Average Error: 15.1 → 0.0
Time: 1.5s
Precision: binary64
\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[\sqrt{\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|} \]
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b) :precision binary64 (sqrt (fabs (fma b (/ (/ b a) a) -1.0))))
double code(double a, double b) {
	return sqrt(fabs((((a * a) - (b * b)) / (a * a))));
}
↓
double code(double a, double b) {
	return sqrt(fabs(fma(b, ((b / a) / a), -1.0)));
}

function code(a, b)
	return sqrt(abs(Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a))))
end
↓
function code(a, b)
	return sqrt(abs(fma(b, Float64(Float64(b / a) / a), -1.0)))
end

code[a_, b_] := N[Sqrt[N[Abs[N[(N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]
↓
code[a_, b_] := N[Sqrt[N[Abs[N[(b * N[(N[(b / a), $MachinePrecision] / a), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]

  2. Simplified15.1

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

  3. Taylor expanded in b around 0 15.1

    \[\leadsto \sqrt{\left|\mathsf{fma}\left(b, \color{blue}{\frac{b}{{a}^{2}}}, -1\right)\right|} \]

  4. Simplified0.0

    \[\leadsto \sqrt{\left|\mathsf{fma}\left(b, \color{blue}{\frac{\frac{b}{a}}{a}}, -1\right)\right|} \]

  5. Final simplification0.0

    \[\leadsto \sqrt{\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|} \]

Reproduce

herbie shell --seed 2022151 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (and (and (<= 0.0 b) (<= b a)) (<= a 1.0))
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))