Hyperbolic arc-(co)secant

Average Accuracy: 99.9% → 99.9%

Time: 7.2s

Precision: binary64

Cost: 13504

Math

\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]

↓

\[\mathsf{log1p}\left(\frac{1 + \sqrt{1 - x \cdot x}}{x} + -1\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))

↓

(FPCore (x)
 :precision binary64
 (log1p (+ (/ (+ 1.0 (sqrt (- 1.0 (* x x)))) x) -1.0)))

C

double code(double x) {
	return log(((1.0 / x) + (sqrt((1.0 - (x * x))) / x)));
}

↓

double code(double x) {
	return log1p((((1.0 + sqrt((1.0 - (x * x)))) / x) + -1.0));
}

Java

public static double code(double x) {
	return Math.log(((1.0 / x) + (Math.sqrt((1.0 - (x * x))) / x)));
}

↓

public static double code(double x) {
	return Math.log1p((((1.0 + Math.sqrt((1.0 - (x * x)))) / x) + -1.0));
}

Python

def code(x):
	return math.log(((1.0 / x) + (math.sqrt((1.0 - (x * x))) / x)))

↓

def code(x):
	return math.log1p((((1.0 + math.sqrt((1.0 - (x * x)))) / x) + -1.0))

Julia

function code(x)
	return log(Float64(Float64(1.0 / x) + Float64(sqrt(Float64(1.0 - Float64(x * x))) / x)))
end

↓

function code(x)
	return log1p(Float64(Float64(Float64(1.0 + sqrt(Float64(1.0 - Float64(x * x)))) / x) + -1.0))
end

Wolfram

code[x_] := N[Log[N[(N[(1.0 / x), $MachinePrecision] + N[(N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := N[Log[1 + N[(N[(N[(1.0 + N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]

2. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\sqrt{1 - x \cdot x}}{x} + \left(\frac{1}{x} - 1\right)\right)} \]
   Proof

   [Start] 99.9	\[ \log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
   log1p-expm1-u [=>] 99.9	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\right)\right)} \]
   expm1-udef [=>] 99.9	\[ \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)} - 1}\right) \]
   add-exp-log [<=] 99.9	\[ \mathsf{log1p}\left(\color{blue}{\left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)} - 1\right) \]
   +-commutative [=>] 99.9	\[ \mathsf{log1p}\left(\color{blue}{\left(\frac{\sqrt{1 - x \cdot x}}{x} + \frac{1}{x}\right)} - 1\right) \]
   associate--l+ [=>] 99.9	\[ \mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1 - x \cdot x}}{x} + \left(\frac{1}{x} - 1\right)}\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1 + \sqrt{1 - x \cdot x}}{x} + -1\right)} \]
   Proof

   [Start] 99.9	\[ \mathsf{log1p}\left(\frac{\sqrt{1 - x \cdot x}}{x} + \left(\frac{1}{x} - 1\right)\right) \]
   sub-neg [=>] 99.9	\[ \mathsf{log1p}\left(\frac{\sqrt{1 - x \cdot x}}{x} + \color{blue}{\left(\frac{1}{x} + \left(-1\right)\right)}\right) \]
   metadata-eval [=>] 99.9	\[ \mathsf{log1p}\left(\frac{\sqrt{1 - x \cdot x}}{x} + \left(\frac{1}{x} + \color{blue}{-1}\right)\right) \]
   associate-+r+ [=>] 99.9	\[ \mathsf{log1p}\left(\color{blue}{\left(\frac{\sqrt{1 - x \cdot x}}{x} + \frac{1}{x}\right) + -1}\right) \]
   *-rgt-identity [<=] 99.9	\[ \mathsf{log1p}\left(\left(\frac{\color{blue}{\sqrt{1 - x \cdot x} \cdot 1}}{x} + \frac{1}{x}\right) + -1\right) \]
   associate-*r/ [<=] 99.9	\[ \mathsf{log1p}\left(\left(\color{blue}{\sqrt{1 - x \cdot x} \cdot \frac{1}{x}} + \frac{1}{x}\right) + -1\right) \]
   distribute-lft1-in [=>] 99.9	\[ \mathsf{log1p}\left(\color{blue}{\left(\sqrt{1 - x \cdot x} + 1\right) \cdot \frac{1}{x}} + -1\right) \]
   +-commutative [<=] 99.9	\[ \mathsf{log1p}\left(\color{blue}{\left(1 + \sqrt{1 - x \cdot x}\right)} \cdot \frac{1}{x} + -1\right) \]
   associate-*r/ [=>] 99.9	\[ \mathsf{log1p}\left(\color{blue}{\frac{\left(1 + \sqrt{1 - x \cdot x}\right) \cdot 1}{x}} + -1\right) \]
   *-commutative [<=] 99.9	\[ \mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \left(1 + \sqrt{1 - x \cdot x}\right)}}{x} + -1\right) \]
   *-lft-identity [=>] 99.9	\[ \mathsf{log1p}\left(\frac{\color{blue}{1 + \sqrt{1 - x \cdot x}}}{x} + -1\right) \]

4. Final simplification 99.9%

   \[\leadsto \mathsf{log1p}\left(\frac{1 + \sqrt{1 - x \cdot x}}{x} + -1\right) \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13376

\[\log \left(\frac{1 + \sqrt{1 - x \cdot x}}{x}\right) \]

Alternative 2
Accuracy	99.5%
Cost	6976

\[\log \left(x \cdot -0.5 + 2 \cdot \frac{1}{x}\right) \]

Alternative 3
Accuracy	99.1%
Cost	6656

\[-\log \left(x \cdot 0.5\right) \]

Alternative 4
Accuracy	99.0%
Cost	6592

\[\log \left(\frac{2}{x}\right) \]

Reproduce

herbie shell --seed 2023131 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))