Hyperbolic arc-cosine

Average Error: 31.5 → 0.5

Time: 3.5s

Precision: binary64

Cost: 6656

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (- (log (/ 0.5 x))))

C

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

↓

double code(double x) {
	return -log((0.5 / x));
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = -log((0.5d0 / x))
end function

Java

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

↓

public static double code(double x) {
	return -Math.log((0.5 / x));
}

Python

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

↓

def code(x):
	return -math.log((0.5 / x))

Julia

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

↓

function code(x)
	return Float64(-log(Float64(0.5 / x)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = -log((0.5 / x));
end

Wolfram

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

↓

code[x_] := (-N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 31.5

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

2. Applied egg-rr 63.0

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

3. Simplified 63.0

   \[\leadsto \color{blue}{-\log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)} \]
   Proof
   (neg.f64 (log.f64 (-.f64 x (sqrt.f64 (fma.f64 x x -1))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> neg-sub0_binary64 (-.f64 0 (log.f64 (-.f64 x (sqrt.f64 (fma.f64 x x -1)))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= metadata-eval (log.f64 1)) (log.f64 (-.f64 x (sqrt.f64 (fma.f64 x x -1))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (log.f64 (Rewrite<= metadata-eval (+.f64 1 0))) (log.f64 (-.f64 x (sqrt.f64 (fma.f64 x x -1))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite=> log1p-def_binary64 (log1p.f64 0)) (log.f64 (-.f64 x (sqrt.f64 (fma.f64 x x -1))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (log1p.f64 (Rewrite<= mul0-rgt_binary64 (*.f64 x 0))) (log.f64 (-.f64 x (sqrt.f64 (fma.f64 x x -1))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (log1p.f64 (*.f64 x (Rewrite<= +-inverses_binary64 (-.f64 x x)))) (log.f64 (-.f64 x (sqrt.f64 (fma.f64 x x -1))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (*.f64 x (-.f64 x x))))) (log.f64 (-.f64 x (sqrt.f64 (fma.f64 x x -1))))): 0 points increase in error, 0 points decrease in error

4. Taylor expanded in x around inf 0.5

   \[\leadsto -\log \color{blue}{\left(\frac{0.5}{x}\right)} \]

5. Final simplification 0.5

   \[\leadsto -\log \left(\frac{0.5}{x}\right) \]

Alternatives

Alternative 1
Error	0.6
Cost	6592

\[\log \left(x + x\right) \]

Alternative 2
Error	62.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))