Hyperbolic arc-(co)secant

Average Error: 0.07% → 0.07%

Time: 4.0s

Precision: binary64

Cost: 13632

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (log1p (+ (/ (sqrt (- 1.0 (* x x))) x) (+ (/ 1.0 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(((sqrt((1.0 - (x * x))) / x) + ((1.0 / 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(((Math.sqrt((1.0 - (x * x))) / x) + ((1.0 / 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(((math.sqrt((1.0 - (x * x))) / x) + ((1.0 / 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(sqrt(Float64(1.0 - Float64(x * x))) / x) + Float64(Float64(1.0 / 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[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 0.07

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

2. Applied egg-rr 0.07

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

3. Final simplification 0.07

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

Alternatives

Alternative 1
Error	0.07%
Cost	13504

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

Alternative 2
Error	0.47%
Cost	6976

\[\log \left(\frac{2}{x}\right) + \left(x \cdot x\right) \cdot -0.25 \]

Alternative 3
Error	0.92%
Cost	6656

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

Alternative 4
Error	0.98%
Cost	6592

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

Alternative 5
Error	97.32%
Cost	320

\[\left(x \cdot x\right) \cdot -0.25 \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))