Hyperbolic arc-cosine

Average Error: 32.5 → 0.0

Time: 4.9s

Precision: binary64

Cost: 20164

Math

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

↓

\[\begin{array}{l} t_0 := x + \sqrt{x \cdot x - 1}\\ \mathbf{if}\;t_0 \leq 4000:\\ \;\;\;\;\log t_0\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x - \frac{0.5}{x}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (sqrt (- (* x x) 1.0)))))
   (if (<= t_0 4000.0) (log t_0) (log (+ x (- x (/ 0.5 x)))))))

C

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

↓

double code(double x) {
	double t_0 = x + sqrt(((x * x) - 1.0));
	double tmp;
	if (t_0 <= 4000.0) {
		tmp = log(t_0);
	} else {
		tmp = log((x + (x - (0.5 / x))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + sqrt(((x * x) - 1.0d0))
    if (t_0 <= 4000.0d0) then
        tmp = log(t_0)
    else
        tmp = log((x + (x - (0.5d0 / x))))
    end if
    code = tmp
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) {
	double t_0 = x + Math.sqrt(((x * x) - 1.0));
	double tmp;
	if (t_0 <= 4000.0) {
		tmp = Math.log(t_0);
	} else {
		tmp = Math.log((x + (x - (0.5 / x))));
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = x + math.sqrt(((x * x) - 1.0))
	tmp = 0
	if t_0 <= 4000.0:
		tmp = math.log(t_0)
	else:
		tmp = math.log((x + (x - (0.5 / x))))
	return tmp

Julia

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

↓

function code(x)
	t_0 = Float64(x + sqrt(Float64(Float64(x * x) - 1.0)))
	tmp = 0.0
	if (t_0 <= 4000.0)
		tmp = log(t_0);
	else
		tmp = log(Float64(x + Float64(x - Float64(0.5 / x))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = x + sqrt(((x * x) - 1.0));
	tmp = 0.0;
	if (t_0 <= 4000.0)
		tmp = log(t_0);
	else
		tmp = log((x + (x - (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4000.0], N[Log[t$95$0], $MachinePrecision], N[Log[N[(x + N[(x - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) 1))) < 4e3

   1. Initial program 0.1

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

   if 4e3 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) 1)))

   1. Initial program 32.9

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

   2. Taylor expanded in x around inf 0.0

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

   3. Simplified 0.0

      \[\leadsto \log \left(x + \color{blue}{\left(x - \frac{0.5}{x}\right)}\right) \]
      Proof
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	0.3
Cost	6848

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

Alternative 2
Error	0.6
Cost	6592

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

Reproduce

herbie shell --seed 2023010 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))