problem 3.4.6)

Average Error: 32.3 → 1.4

Time: 18.1s

Precision: binary64

Cost: 58888

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;t_1 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left({e}^{\left(\sqrt[3]{{t_0}^{2}}\right)}\right)}^{\left(\sqrt[3]{t_0}\right)} - t_1\\ \end{array} \]

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 (/ (log1p x) n)) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-17)
     (* t_1 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) 0.2)
       (/ (log1p (/ 1.0 x)) n)
       (- (pow (pow E (cbrt (pow t_0 2.0))) (cbrt t_0)) t_1)))))

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 = log1p(x) / n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-17) {
		tmp = t_1 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 0.2) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = pow(pow(((double) M_E), cbrt(pow(t_0, 2.0))), cbrt(t_0)) - t_1;
	}
	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 t_0 = Math.log1p(x) / n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-17) {
		tmp = t_1 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 0.2) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.pow(Math.pow(Math.E, Math.cbrt(Math.pow(t_0, 2.0))), Math.cbrt(t_0)) - t_1;
	}
	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 = Float64(log1p(x) / n)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-17)
		tmp = Float64(t_1 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= 0.2)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(((exp(1) ^ cbrt((t_0 ^ 2.0))) ^ cbrt(t_0)) - t_1);
	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[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-17], N[(t$95$1 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.2], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Power[N[Power[E, N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision], N[Power[t$95$0, 1/3], $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2.00000000000000014e-17

   1. Initial program 3.7

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

   2. Taylor expanded in x around inf 2.7

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

   3. Simplified 2.7

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
      Proof
      (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 x)))) n)) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (/.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 x)))) n)) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (*.f64 -1 (log.f64 x)) n)))) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (neg.f64 (/.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (log.f64 x))) n))) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (neg.f64 (/.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 x))) n))) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n)))) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (Rewrite<= *-commutative_binary64 (*.f64 n x))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 2.7

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

   if -2.00000000000000014e-17 < (/.f64 1 n) < 0.20000000000000001

   1. Initial program 44.4

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

   2. Taylor expanded in n around inf 14.8

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

   3. Simplified 14.8

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
      Proof
      (/.f64 (-.f64 (log1p.f64 x) (log.f64 x)) n): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 x))) (log.f64 x)) n): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 (log.f64 (+.f64 1 x)) 0)) (log.f64 x)) n): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (Rewrite=> +-rgt-identity_binary64 (log.f64 (+.f64 1 x))) (log.f64 x)) n): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 14.7

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

   5. Taylor expanded in n around 0 14.7

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

   6. Simplified 1.1

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}} \]
      Proof
      (/.f64 (log1p.f64 (/.f64 1 x)) n): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (/.f64 1 x)))) n): 50 points increase in error, 53 points decrease in error
      (/.f64 (log.f64 (+.f64 (Rewrite<= lft-mult-inverse_binary64 (*.f64 (/.f64 1 x) x)) (/.f64 1 x))) n): 26 points increase in error, 0 points decrease in error
      (/.f64 (log.f64 (+.f64 (*.f64 (/.f64 1 x) x) (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 1 x) 1)))) n): 0 points increase in error, 0 points decrease in error
      (/.f64 (log.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 (/.f64 1 x) (+.f64 x 1)))) n): 0 points increase in error, 0 points decrease in error
      (/.f64 (log.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1 (+.f64 x 1)) x))) n): 0 points increase in error, 26 points decrease in error
      (/.f64 (log.f64 (/.f64 (Rewrite=> *-lft-identity_binary64 (+.f64 x 1)) x)) n): 0 points increase in error, 0 points decrease in error
      (/.f64 (log.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 x)) x)) n): 0 points increase in error, 0 points decrease in error

   if 0.20000000000000001 < (/.f64 1 n)

   1. Initial program 4.2

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

   2. Taylor expanded in n around 0 4.2

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

   3. Simplified 0.9

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
      Proof
      (exp.f64 (/.f64 (log1p.f64 x) n)): 0 points increase in error, 0 points decrease in error
      (exp.f64 (/.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 x))) n)): 4 points increase in error, 0 points decrease in error

   4. Applied egg-rr 0.9

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left({e}^{\left(\sqrt[3]{{\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)}^{2}}\right)}\right)}^{\left(\sqrt[3]{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	13832

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;t_0 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \]

Alternative 2
Error	1.6
Cost	8200

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;t_0 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{x}{n}\right) + 0.5 \cdot \frac{\frac{x}{\frac{n}{x}}}{n}\right) - t_0\\ \end{array} \]

Alternative 3
Error	1.6
Cost	7560

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;t_0 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \end{array} \]

Alternative 4
Error	1.8
Cost	7304

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5
Error	2.7
Cost	7304

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -5.4 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -2.8047661122988896 \cdot 10^{-286}:\\ \;\;\;\;t_1 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq 5.2:\\ \;\;\;\;1 - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	2.5
Cost	7180

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -3.4 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -2.8047661122988896 \cdot 10^{-286}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;n \leq 5.2:\\ \;\;\;\;1 - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	7.7
Cost	6984

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -2.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -2.8047661122988896 \cdot 10^{-286}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	16.3
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} + \frac{\frac{\frac{-0.5}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+98}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 9
Error	34.8
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -2.45 \cdot 10^{-247}:\\ \;\;\;\;\left(1 + \frac{1}{n \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	28.3
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -2.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	40.4
Cost	320

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

Alternative 12
Error	40.0
Cost	320

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

Alternative 13
Error	40.0
Cost	320

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

Alternative 14
Error	61.1
Cost	192

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

Reproduce

herbie shell --seed 2022330 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))