Eccentricity of an ellipse

Average Accuracy: 77.1% → 100.0%

Time: 3.9s

Precision: binary64

Cost: 13440

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

Math

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

↓

\[e^{\mathsf{log1p}\left(\frac{\frac{-b}{a}}{\frac{a}{b}}\right) \cdot 0.5} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (exp (* (log1p (/ (/ (- b) a) (/ a b))) 0.5)))

C

double code(double a, double b) {
	return sqrt(fabs((((a * a) - (b * b)) / (a * a))));
}

↓

double code(double a, double b) {
	return exp((log1p(((-b / a) / (a / b))) * 0.5));
}

Java

public static double code(double a, double b) {
	return Math.sqrt(Math.abs((((a * a) - (b * b)) / (a * a))));
}

↓

public static double code(double a, double b) {
	return Math.exp((Math.log1p(((-b / a) / (a / b))) * 0.5));
}

Python

def code(a, b):
	return math.sqrt(math.fabs((((a * a) - (b * b)) / (a * a))))

↓

def code(a, b):
	return math.exp((math.log1p(((-b / a) / (a / b))) * 0.5))

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 exp(Float64(log1p(Float64(Float64(Float64(-b) / a) / Float64(a / b))) * 0.5))
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[Exp[N[(N[Log[1 + N[(N[((-b) / a), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 77.1%

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

2. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}} \]
   Proof

   [Start] 77.1	\[ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   div-sub [=>] 77.1	\[ \sqrt{\left|\color{blue}{\frac{a \cdot a}{a \cdot a} - \frac{b \cdot b}{a \cdot a}}\right|} \]
   *-inverses [=>] 77.1	\[ \sqrt{\left|\color{blue}{1} - \frac{b \cdot b}{a \cdot a}\right|} \]
   times-frac [=>] 100.0	\[ \sqrt{\left|1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right|} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5}} \]
   Proof

   [Start] 100.0	\[ \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|} \]
   pow1/2 [=>] 100.0	\[ \color{blue}{{\left(\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|\right)}^{0.5}} \]
   pow-to-exp [=>] 100.0	\[ \color{blue}{e^{\log \left(\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|\right) \cdot 0.5}} \]
   add-sqr-sqrt [=>] 100.0	\[ e^{\log \left(\left|\color{blue}{\sqrt{1 - \frac{b}{a} \cdot \frac{b}{a}} \cdot \sqrt{1 - \frac{b}{a} \cdot \frac{b}{a}}}\right|\right) \cdot 0.5} \]
   fabs-sqr [=>] 100.0	\[ e^{\log \color{blue}{\left(\sqrt{1 - \frac{b}{a} \cdot \frac{b}{a}} \cdot \sqrt{1 - \frac{b}{a} \cdot \frac{b}{a}}\right)} \cdot 0.5} \]
   add-sqr-sqrt [<=] 100.0	\[ e^{\log \color{blue}{\left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)} \cdot 0.5} \]
   sub-neg [=>] 100.0	\[ e^{\log \color{blue}{\left(1 + \left(-\frac{b}{a} \cdot \frac{b}{a}\right)\right)} \cdot 0.5} \]
   log1p-def [=>] 100.0	\[ e^{\color{blue}{\mathsf{log1p}\left(-\frac{b}{a} \cdot \frac{b}{a}\right)} \cdot 0.5} \]
   pow2 [=>] 100.0	\[ e^{\mathsf{log1p}\left(-\color{blue}{{\left(\frac{b}{a}\right)}^{2}}\right) \cdot 0.5} \]

4. Applied egg-rr 100.0%

   \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}}\right) \cdot 0.5} \]
   Proof

   [Start] 100.0	\[ e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5} \]
   unpow2 [=>] 100.0	\[ e^{\mathsf{log1p}\left(-\color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right) \cdot 0.5} \]
   clear-num [=>] 100.0	\[ e^{\mathsf{log1p}\left(-\frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right) \cdot 0.5} \]
   un-div-inv [=>] 100.0	\[ e^{\mathsf{log1p}\left(-\color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}}\right) \cdot 0.5} \]

5. Final simplification 100.0%

   \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-b}{a}}{\frac{a}{b}}\right) \cdot 0.5} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13248

\[{\left(1 - {\left(\frac{b}{a}\right)}^{2}\right)}^{0.5} \]

Alternative 2
Accuracy	100.0%
Cost	6976

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

Alternative 3
Accuracy	99.1%
Cost	704

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

Alternative 4
Accuracy	97.9%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023146 
(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)))))