Average Error: 32.2 → 0.3
Time: 1.5s
Precision: binary64
Cost: 6848
\[x \geq 1\]
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log \left(x \cdot 2 + \frac{-0.5}{x}\right) \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x) :precision binary64 (log (+ (* x 2.0) (/ -0.5 x))))
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log(((x * 2.0) + (-0.5 / x)));
}

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(((x * 2.0d0) + ((-0.5d0) / x)))
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(((x * 2.0) + (-0.5 / x)));
}

def code(x):
	return math.log((x + math.sqrt(((x * x) - 1.0))))
↓
def code(x):
	return math.log(((x * 2.0) + (-0.5 / x)))

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

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) - 1.0))));
end
↓
function tmp = code(x)
	tmp = log(((x * 2.0) + (-0.5 / x)));
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[(x * 2.0), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.2
Target0.1
Herbie0.3
\[\log \left(x + \sqrt{x - 1} \cdot \sqrt{x + 1}\right) \]

Derivation

  1. Initial program 32.2

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

  2. Taylor expanded in x around inf 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Alternatives

Alternative 1
Error0.6
Cost6592
\[\log \left(x + x\right) \]

Error

Reproduce

herbie shell --seed 2022330 
(FPCore (x)
  :name "Rust f64::acosh"
  :precision binary64
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

  (log (+ x (sqrt (- (* x x) 1.0)))))