Average Error: 15.1 → 0.0
Time: 3.9s
Precision: binary64
\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[{\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{0.5} \]
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b) :precision binary64 (pow (fabs (fma b (/ (/ b a) a) -1.0)) 0.5))
double code(double a, double b) {
	return sqrt(fabs((((a * a) - (b * b)) / (a * a))));
}
↓
double code(double a, double b) {
	return pow(fabs(fma(b, ((b / a) / a), -1.0)), 0.5);
}

function code(a, b)
	return sqrt(abs(Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a))))
end
↓
function code(a, b)
	return abs(fma(b, Float64(Float64(b / a) / a), -1.0)) ^ 0.5
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[Power[N[Abs[N[(b * N[(N[(b / a), $MachinePrecision] / a), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], 0.5], $MachinePrecision]

\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
{\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{0.5}

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. Applied associate-/r*_binary640.0

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

  4. Applied add-sqr-sqrt_binary640.0

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

  5. Applied pow1/2_binary640.0

    \[\leadsto \sqrt{\sqrt{\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|}} \cdot \sqrt{\color{blue}{{\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{0.5}}} \]

  6. Applied sqrt-pow1_binary640.0

    \[\leadsto \sqrt{\sqrt{\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|}} \cdot \color{blue}{{\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

  7. Applied pow1/2_binary640.0

    \[\leadsto \sqrt{\color{blue}{{\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{0.5}}} \cdot {\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

  8. Applied sqrt-pow1_binary640.0

    \[\leadsto \color{blue}{{\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{\left(\frac{0.5}{2}\right)}} \cdot {\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

  9. Applied pow-prod-up_binary640.0

    \[\leadsto \color{blue}{{\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{\left(\frac{0.5}{2} + \frac{0.5}{2}\right)}} \]

  10. Simplified0.0

    \[\leadsto {\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{\color{blue}{0.5}} \]

  11. Final simplification0.0

    \[\leadsto {\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{0.5} \]

Reproduce

herbie shell --seed 2022131 
(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)))))