2nthrt (problem 3.4.6)

Average Error: 33.0 → 1.5

Time: 32.6s

Precision: binary64

Cost: 13640

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;-\mathsf{expm1}\left(t_0\right)\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;-2 \cdot \frac{\log \left(\sqrt{\frac{x}{x + 1}}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t_0}}{x \cdot n}\\ \end{array} \]

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 1.95e-16)
     (- (expm1 t_0))
     (if (<= x 5000000.0)
       (* -2.0 (/ (log (sqrt (/ x (+ x 1.0)))) n))
       (/ (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 <= 1.95e-16) {
		tmp = -expm1(t_0);
	} else if (x <= 5000000.0) {
		tmp = -2.0 * (log(sqrt((x / (x + 1.0)))) / n);
	} 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 <= 1.95e-16) {
		tmp = -Math.expm1(t_0);
	} else if (x <= 5000000.0) {
		tmp = -2.0 * (Math.log(Math.sqrt((x / (x + 1.0)))) / n);
	} 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 <= 1.95e-16:
		tmp = -math.expm1(t_0)
	elif x <= 5000000.0:
		tmp = -2.0 * (math.log(math.sqrt((x / (x + 1.0)))) / n)
	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 <= 1.95e-16)
		tmp = Float64(-expm1(t_0));
	elseif (x <= 5000000.0)
		tmp = Float64(-2.0 * Float64(log(sqrt(Float64(x / Float64(x + 1.0)))) / n));
	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, 1.95e-16], (-N[(Exp[t$95$0] - 1), $MachinePrecision]), If[LessEqual[x, 5000000.0], N[(-2.0 * N[(N[Log[N[Sqrt[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[Exp[t$95$0], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.94999999999999989e-16

   1. Initial program 46.1

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

   2. Taylor expanded in x around 0 48.7

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

   3. Simplified 48.7

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}, x \cdot x, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
      Proof
      (+.f64 1 (fma.f64 (+.f64 (/.f64 1/2 (*.f64 n n)) (/.f64 -1/2 n)) (*.f64 x x) (/.f64 x n))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (fma.f64 (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (*.f64 n n)) (/.f64 -1/2 n)) (*.f64 x x) (/.f64 x n))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (fma.f64 (+.f64 (/.f64 (*.f64 1/2 1) (Rewrite<= unpow2_binary64 (pow.f64 n 2))) (/.f64 -1/2 n)) (*.f64 x x) (/.f64 x n))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (fma.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2)))) (/.f64 -1/2 n)) (*.f64 x x) (/.f64 x n))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (fma.f64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (/.f64 (Rewrite<= metadata-eval (neg.f64 1/2)) n)) (*.f64 x x) (/.f64 x n))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (fma.f64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1/2 n)))) (*.f64 x x) (/.f64 x n))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (fma.f64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) n))) (*.f64 x x) (/.f64 x n))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (fma.f64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 n))))) (*.f64 x x) (/.f64 x n))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (fma.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n)))) (*.f64 x x) (/.f64 x n))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (fma.f64 (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (/.f64 x n))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (pow.f64 x 2)) (/.f64 x n)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (*.f64 (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (pow.f64 x 2))) (/.f64 x n))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 x n) (+.f64 1 (*.f64 (-.f64 (*.f64 1/2 (/.f64 1 (pow.f64 n 2))) (*.f64 1/2 (/.f64 1 n))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in x around 0 46.1

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

   5. Simplified 0.9

      \[\leadsto \color{blue}{-\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]
      Proof
      (neg.f64 (expm1.f64 (/.f64 (log.f64 x) n))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (/.f64 (log.f64 x) n)) 1))): 90 points increase in error, 106 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 1.94999999999999989e-16 < x < 5e6

   1. Initial program 48.0

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

   2. Taylor expanded in n around inf 15.3

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

   3. Applied egg-rr 14.6

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

   4. Applied egg-rr 14.7

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

   5. Taylor expanded in n around 0 14.7

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

   if 5e6 < 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 1.1

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

   3. Simplified 1.1

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;-2 \cdot \frac{\log \left(\sqrt{\frac{x}{x + 1}}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	13512

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;-\mathsf{expm1}\left(t_0\right)\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t_0}}{x \cdot n}\\ \end{array} \]

Alternative 2
Error	9.5
Cost	13188

\[\begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{elif}\;x \leq 1080000000:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+14}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \end{array} \]

Alternative 3
Error	11.5
Cost	9108

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{x}{n} + \left(x \cdot \frac{x}{n}\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4
Error	11.5
Cost	8340

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5
Error	11.7
Cost	8084

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -20000000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 6
Error	11.7
Cost	8084

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 7
Error	15.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 0.088:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 8
Error	15.9
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.088:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 9
Error	29.9
Cost	1348

\[\begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ \mathbf{if}\;n \leq -175291.3910876423:\\ \;\;\;\;t_0 + \frac{\frac{\frac{1}{x}}{n}}{x} \cdot \left(\frac{0.3333333333333333}{x} + -0.5\right)\\ \mathbf{elif}\;n \leq 1.96 \cdot 10^{-69}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	29.9
Cost	1220

\[\begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ \mathbf{if}\;n \leq -175291.3910876423:\\ \;\;\;\;t_0 + \frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;n \leq 1.96 \cdot 10^{-69}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	29.6
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ \mathbf{if}\;n \leq -175291.3910876423:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.96 \cdot 10^{-69}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	61.0
Cost	192

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

Alternative 13
Error	39.7
Cost	192

\[\frac{0}{n} \]

Reproduce

herbie shell --seed 2022297 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))