?

Average Accuracy: 50.9% → 99.9%
Time: 9.7s
Precision: binary64

?

\[x \geq 1\]
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log \left(x + x \cdot \sqrt{\frac{-1}{{x}^{2}} - -1}\right) \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (log (+ x (* x (sqrt (- (/ -1.0 (pow x 2.0)) -1.0))))))
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log((x + (x * sqrt(((-1.0 / pow(x, 2.0)) - -1.0)))));
}

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((x + (x * sqrt((((-1.0d0) / (x ** 2.0d0)) - (-1.0d0))))))
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((x + (x * Math.sqrt(((-1.0 / Math.pow(x, 2.0)) - -1.0)))));
}

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

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

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

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

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original50.9%
Target99.9%
Herbie99.9%
\[\log \left(x + \sqrt{x - 1} \cdot \sqrt{x + 1}\right) \]

Derivation?

  1. Initial program 50.9%

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

  2. Simplified50.9%

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

  3. Applied egg-rr99.9%

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

  4. Simplified99.9%

    \[\leadsto \log \left(x + \color{blue}{x \cdot \sqrt{\frac{-1}{{x}^{2}} - -1}}\right) \]
    Proof

Reproduce?

herbie shell --seed 2023151 
(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)))))