Average Error: 31.5 → 0.2
Time: 6.6s
Precision: binary64
Cost: 14080
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log \left(2 \cdot x + \left(\frac{\frac{-0.125}{x \cdot x}}{x} + \left(\frac{-0.5}{x} - \frac{0.0625}{{x}^{5}}\right)\right)\right) \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (log
  (+
   (* 2.0 x)
   (+ (/ (/ -0.125 (* x x)) x) (- (/ -0.5 x) (/ 0.0625 (pow x 5.0)))))))
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log(((2.0 * x) + (((-0.125 / (x * x)) / x) + ((-0.5 / x) - (0.0625 / pow(x, 5.0))))));
}

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(((2.0d0 * x) + ((((-0.125d0) / (x * x)) / x) + (((-0.5d0) / x) - (0.0625d0 / (x ** 5.0d0))))))
end function

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(((2.0 * x) + (((-0.125 / (x * x)) / x) + ((-0.5 / x) - (0.0625 / Math.pow(x, 5.0))))));
}

def code(x):
	return math.log((x + math.sqrt(((x * x) - 1.0))))
↓
def code(x):
	return math.log(((2.0 * x) + (((-0.125 / (x * x)) / x) + ((-0.5 / x) - (0.0625 / math.pow(x, 5.0))))))

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) - 1.0))))
end
↓
function code(x)
	return log(Float64(Float64(2.0 * x) + Float64(Float64(Float64(-0.125 / Float64(x * x)) / x) + Float64(Float64(-0.5 / x) - Float64(0.0625 / (x ^ 5.0))))))
end

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) - 1.0))));
end
↓
function tmp = code(x)
	tmp = log(((2.0 * x) + (((-0.125 / (x * x)) / x) + ((-0.5 / x) - (0.0625 / (x ^ 5.0))))));
end

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := N[Log[N[(N[(2.0 * x), $MachinePrecision] + N[(N[(N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(-0.5 / x), $MachinePrecision] - N[(0.0625 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(2 \cdot x + \left(\frac{\frac{-0.125}{x \cdot x}}{x} + \left(\frac{-0.5}{x} - \frac{0.0625}{{x}^{5}}\right)\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.5

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

  2. Taylor expanded in x around inf 0.2

    \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right)} \]

  3. Taylor expanded in x around inf 0.2

    \[\leadsto \log \left(2 \cdot x - \color{blue}{\left(0.0625 \cdot \frac{1}{{x}^{5}} + \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right)}\right) \]

  4. Simplified0.2

    \[\leadsto \log \left(2 \cdot x - \color{blue}{\left(\frac{0.125}{{x}^{3}} + \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)}\right) \]
    Proof
    (+.f64 (/.f64 1/8 (pow.f64 x 3)) (+.f64 (/.f64 1/16 (pow.f64 x 5)) (/.f64 1/2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/8 1)) (pow.f64 x 3)) (+.f64 (/.f64 1/16 (pow.f64 x 5)) (/.f64 1/2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3)))) (+.f64 (/.f64 1/16 (pow.f64 x 5)) (/.f64 1/2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))) (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/16 1)) (pow.f64 x 5)) (/.f64 1/2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/16 (/.f64 1 (pow.f64 x 5)))) (/.f64 1/2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))) (+.f64 (*.f64 1/16 (/.f64 1 (pow.f64 x 5))) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))) (+.f64 (*.f64 1/16 (/.f64 1 (pow.f64 x 5))) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))) (*.f64 1/16 (/.f64 1 (pow.f64 x 5)))) (*.f64 1/2 (/.f64 1 x)))): 3 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/16 (/.f64 1 (pow.f64 x 5))) (*.f64 1/8 (/.f64 1 (pow.f64 x 3))))) (*.f64 1/2 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 1/16 (/.f64 1 (pow.f64 x 5))) (+.f64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))) (*.f64 1/2 (/.f64 1 x))))): 0 points increase in error, 3 points decrease in error

  5. Applied egg-rr0.2

    \[\leadsto \log \left(2 \cdot x - \left(\color{blue}{\frac{0.25}{x \cdot x} \cdot \frac{0.5}{x}} + \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\right) \]

  6. Applied egg-rr0.2

    \[\leadsto \log \left(2 \cdot x - \left(\color{blue}{\frac{\frac{0.125}{x \cdot x}}{x}} + \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\right) \]

  7. Final simplification0.2

    \[\leadsto \log \left(2 \cdot x + \left(\frac{\frac{-0.125}{x \cdot x}}{x} + \left(\frac{-0.5}{x} - \frac{0.0625}{{x}^{5}}\right)\right)\right) \]

Alternatives

Alternative 1
Error0.3
Cost6848
\[\log \left(2 \cdot x + \frac{-0.5}{x}\right) \]
Alternative 2
Error0.6
Cost6592
\[\log \left(x + x\right) \]

Error

Reproduce

herbie shell --seed 2022300 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))