2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 91.3%

Time: 18.9s

Precision: binary64

Cost: 13188

• could not determine a ground truth

  1. x = -2
  2. n = 4.94066e-324

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} \cdot {x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \]

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 0.6)
   (- (expm1 (/ (log x) n)))
   (/ (* (/ 1.0 n) (pow x (/ 1.0 n))) x)))

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 <= 0.6) {
		tmp = -expm1((log(x) / n));
	} else {
		tmp = ((1.0 / n) * pow(x, (1.0 / n))) / x;
	}
	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 <= 0.6) {
		tmp = -Math.expm1((Math.log(x) / n));
	} else {
		tmp = ((1.0 / n) * Math.pow(x, (1.0 / n))) / x;
	}
	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 <= 0.6:
		tmp = -math.expm1((math.log(x) / n))
	else:
		tmp = ((1.0 / n) * math.pow(x, (1.0 / n))) / x
	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 <= 0.6)
		tmp = Float64(-expm1(Float64(log(x) / n)));
	else
		tmp = Float64(Float64(Float64(1.0 / n) * (x ^ Float64(1.0 / n))) / x);
	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, 0.6], (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]), N[(N[(N[(1.0 / n), $MachinePrecision] * N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.599999999999999978

   1. Initial program 45.9%

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

   2. Taylor expanded in x around 0 45.2%

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

   3. Simplified 45.2%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]
      Step-by-step derivation

      [Start] 45.2	\[ 1 - e^{\frac{\log x}{n}} \]
      *-rgt-identity [<=] 45.2	\[ 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]
      associate-*r/ [<=] 45.2	\[ 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]
      unpow-1 [<=] 45.2	\[ 1 - e^{\log x \cdot \color{blue}{{n}^{-1}}} \]
      exp-to-pow [=>] 45.2	\[ 1 - \color{blue}{{x}^{\left({n}^{-1}\right)}} \]
      unpow-1 [=>] 45.2	\[ 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   4. Taylor expanded in x around 0 45.2%

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

   5. Simplified 89.7%

      \[\leadsto \color{blue}{-\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]
      Step-by-step derivation

      [Start] 45.2	\[ 1 - e^{\frac{\log x}{n}} \]
      sub-neg [=>] 45.2	\[ \color{blue}{1 + \left(-e^{\frac{\log x}{n}}\right)} \]
      +-commutative [=>] 45.2	\[ \color{blue}{\left(-e^{\frac{\log x}{n}}\right) + 1} \]
      neg-sub0 [=>] 45.2	\[ \color{blue}{\left(0 - e^{\frac{\log x}{n}}\right)} + 1 \]
      metadata-eval [<=] 45.2	\[ \left(\color{blue}{\log 1} - e^{\frac{\log x}{n}}\right) + 1 \]
      associate-+l- [=>] 45.2	\[ \color{blue}{\log 1 - \left(e^{\frac{\log x}{n}} - 1\right)} \]
      metadata-eval [=>] 45.2	\[ \color{blue}{0} - \left(e^{\frac{\log x}{n}} - 1\right) \]
      sub0-neg [=>] 45.2	\[ \color{blue}{-\left(e^{\frac{\log x}{n}} - 1\right)} \]
      expm1-def [=>] 89.7	\[ -\color{blue}{\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 0.599999999999999978 < x

   1. Initial program 59.8%

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

   2. Taylor expanded in x around inf 97.3%

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

   3. Simplified 97.3%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
      Step-by-step derivation

      [Start] 97.3	\[ \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} \]
      log-rec [=>] 97.3	\[ \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]
      mul-1-neg [<=] 97.3	\[ \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]
      mul-1-neg [=>] 97.3	\[ \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]
      distribute-frac-neg [=>] 97.3	\[ \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]
      neg-mul-1 [<=] 97.3	\[ \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]
      remove-double-neg [=>] 97.3	\[ \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]
      *-rgt-identity [<=] 97.3	\[ \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]
      associate-*r/ [<=] 97.3	\[ \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]
      unpow-1 [<=] 97.3	\[ \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]
      exp-to-pow [=>] 97.3	\[ \frac{\color{blue}{{x}^{\left({n}^{-1}\right)}}}{n \cdot x} \]
      unpow-1 [=>] 97.3	\[ \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]
      *-commutative [=>] 97.3	\[ \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}} \]
      Step-by-step derivation

      [Start] 97.3	\[ \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} \]
      *-un-lft-identity [=>] 97.3	\[ \frac{\color{blue}{1 \cdot {x}^{\left(\frac{1}{n}\right)}}}{x \cdot n} \]
      times-frac [=>] 99.2	\[ \color{blue}{\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}} \]

   5. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]
      Step-by-step derivation

      [Start] 99.2	\[ \frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n} \]
      associate-*l/ [=>] 99.3	\[ \color{blue}{\frac{1 \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]
      *-un-lft-identity [<=] 99.3	\[ \frac{\color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}}{x} \]

   6. Applied egg-rr 99.3%

      \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{n}}}{x} \]
      Step-by-step derivation

      [Start] 99.3	\[ \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x} \]
      div-inv [=>] 99.3	\[ \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{n}}}{x} \]

3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} \cdot {x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	69.9%
Cost	7436

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 8.8 \cdot 10^{-281}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-152}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{elif}\;x \leq 0.007:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} \cdot t_0}{x}\\ \end{array} \]

Alternative 2
Accuracy	69.9%
Cost	7308

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 9 \cdot 10^{-281}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-152}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 0.031:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \end{array} \]

Alternative 3
Accuracy	69.9%
Cost	7308

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 2.1 \cdot 10^{-281}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-152}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{elif}\;x \leq 0.0048:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \end{array} \]

Alternative 4
Accuracy	56.8%
Cost	7116

\[\begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-281}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-152}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+29}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+252}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Accuracy	58.1%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+252}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Accuracy	57.9%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+30}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+252}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Accuracy	45.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;n \leq -3.2 \lor \neg \left(n \leq -8.6 \cdot 10^{-194}\right):\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Accuracy	45.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;n \leq -2.8 \lor \neg \left(n \leq -1.5 \cdot 10^{-190}\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Accuracy	30.4%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023161 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))