2nthrt (problem 3.4.6)

Average Error: 33.2 → 1.2

Time: 17.0s

Precision: binary64

Cost: 13832

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-11)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-9)
     (/ (log1p (/ 1.0 x)) n)
     (- (exp (/ x n)) (pow x (/ 1.0 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 ((1.0 / n) <= -5e-11) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-9) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / 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 ((1.0 / n) <= -5e-11) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-9) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	return math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))

↓

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-11:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e-9:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / 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 (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-9)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / 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[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-9], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5.00000000000000018e-11

   1. Initial program 2.2

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

   2. Taylor expanded in x around inf 2.5

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

   3. Simplified 2.5

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 5.0000000000000001e-9

   1. Initial program 45.3

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

   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 0.6

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}} \]
      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): 44 points increase in error, 45 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): 1 points increase in error, 1 points decrease in error
      (/.f64 (log.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1 (+.f64 x 1)) x))) n): 1 points increase in error, 27 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 5.0000000000000001e-9 < (/.f64 1 n)

   1. Initial program 5.8

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

   2. Taylor expanded in n around 0 5.8

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

   3. Simplified 2.7

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
      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)): 1 points increase in error, 0 points decrease in error

   4. Taylor expanded in x around 0 2.9

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	13508

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2
Error	1.5
Cost	7436

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -8:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.8773133851961986 \cdot 10^{-298}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 3200000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.6
Cost	7180

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -8.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.8773133851961986 \cdot 10^{-298}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 28000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	6.8
Cost	6984

\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -6.2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -4.100026628795156 \cdot 10^{-309}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.6
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	29.5
Cost	584

\[\begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ \mathbf{if}\;n \leq -13.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.42 \cdot 10^{-84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	29.1
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ \mathbf{if}\;n \leq -10:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.42 \cdot 10^{-84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	29.1
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -7.4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.42 \cdot 10^{-84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	39.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))