Average Error: 0.0 → 0.0
Time: 8.2s
Precision: binary64
Cost: 19712
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
↓
\[\log \left(\frac{1 + \sqrt{-\mathsf{fma}\left(x, x, -1\right)}}{x}\right) \]
(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))
↓
(FPCore (x) :precision binary64 (log (/ (+ 1.0 (sqrt (- (fma x x -1.0)))) x)))
double code(double x) {
	return log(((1.0 / x) + (sqrt((1.0 - (x * x))) / x)));
}
↓
double code(double x) {
	return log(((1.0 + sqrt(-fma(x, x, -1.0))) / x));
}

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

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

\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1 + \sqrt{-\mathsf{fma}\left(x, x, -1\right)}}{x}\right)

Error

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

Alternatives

Alternative 1
Error0.0
Cost13504
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
Alternative 2
Error0.3
Cost6976
\[\log \left(\frac{2 + \left(x \cdot x\right) \cdot -0.5}{x}\right) \]
Alternative 3
Error0.5
Cost6592
\[\log \left(\frac{2}{x}\right) \]

Error

Reproduce

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