Average Error: 0.0 → 0.0
Time: 1.4s
Precision: binary64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
↓
\[\log \left(\frac{\frac{\mathsf{fma}\left(x, \sqrt{1 - x \cdot x}, x\right)}{x}}{x}\right) \]
(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))
↓
(FPCore (x)
 :precision binary64
 (log (/ (/ (fma x (sqrt (- 1.0 (* x x))) x) x) x)))
double code(double x) {
	return log(((1.0 / x) + (sqrt((1.0 - (x * x))) / x)));
}
↓
double code(double x) {
	return log(((fma(x, sqrt((1.0 - (x * x))), x) / x) / 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(fma(x, sqrt(Float64(1.0 - Float64(x * x))), x) / x) / 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[(N[(x * N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]

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

Error

Derivation

  1. Initial program 0.0

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

  2. Applied egg-rr0.0

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

  3. Final simplification0.0

    \[\leadsto \log \left(\frac{\frac{\mathsf{fma}\left(x, \sqrt{1 - x \cdot x}, x\right)}{x}}{x}\right) \]

Reproduce

herbie shell --seed 2022206 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))