2nthrt (problem 3.4.6)

Average Error: 32.4 → 1.8

Time: 24.0s

Precision: binary64

Cost: 13380

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;-\mathsf{expm1}\left(t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t_0}}{x \cdot n}\\ \end{array} \]

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 (/ (log x) n)))
   (if (<= x 1.0) (- (expm1 t_0)) (/ (exp t_0) (* x 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 t_0 = log(x) / n;
	double tmp;
	if (x <= 1.0) {
		tmp = -expm1(t_0);
	} else {
		tmp = exp(t_0) / (x * 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 t_0 = Math.log(x) / n;
	double tmp;
	if (x <= 1.0) {
		tmp = -Math.expm1(t_0);
	} else {
		tmp = Math.exp(t_0) / (x * 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):
	t_0 = math.log(x) / n
	tmp = 0
	if x <= 1.0:
		tmp = -math.expm1(t_0)
	else:
		tmp = math.exp(t_0) / (x * 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)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-expm1(t_0));
	else
		tmp = Float64(exp(t_0) / Float64(x * 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_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 1.0], (-N[(Exp[t$95$0] - 1), $MachinePrecision]), N[(N[Exp[t$95$0], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 46.8

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

   2. Taylor expanded in x around 0 46.8

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

   3. Taylor expanded in x around inf 46.8

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

   4. Simplified 1.9

      \[\leadsto \color{blue}{-\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]
      Proof
      (neg.f64 (expm1.f64 (/.f64 (log.f64 x) n))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (expm1.f64 (/.f64 (Rewrite<= +-lft-identity_binary64 (+.f64 0 (log.f64 x))) n))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (expm1.f64 (/.f64 (+.f64 (Rewrite<= +-inverses_binary64 (-.f64 (log.f64 -1) (log.f64 -1))) (log.f64 x)) n))): 256 points increase in error, 0 points decrease in error
      (neg.f64 (expm1.f64 (/.f64 (Rewrite<= associate--r-_binary64 (-.f64 (log.f64 -1) (-.f64 (log.f64 -1) (log.f64 x)))) n))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (expm1.f64 (/.f64 (-.f64 (log.f64 -1) (Rewrite<= log-div_binary64 (log.f64 (/.f64 -1 x)))) n))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (expm1.f64 (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (log.f64 -1) (neg.f64 (log.f64 (/.f64 -1 x))))) n))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (expm1.f64 (/.f64 (+.f64 (log.f64 -1) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 -1 x))))) n))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (expm1.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))) n))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (/.f64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)) n)) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (exp.f64 (/.f64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)) n)) 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= metadata-eval (log.f64 1)) (-.f64 (exp.f64 (/.f64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)) n)) 1)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (log.f64 1) (exp.f64 (/.f64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)) n))) 1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (Rewrite=> metadata-eval 0) (exp.f64 (/.f64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)) n))) 1): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (exp.f64 (/.f64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)) n)))) 1): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 (exp.f64 (/.f64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)) n))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 1 (exp.f64 (/.f64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)) n)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (exp.f64 (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (log.f64 -1) (*.f64 -1 (log.f64 (/.f64 -1 x))))) n))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (exp.f64 (/.f64 (+.f64 (log.f64 -1) (Rewrite=> mul-1-neg_binary64 (neg.f64 (log.f64 (/.f64 -1 x))))) n))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (exp.f64 (/.f64 (Rewrite=> unsub-neg_binary64 (-.f64 (log.f64 -1) (log.f64 (/.f64 -1 x)))) n))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (exp.f64 (/.f64 (-.f64 (log.f64 -1) (Rewrite=> log-div_binary64 (-.f64 (log.f64 -1) (log.f64 x)))) n))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (exp.f64 (/.f64 (Rewrite=> associate--r-_binary64 (+.f64 (-.f64 (log.f64 -1) (log.f64 -1)) (log.f64 x))) n))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (exp.f64 (/.f64 (+.f64 (Rewrite=> +-inverses_binary64 0) (log.f64 x)) n))): 0 points increase in error, 256 points decrease in error
      (-.f64 1 (exp.f64 (/.f64 (Rewrite=> +-lft-identity_binary64 (log.f64 x)) n))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (exp.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 x)))) n))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (exp.f64 (/.f64 (neg.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 x)))) n))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (exp.f64 (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 1 x)))) n))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (exp.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))))): 0 points increase in error, 0 points decrease in error

   if 1 < x

   1. Initial program 20.5

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

   2. Taylor expanded in x around inf 11.8

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

   3. Simplified 11.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{\log x}{n}}}{x \cdot x}, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{e^{\frac{\log x}{n}}}{x \cdot n}\right)} \]
      Proof
      (fma.f64 (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 x)))) n)) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (/.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 x)))) n)) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (*.f64 -1 (log.f64 x)) n)))) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (neg.f64 (/.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (log.f64 x))) n))) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (neg.f64 (/.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 x))) n))) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n)))) (*.f64 x x)) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (Rewrite<= unpow2_binary64 (pow.f64 x 2))) (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (*.f64 n n)) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (/.f64 (*.f64 1/2 1) (Rewrite<= unpow2_binary64 (pow.f64 n 2))) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2)))) (/.f64 -1/2 n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (/.f64 (Rewrite<= metadata-eval (neg.f64 1/2)) n)) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1/2 n)))) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) n))) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 n))))) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n)))) (/.f64 (exp.f64 (/.f64 (log.f64 x) n)) (*.f64 x n))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.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
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.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
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.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
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.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
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.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
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.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
      (fma.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (/.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
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (pow.f64 x 2)) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n)))) (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (*.f64 n x)))): 1 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n)))) (pow.f64 x 2))) (/.f64 (exp.f64 (*.f64 -1 (/.f64 (log.f64 (/.f64 1 x)) n))) (*.f64 n x))): 6 points increase in error, 7 points decrease in error

   4. Taylor expanded in x around inf 1.8

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

   5. Simplified 1.8

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
      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 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (log.f64 x) n))))) (*.f64 x n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (neg.f64 (Rewrite<= distribute-frac-neg_binary64 (/.f64 (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 simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	13384

\[\begin{array}{l} \mathbf{if}\;x \leq 0.00049:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{1}{x \cdot n} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 2
Error	10.1
Cost	13188

\[\begin{array}{l} t_0 := -2 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 0.00049:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{x + t_0}{x \cdot t_0}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n}}{x \cdot x}\\ \end{array} \]

Alternative 3
Error	15.8
Cost	6852

\[\begin{array}{l} t_0 := -2 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 0.00049:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{x + t_0}{x \cdot t_0}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n}}{x \cdot x}\\ \end{array} \]

Alternative 4
Error	16.0
Cost	6788

\[\begin{array}{l} t_0 := -2 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 0.00049:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{x + t_0}{x \cdot t_0}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n}}{x \cdot x}\\ \end{array} \]

Alternative 5
Error	34.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;n \leq -199448363.30931035:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq 0:\\ \;\;\;\;\frac{\frac{-0.5}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 6
Error	40.1
Cost	320

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

Alternative 7
Error	61.1
Cost	192

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

Reproduce

herbie shell --seed 2022291 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))