Eccentricity of an ellipse

Average Error: 14.6 → 0.0

Time: 6.8s

Precision: binary64

Cost: 19648

\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]

Math

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

↓

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

FPCore

(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))

↓

(FPCore (a b) :precision binary64 (sqrt (fabs (fma (/ b a) (/ b a) -1.0))))

C

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 / a), (b / a), -1.0)));
}

Julia

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(Float64(b / a), Float64(b / a), -1.0)))
end

Wolfram

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[(N[(b / a), $MachinePrecision] * N[(b / a), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 14.6

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

2. Simplified 0.0

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

   [Start] 14.6	\[ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   div-sub [=>] 14.6	\[ \sqrt{\left|\color{blue}{\frac{a \cdot a}{a \cdot a} - \frac{b \cdot b}{a \cdot a}}\right|} \]
   fabs-sub [=>] 14.6	\[ \sqrt{\color{blue}{\left|\frac{b \cdot b}{a \cdot a} - \frac{a \cdot a}{a \cdot a}\right|}} \]
   times-frac [=>] 14.5	\[ \sqrt{\left|\color{blue}{\frac{b}{a} \cdot \frac{b}{a}} - \frac{a \cdot a}{a \cdot a}\right|} \]
   fma-neg [=>] 14.5	\[ \sqrt{\left|\color{blue}{\mathsf{fma}\left(\frac{b}{a}, \frac{b}{a}, -\frac{a \cdot a}{a \cdot a}\right)}\right|} \]
   *-inverses [=>] 0.0	\[ \sqrt{\left|\mathsf{fma}\left(\frac{b}{a}, \frac{b}{a}, -\color{blue}{1}\right)\right|} \]
   metadata-eval [=>] 0.0	\[ \sqrt{\left|\mathsf{fma}\left(\frac{b}{a}, \frac{b}{a}, \color{blue}{-1}\right)\right|} \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	6976

\[\sqrt{1 - \frac{\frac{b}{a}}{\frac{a}{b}}} \]

Alternative 2
Error	0.6
Cost	704

\[1 + \frac{b}{a} \cdot \frac{b \cdot -0.5}{a} \]

Alternative 3
Error	1.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023012 
(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)))))