Hyperbolic arc-(co)secant

Average Error: 0.0 → 0.0

Time: 6.0s

Precision: binary64

Cost: 13440

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

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.log((x / (1.0 + Math.sqrt((1.0 - (x * x))))));
}

Python

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

↓

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

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

MATLAB

function tmp = code(x)
	tmp = log(((1.0 / x) + (sqrt((1.0 - (x * x))) / x)));
end

↓

function tmp = code(x)
	tmp = -log((x / (1.0 + sqrt((1.0 - (x * x))))));
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[N[(x / N[(1.0 + N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ \log \left(\frac{\sqrt{1 - x \cdot x}}{x} + {x}^{-1}\right) + 0 \]
   +-rgt-identity [=>] 0.0	\[ \color{blue}{\log \left(\frac{\sqrt{1 - x \cdot x}}{x} + {x}^{-1}\right)} \]
   *-lft-identity [<=] 0.0	\[ \log \left(\color{blue}{1 \cdot \frac{\sqrt{1 - x \cdot x}}{x}} + {x}^{-1}\right) \]
   *-commutative [<=] 0.0	\[ \log \left(\color{blue}{\frac{\sqrt{1 - x \cdot x}}{x} \cdot 1} + {x}^{-1}\right) \]
   associate-*l/ [=>] 0.0	\[ \log \left(\color{blue}{\frac{\sqrt{1 - x \cdot x} \cdot 1}{x}} + {x}^{-1}\right) \]
   associate-*r/ [<=] 0.0	\[ \log \left(\color{blue}{\sqrt{1 - x \cdot x} \cdot \frac{1}{x}} + {x}^{-1}\right) \]
   unpow-1 [<=] 0.0	\[ \log \left(\sqrt{1 - x \cdot x} \cdot \color{blue}{{x}^{-1}} + {x}^{-1}\right) \]
   distribute-lft1-in [=>] 0.0	\[ \log \color{blue}{\left(\left(\sqrt{1 - x \cdot x} + 1\right) \cdot {x}^{-1}\right)} \]
   +-commutative [<=] 0.0	\[ \log \left(\color{blue}{\left(1 + \sqrt{1 - x \cdot x}\right)} \cdot {x}^{-1}\right) \]
   unpow-1 [=>] 0.0	\[ \log \left(\left(1 + \sqrt{1 - x \cdot x}\right) \cdot \color{blue}{\frac{1}{x}}\right) \]
   associate-*r/ [=>] 0.0	\[ \log \color{blue}{\left(\frac{\left(1 + \sqrt{1 - x \cdot x}\right) \cdot 1}{x}\right)} \]
   +-commutative [=>] 0.0	\[ \log \left(\frac{\color{blue}{\left(\sqrt{1 - x \cdot x} + 1\right)} \cdot 1}{x}\right) \]
   distribute-rgt1-in [<=] 0.0	\[ \log \left(\frac{\color{blue}{1 + \sqrt{1 - x \cdot x} \cdot 1}}{x}\right) \]
   *-rgt-identity [=>] 0.0	\[ \log \left(\frac{1 + \color{blue}{\sqrt{1 - x \cdot x}}}{x}\right) \]

4. Applied egg-rr 0.0

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

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	13376

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

Alternative 2
Error	0.3
Cost	7040

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

Alternative 3
Error	0.4
Cost	6976

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

Alternative 4
Error	0.4
Cost	6976

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

Alternative 5
Error	0.6
Cost	6656

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

Alternative 6
Error	0.7
Cost	6592

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

Reproduce

herbie shell --seed 2023056 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))