2nthrt (problem 3.4.6)

Average Error: 33.3 → 1.4

Time: 28.4s

Precision: binary64

Cost: 13188

Math

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

↓

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

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (- (expm1 (/ (log x) n)))
   (* (pow x (/ 1.0 n)) (/ (/ 1.0 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 tmp;
	if (x <= 1.0) {
		tmp = -expm1((log(x) / n));
	} else {
		tmp = pow(x, (1.0 / n)) * ((1.0 / 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 tmp;
	if (x <= 1.0) {
		tmp = -Math.expm1((Math.log(x) / n));
	} else {
		tmp = Math.pow(x, (1.0 / n)) * ((1.0 / 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):
	tmp = 0
	if x <= 1.0:
		tmp = -math.expm1((math.log(x) / n))
	else:
		tmp = math.pow(x, (1.0 / n)) * ((1.0 / 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)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-expm1(Float64(log(x) / n)));
	else
		tmp = Float64((x ^ Float64(1.0 / n)) * Float64(Float64(1.0 / 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_] := If[LessEqual[x, 1.0], (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]), N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 47.6

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

   2. Taylor expanded in x around 0 47.6

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

   3. Taylor expanded in x around inf 47.6

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

   4. Simplified 2.0

      \[\leadsto \color{blue}{-\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]
      Proof
      (neg.f64 (expm1.f64 (/.f64 (log.f64 x) n))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (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 21.2

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

   2. Taylor expanded in x around inf 1.7

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

   3. Simplified 1.7

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

   4. Applied egg-rr 0.9

      \[\leadsto \color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

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

Alternatives

Alternative 1
Error	12.9
Cost	8344

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ t_1 := t_0 + \frac{\frac{1}{x \cdot x}}{n} \cdot \left(\left(\frac{0.3333333333333333}{x} + -0.5\right) + \frac{-0.25}{x \cdot x}\right)\\ t_2 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -4 \cdot 10^{-97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2
Error	18.0
Cost	8216

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ t_1 := t_0 + \frac{\frac{1}{x \cdot x}}{n} \cdot \left(\left(\frac{0.3333333333333333}{x} + -0.5\right) + \frac{-0.25}{x \cdot x}\right)\\ t_2 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -4 \cdot 10^{-97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log x}{n}\\ \end{array} \]

Alternative 3
Error	8.5
Cost	7436

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.01 \cdot 10^{-286}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-242}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 10.2:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 4
Error	8.5
Cost	7308

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.01 \cdot 10^{-286}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-242}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 10.2:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \end{array} \]

Alternative 5
Error	15.8
Cost	6852

\[\begin{array}{l} t_0 := \frac{-0.25}{x \cdot x}\\ \mathbf{if}\;x \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} + \frac{\frac{1}{x \cdot x}}{n} \cdot \left(\left(\frac{0.3333333333333333}{x} + -0.5\right) + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{\frac{1}{n}}} \cdot \left(\frac{0.3333333333333333}{x} + \left(-0.5 + t_0\right)\right)\\ \end{array} \]

Alternative 6
Error	15.9
Cost	6788

\[\begin{array}{l} t_0 := \frac{-0.25}{x \cdot x}\\ \mathbf{if}\;x \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} + \frac{\frac{1}{x \cdot x}}{n} \cdot \left(\left(\frac{0.3333333333333333}{x} + -0.5\right) + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{\frac{1}{n}}} \cdot \left(\frac{0.3333333333333333}{x} + \left(-0.5 + t_0\right)\right)\\ \end{array} \]

Alternative 7
Error	33.4
Cost	1740

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -388.04321068177927:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-92}:\\ \;\;\;\;-1 + \left(1 + t_0\right)\\ \mathbf{elif}\;n \leq 0:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{\frac{1}{n}}} \cdot \left(\frac{0.3333333333333333}{x} + \left(-0.5 + \frac{-0.25}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	35.3
Cost	1352

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -388.04321068177927:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 545447863634762.9:\\ \;\;\;\;-1 + \left(1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(1 + \frac{0.3333333333333333}{x \cdot x}\right) + \frac{-0.5}{x}\right)\\ \end{array} \]

Alternative 9
Error	29.1
Cost	1352

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -388.04321068177927:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(1 + \frac{0.3333333333333333}{x \cdot x}\right) + \frac{-0.5}{x}\right)\\ \end{array} \]

Alternative 10
Error	35.0
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -388.04321068177927:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-275}:\\ \;\;\;\;-1 + \left(1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	40.0
Cost	320

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

Reproduce

herbie shell --seed 2022295 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))