2nthrt (problem 3.4.6)

Average Error: 32.7 → 12.1

Time: 33.1s

Precision: binary64

Cost: 39504

Math

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

↓

\[\begin{array}{l} t_0 := e^{\frac{\log x}{n}}\\ \mathbf{if}\;n \leq -1.7257384608900953 \cdot 10^{+85}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -1.0495287479582049 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1}{n}}{x + 0.5}\\ \mathbf{elif}\;n \leq -9.764986778507757 \cdot 10^{+23}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 120.07868581254682:\\ \;\;\;\;-\log \left(\frac{e^{{x}^{\left(\frac{1}{n}\right)}}}{e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}}\right)\\ \mathbf{elif}\;n \leq 9.111207859664332 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t_0}{x \cdot x}, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{t_0}{n \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

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 (exp (/ (log x) n))))
   (if (<= n -1.7257384608900953e+85)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= n -1.0495287479582049e+49)
       (/ (/ 1.0 n) (+ x 0.5))
       (if (<= n -9.764986778507757e+23)
         (/ (- x (log x)) n)
         (if (<= n 120.07868581254682)
           (- (log (/ (exp (pow x (/ 1.0 n))) (exp (exp (/ (log1p x) n))))))
           (if (<= n 9.111207859664332e+52)
             (fma
              (/ t_0 (* x x))
              (+ (/ 0.5 (* n n)) (/ -0.5 n))
              (/ t_0 (* n x)))
             (/ (- (log1p x) (log 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 = exp((log(x) / n));
	double tmp;
	if (n <= -1.7257384608900953e+85) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if (n <= -1.0495287479582049e+49) {
		tmp = (1.0 / n) / (x + 0.5);
	} else if (n <= -9.764986778507757e+23) {
		tmp = (x - log(x)) / n;
	} else if (n <= 120.07868581254682) {
		tmp = -log((exp(pow(x, (1.0 / n))) / exp(exp((log1p(x) / n)))));
	} else if (n <= 9.111207859664332e+52) {
		tmp = fma((t_0 / (x * x)), ((0.5 / (n * n)) + (-0.5 / n)), (t_0 / (n * x)));
	} else {
		tmp = (log1p(x) - log(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 = exp(Float64(log(x) / n))
	tmp = 0.0
	if (n <= -1.7257384608900953e+85)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (n <= -1.0495287479582049e+49)
		tmp = Float64(Float64(1.0 / n) / Float64(x + 0.5));
	elseif (n <= -9.764986778507757e+23)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (n <= 120.07868581254682)
		tmp = Float64(-log(Float64(exp((x ^ Float64(1.0 / n))) / exp(exp(Float64(log1p(x) / n))))));
	elseif (n <= 9.111207859664332e+52)
		tmp = fma(Float64(t_0 / Float64(x * x)), Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)), Float64(t_0 / Float64(n * x)));
	else
		tmp = Float64(Float64(log1p(x) - log(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[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.7257384608900953e+85], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -1.0495287479582049e+49], N[(N[(1.0 / n), $MachinePrecision] / N[(x + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -9.764986778507757e+23], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 120.07868581254682], (-N[Log[N[(N[Exp[N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Exp[N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[n, 9.111207859664332e+52], N[(N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if n < -1.72573846089009533e85

   1. Initial program 40.6

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

   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

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
      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 (+.f64 (log.f64 (+.f64 1 x)) (Rewrite<= metadata-eval (log.f64 1))) (log.f64 x)) n): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (+.f64 (log.f64 (+.f64 1 x)) (Rewrite=> metadata-eval 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 10.9

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

   if -1.72573846089009533e85 < n < -1.04952874795820485e49

   1. Initial program 54.8

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

   2. Taylor expanded in n around inf 22.2

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

   3. Simplified 22.2

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
      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 (+.f64 (log.f64 (+.f64 1 x)) (Rewrite<= metadata-eval (log.f64 1))) (log.f64 x)) n): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (+.f64 (log.f64 (+.f64 1 x)) (Rewrite=> metadata-eval 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 22.3

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]

   5. Applied egg-rr 22.3

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{\frac{1}{\mathsf{log1p}\left(x\right) - \log x}}} \]

   6. Taylor expanded in x around inf 28.3

      \[\leadsto \frac{\frac{1}{n}}{\color{blue}{0.5 + x}} \]

   7. Simplified 28.3

      \[\leadsto \frac{\frac{1}{n}}{\color{blue}{x + 0.5}} \]
      Proof
      (+.f64 x 1/2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 1/2 x)): 0 points increase in error, 0 points decrease in error

   if -1.04952874795820485e49 < n < -9.76498677850775652e23

   1. Initial program 58.8

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

   2. Taylor expanded in n around inf 28.5

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

   3. Simplified 28.5

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
      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 (+.f64 (log.f64 (+.f64 1 x)) (Rewrite<= metadata-eval (log.f64 1))) (log.f64 x)) n): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (+.f64 (log.f64 (+.f64 1 x)) (Rewrite=> metadata-eval 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. Taylor expanded in x around 0 32.9

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \log x}}{n} \]

   5. Simplified 32.9

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]
      Proof
      (-.f64 x (log.f64 x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 x (neg.f64 (log.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 x)))): 0 points increase in error, 0 points decrease in error

   if -9.76498677850775652e23 < n < 120.078685812546823

   1. Initial program 5.6

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

   2. Applied egg-rr 5.0

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   3. Applied egg-rr 5.0

      \[\leadsto \color{blue}{0 - \log \left(\frac{e^{{x}^{\left(\frac{1}{n}\right)}}}{e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}}\right)} \]

   if 120.078685812546823 < n < 9.1112078596643319e52

   1. Initial program 52.1

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

   2. Taylor expanded in x around inf 32.3

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

   3. Simplified 32.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{\log x}{n}}}{x \cdot x}, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{e^{\frac{\log x}{n}}}{x \cdot n}\right)} \]
      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, 2 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))): 5 points increase in error, 10 points decrease in error

   if 9.1112078596643319e52 < n

   1. Initial program 43.2

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

   2. Taylor expanded in n around inf 12.0

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

   3. Simplified 12.0

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
      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 (+.f64 (log.f64 (+.f64 1 x)) (Rewrite<= metadata-eval (log.f64 1))) (log.f64 x)) n): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (+.f64 (log.f64 (+.f64 1 x)) (Rewrite=> metadata-eval 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
3. Recombined 6 regimes into one program.

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.7257384608900953 \cdot 10^{+85}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -1.0495287479582049 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1}{n}}{x + 0.5}\\ \mathbf{elif}\;n \leq -9.764986778507757 \cdot 10^{+23}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 120.07868581254682:\\ \;\;\;\;-\log \left(\frac{e^{{x}^{\left(\frac{1}{n}\right)}}}{e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}}\right)\\ \mathbf{elif}\;n \leq 9.111207859664332 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{\frac{\log x}{n}}}{x \cdot x}, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{e^{\frac{\log x}{n}}}{n \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.1
Cost	34132

\[\begin{array}{l} t_0 := e^{\frac{\log x}{n}}\\ \mathbf{if}\;n \leq -1.7257384608900953 \cdot 10^{+85}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -1.0495287479582049 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1}{n}}{x + 0.5}\\ \mathbf{elif}\;n \leq -9.764986778507757 \cdot 10^{+23}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 120.07868581254682:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;n \leq 9.111207859664332 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t_0}{x \cdot x}, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{t_0}{n \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 2
Error	12.0
Cost	33040

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -1.7257384608900953 \cdot 10^{+85}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -1.0495287479582049 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1}{n}}{x + 0.5}\\ \mathbf{elif}\;n \leq -9.764986778507757 \cdot 10^{+23}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 120.07868581254682:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0}\right)\\ \mathbf{elif}\;n \leq 9.111207859664332 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{t_0}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 3
Error	11.9
Cost	20240

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -1.7257384608900953 \cdot 10^{+85}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -1.0495287479582049 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1}{n}}{x + 0.5}\\ \mathbf{elif}\;n \leq -9.764986778507757 \cdot 10^{+23}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 120.07868581254682:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \mathbf{elif}\;n \leq 9.111207859664332 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{t_0}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 4
Error	12.2
Cost	14032

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -1.7257384608900953 \cdot 10^{+85}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -1.0495287479582049 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1}{n}}{x + 0.5}\\ \mathbf{elif}\;n \leq -9.764986778507757 \cdot 10^{+23}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 120.07868581254682:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t_0\\ \mathbf{elif}\;n \leq 9.111207859664332 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{t_0}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 5
Error	6.8
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 580000:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \]

Alternative 6
Error	7.1
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 580000:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]

Alternative 7
Error	15.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 220000000:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{n}}{x + 0.5}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+134}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	16.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 580000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{n}}{x + 0.5}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+134}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	16.2
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.28:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{n}}{x + 0.5}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+134}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Error	26.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x + 0.5}\\ \end{array} \]

Alternative 11
Error	28.8
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ \mathbf{if}\;n \leq -0.48188130043467237:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-37}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	28.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;n \leq -0.48188130043467237:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-37}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 13
Error	38.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022306 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))