2nthrt (problem 3.4.6)

Average Error: 32.9 → 7.4

Time: 31.0s

Precision: binary64

Cost: 20036

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 1.95:\\ \;\;\;\;\frac{\log x + \left(\log x \cdot -2 + \log \left(x + 1\right)\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n} + 0}}{x \cdot n}\\ \end{array} \]

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (if (<= x 1.95)
   (/ (+ (log x) (+ (* (log x) -2.0) (log (+ x 1.0)))) n)
   (/ (exp (+ (/ (log x) n) 0.0)) (* 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 <= 1.95) {
		tmp = (log(x) + ((log(x) * -2.0) + log((x + 1.0)))) / n;
	} else {
		tmp = exp(((log(x) / n) + 0.0)) / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
end function

↓

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.95d0) then
        tmp = (log(x) + ((log(x) * (-2.0d0)) + log((x + 1.0d0)))) / n
    else
        tmp = exp(((log(x) / n) + 0.0d0)) / (x * n)
    end if
    code = tmp
end function

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 <= 1.95) {
		tmp = (Math.log(x) + ((Math.log(x) * -2.0) + Math.log((x + 1.0)))) / n;
	} else {
		tmp = Math.exp(((Math.log(x) / n) + 0.0)) / (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 <= 1.95:
		tmp = (math.log(x) + ((math.log(x) * -2.0) + math.log((x + 1.0)))) / n
	else:
		tmp = math.exp(((math.log(x) / n) + 0.0)) / (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 <= 1.95)
		tmp = Float64(Float64(log(x) + Float64(Float64(log(x) * -2.0) + log(Float64(x + 1.0)))) / n);
	else
		tmp = Float64(exp(Float64(Float64(log(x) / n) + 0.0)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
end

↓

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.95)
		tmp = (log(x) + ((log(x) * -2.0) + log((x + 1.0)))) / n;
	else
		tmp = exp(((log(x) / n) + 0.0)) / (x * n);
	end
	tmp_2 = 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, 1.95], N[(N[(N[Log[x], $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] + N[Log[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] + 0.0), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.94999999999999996

   1. Initial program 46.8

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

   2. Taylor expanded in n around inf 14.0

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

   3. Simplified 14.0

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

      [Start] 14.0	\[ \frac{\log \left(1 + x\right) - \log x}{n} \]
      rational_best-simplify-1 [=>] 14.0	\[ \frac{\log \color{blue}{\left(x + 1\right)} - \log x}{n} \]
      rational_best-simplify-16 [=>] 14.0	\[ \frac{\log \color{blue}{\left(x - -1\right)} - \log x}{n} \]

   4. Applied egg-rr 14.0

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

   5. Simplified 14.0

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

      [Start] 14.0	\[ \frac{\left(\log \left(x + 1\right) + \log x\right) + \left(-\log x \cdot 2\right)}{n} \]
      rational_best-simplify-1 [<=] 14.0	\[ \frac{\color{blue}{\left(-\log x \cdot 2\right) + \left(\log \left(x + 1\right) + \log x\right)}}{n} \]
      rational_best-simplify-43 [=>] 14.0	\[ \frac{\color{blue}{\log x + \left(\log \left(x + 1\right) + \left(-\log x \cdot 2\right)\right)}}{n} \]
      rational_best-simplify-16 [=>] 14.0	\[ \frac{\log x + \left(\log \color{blue}{\left(x - -1\right)} + \left(-\log x \cdot 2\right)\right)}{n} \]

   6. Applied egg-rr 14.0

      \[\leadsto \frac{\log x + \color{blue}{\left(\left(\log \left(x + 1\right) + \log x \cdot -2\right) - 0\right)}}{n} \]

   7. Simplified 14.0

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

      [Start] 14.0	\[ \frac{\log x + \left(\left(\log \left(x + 1\right) + \log x \cdot -2\right) - 0\right)}{n} \]
      rational_best-simplify-6 [=>] 14.0	\[ \frac{\log x + \color{blue}{\left(\log \left(x + 1\right) + \log x \cdot -2\right)}}{n} \]
      rational_best-simplify-1 [=>] 14.0	\[ \frac{\log x + \color{blue}{\left(\log x \cdot -2 + \log \left(x + 1\right)\right)}}{n} \]

   if 1.94999999999999996 < 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 1.8

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

   3. Simplified 1.8

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

      [Start] 1.8	\[ \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} \]
      rational_best-simplify-2 [=>] 1.8	\[ \frac{e^{\color{blue}{\frac{\log \left(\frac{1}{x}\right)}{n} \cdot -1}}}{n \cdot x} \]
      rational_best-simplify-12 [=>] 1.8	\[ \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]
      rational_best-simplify-2 [=>] 1.8	\[ \frac{e^{-\frac{\log \left(\frac{1}{x}\right)}{n}}}{\color{blue}{x \cdot n}} \]

   4. Taylor expanded in x around 0 1.8

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

   5. Simplified 1.8

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n} + 0}}{x \cdot n}} \]
      Proof

      [Start] 1.8	\[ \frac{e^{--1 \cdot \frac{\log x}{n}}}{n \cdot x} \]
      rational_best-simplify-11 [=>] 1.8	\[ \frac{e^{\color{blue}{0 - -1 \cdot \frac{\log x}{n}}}}{n \cdot x} \]
      rational_best-simplify-2 [=>] 1.8	\[ \frac{e^{0 - \color{blue}{\frac{\log x}{n} \cdot -1}}}{n \cdot x} \]
      rational_best-simplify-12 [=>] 1.8	\[ \frac{e^{0 - \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]
      rational_best-simplify-11 [=>] 1.8	\[ \frac{e^{0 - \color{blue}{\left(0 - \frac{\log x}{n}\right)}}}{n \cdot x} \]
      rational_best-simplify-46 [=>] 1.8	\[ \frac{e^{\color{blue}{\frac{\log x}{n} + \left(0 - 0\right)}}}{n \cdot x} \]
      metadata-eval [=>] 1.8	\[ \frac{e^{\frac{\log x}{n} + \color{blue}{0}}}{n \cdot x} \]
      rational_best-simplify-2 [=>] 1.8	\[ \frac{e^{\frac{\log x}{n} + 0}}{\color{blue}{x \cdot n}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95:\\ \;\;\;\;\frac{\log x + \left(\log x \cdot -2 + \log \left(x + 1\right)\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n} + 0}}{x \cdot n}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.1
Cost	13908

\[\begin{array}{l} t_0 := \frac{\log \left(x - -1\right) - \log x}{n}\\ t_1 := \left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -8 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -3800000000000:\\ \;\;\;\;\frac{1}{x \cdot n} - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\\ \mathbf{elif}\;n \leq -0.56:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -9.127852981362938 \cdot 10^{-306}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;n \leq 0.8:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	7.4
Cost	13508

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{\log \left(x - -1\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n} + 0}}{x \cdot n}\\ \end{array} \]

Alternative 3
Error	15.8
Cost	7432

\[\begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\frac{\log \left(\frac{1}{x}\right) + x}{n}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+184}:\\ \;\;\;\;\frac{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 4
Error	15.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\log \left(\frac{1}{x}\right) + x}{n}\\ \mathbf{elif}\;x \leq 76000000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 5
Error	15.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 76000000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 6
Error	16.1
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 76000000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 7
Error	29.4
Cost	584

\[\begin{array}{l} t_0 := \frac{1}{x \cdot n}\\ \mathbf{if}\;n \leq -6.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.15 \cdot 10^{-75}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	29.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;n \leq -2.75:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-77}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 9
Error	39.4
Cost	192

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

Reproduce

herbie shell --seed 2023092 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))