2nthrt (problem 3.4.6)

Average Error: 32.9 → 1.8

Time: 25.5s

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 46.5

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

   2. Taylor expanded in x around 0 46.3

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

   3. Taylor expanded in x around 0 46.5

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

   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 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (/.f64 (log.f64 x) n)) 1))): 86 points increase in error, 92 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 < 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
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	13188

\[\begin{array}{l} \mathbf{if}\;x \leq 860:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+186}:\\ \;\;\;\;\left(1 + \frac{-0.5}{x}\right) \cdot \frac{\frac{1}{n}}{x} + \frac{1}{n \cdot {x}^{3}} \cdot \left(0.3333333333333333 + \frac{-0.25}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 2
Error	13.4
Cost	8860

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ t_1 := \frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1000000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{n} + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]

Alternative 3
Error	13.5
Cost	8860

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1000000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-167}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{n} + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]

Alternative 4
Error	13.5
Cost	8668

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ t_1 := \frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1000000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5
Error	16.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} \cdot \left(\frac{-0.5}{x} + \left(1 + \frac{0.3333333333333333}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 6
Error	16.3
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} \cdot \left(\frac{-0.5}{x} + \left(1 + \frac{0.3333333333333333}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 7
Error	33.6
Cost	840

\[\begin{array}{l} t_0 := \frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{if}\;n \leq -8.2 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.1 \cdot 10^{-262}:\\ \;\;\;\;\left(1 + \frac{1}{x \cdot n}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	27.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \end{array} \]

Alternative 9
Error	38.1
Cost	448

\[\frac{1}{n \cdot \left(x + 0.5\right)} \]

Alternative 10
Error	40.7
Cost	320

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

Alternative 11
Error	40.3
Cost	320

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

Alternative 12
Error	40.3
Cost	320

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

Alternative 13
Error	61.0
Cost	192

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

Reproduce

herbie shell --seed 2022308 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))