Average Error: 32.1 → 1.9
Time: 19.5s
Precision: binary64
Cost: 13380
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;-\mathsf{expm1}\left(t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t_0}}{x \cdot n}\\ \end{array} \]
(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 (/ (log x) n)))
   (if (<= x 0.56) (- (expm1 t_0)) (/ (exp t_0) (* x n)))))
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 = log(x) / n;
	double tmp;
	if (x <= 0.56) {
		tmp = -expm1(t_0);
	} else {
		tmp = exp(t_0) / (x * n);
	}
	return tmp;
}

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.log(x) / n;
	double tmp;
	if (x <= 0.56) {
		tmp = -Math.expm1(t_0);
	} else {
		tmp = Math.exp(t_0) / (x * n);
	}
	return tmp;
}

def code(x, n):
	return math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
↓
def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if x <= 0.56:
		tmp = -math.expm1(t_0)
	else:
		tmp = math.exp(t_0) / (x * n)
	return tmp

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(log(x) / n)
	tmp = 0.0
	if (x <= 0.56)
		tmp = Float64(-expm1(t_0));
	else
		tmp = Float64(exp(t_0) / Float64(x * n));
	end
	return tmp
end

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[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 0.56], (-N[(Exp[t$95$0] - 1), $MachinePrecision]), N[(N[Exp[t$95$0], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
t_0 := \frac{\log x}{n}\\
\mathbf{if}\;x \leq 0.56:\\
\;\;\;\;-\mathsf{expm1}\left(t_0\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{t_0}}{x \cdot n}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 0.56000000000000005

    1. Initial program 46.5

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

    2. Taylor expanded in x around 0 46.5

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

    3. Taylor expanded in x around 0 46.5

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]

    4. Simplified2.0

      \[\leadsto \color{blue}{-\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]
      Proof
      (neg.f64 (expm1.f64 (/.f64 (log.f64 x) n))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (/.f64 (log.f64 x) n)) 1))): 85 points increase in error, 88 points decrease in error
      (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (exp.f64 (/.f64 (log.f64 x) n)) 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= metadata-eval (log.f64 1)) (-.f64 (exp.f64 (/.f64 (log.f64 x) n)) 1)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (log.f64 1) (exp.f64 (/.f64 (log.f64 x) n))) 1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (Rewrite=> metadata-eval 0) (exp.f64 (/.f64 (log.f64 x) n))) 1): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (exp.f64 (/.f64 (log.f64 x) n)))) 1): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 (exp.f64 (/.f64 (log.f64 x) n))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 1 (exp.f64 (/.f64 (log.f64 x) n)))): 0 points increase in error, 0 points decrease in error

    if 0.56000000000000005 < x

    1. Initial program 20.6

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

    2. Taylor expanded in x around inf 1.9

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

    3. Simplified1.9

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
      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
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternatives

Alternative 1
Error1.9
Cost13576
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \end{array} \]
Alternative 2
Error1.9
Cost7304
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
Alternative 3
Error7.3
Cost6984
\[\begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -4.4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -8.35739129220041 \cdot 10^{-307}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error16.3
Cost6788
\[\begin{array}{l} \mathbf{if}\;x \leq 0.33:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+97}:\\ \;\;\;\;\frac{1}{x \cdot n + \left(0.25 \cdot \frac{n}{x} + \left(n \cdot 0.5 + \frac{n}{x} \cdot -0.3333333333333333\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 5
Error34.4
Cost712
\[\begin{array}{l} \mathbf{if}\;n \leq -2.1:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -8.35739129220041 \cdot 10^{-307}:\\ \;\;\;\;\frac{-0.5}{n \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
Alternative 6
Error32.3
Cost708
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000:\\ \;\;\;\;\frac{-0.5}{n \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \end{array} \]
Alternative 7
Error26.5
Cost708
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \end{array} \]
Alternative 8
Error40.5
Cost320
\[\frac{1}{x \cdot n} \]
Alternative 9
Error40.1
Cost320
\[\frac{\frac{1}{n}}{x} \]
Alternative 10
Error40.1
Cost320
\[\frac{\frac{1}{x}}{n} \]
Alternative 11
Error61.1
Cost192
\[\frac{x}{n} \]

Error

Reproduce

herbie shell --seed 2022328 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))