Eccentricity of an ellipse

Average Accuracy: 77.3% → 100.0%

Time: 5.0s

Precision: binary64

Cost: 7232

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

Math

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

↓

\[\sqrt{1 + \left(1 + \left(-1 - \frac{\frac{b}{a}}{\frac{a}{b}}\right)\right)} \]

FPCore

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

↓

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

C

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

↓

double code(double a, double b) {
	return sqrt((1.0 + (1.0 + (-1.0 - ((b / a) / (a / b))))));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(abs((((a * a) - (b * b)) / (a * a))))
end function

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt((1.0d0 + (1.0d0 + ((-1.0d0) - ((b / a) / (a / b))))))
end function

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.sqrt((1.0 + (1.0 + (-1.0 - ((b / a) / (a / b))))));
}

Python

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

↓

def code(a, b):
	return math.sqrt((1.0 + (1.0 + (-1.0 - ((b / a) / (a / b))))))

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

MATLAB

function tmp = code(a, b)
	tmp = sqrt(abs((((a * a) - (b * b)) / (a * a))));
end

↓

function tmp = code(a, b)
	tmp = sqrt((1.0 + (1.0 + (-1.0 - ((b / a) / (a / b))))));
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[(1.0 + N[(1.0 + N[(-1.0 - N[(N[(b / a), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 74.2%

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

2. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}} \]
   Step-by-step derivation

   [Start] 74.2	\[ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   div-sub [=>] 74.2	\[ \sqrt{\left|\color{blue}{\frac{a \cdot a}{a \cdot a} - \frac{b \cdot b}{a \cdot a}}\right|} \]
   *-inverses [=>] 74.2	\[ \sqrt{\left|\color{blue}{1} - \frac{b \cdot b}{a \cdot a}\right|} \]
   times-frac [=>] 100.0	\[ \sqrt{\left|1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right|} \]

3. Taylor expanded in b around 0 74.2%

   \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{{b}^{2}}{{a}^{2}}\right|}} \]

4. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}} \]
   Step-by-step derivation

   [Start] 74.2	\[ \sqrt{\left|1 - \frac{{b}^{2}}{{a}^{2}}\right|} \]
   unpow2 [=>] 74.2	\[ \sqrt{\left|1 - \frac{\color{blue}{b \cdot b}}{{a}^{2}}\right|} \]
   unpow2 [=>] 74.2	\[ \sqrt{\left|1 - \frac{b \cdot b}{\color{blue}{a \cdot a}}\right|} \]
   times-frac [=>] 100.0	\[ \sqrt{\left|1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right|} \]
   unpow2 [<=] 100.0	\[ \sqrt{\left|1 - \color{blue}{{\left(\frac{b}{a}\right)}^{2}}\right|} \]
   rem-square-sqrt [<=] 100.0	\[ \sqrt{\left|\color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}}\right|} \]
   fabs-sqr [=>] 100.0	\[ \sqrt{\color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}}} \]
   rem-square-sqrt [=>] 100.0	\[ \sqrt{\color{blue}{1 - {\left(\frac{b}{a}\right)}^{2}}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \sqrt{1 - \color{blue}{\left(\left({\left(\frac{b}{a}\right)}^{2} + 1\right) - 1\right)}} \]
   Step-by-step derivation

   [Start] 100.0	\[ \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \]
   add-sqr-sqrt [=>] 100.0	\[ \sqrt{1 - \color{blue}{\sqrt{{\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{{\left(\frac{b}{a}\right)}^{2}}}} \]
   sqrt-unprod [=>] 100.0	\[ \sqrt{1 - \color{blue}{\sqrt{{\left(\frac{b}{a}\right)}^{2} \cdot {\left(\frac{b}{a}\right)}^{2}}}} \]
   sqr-neg [<=] 100.0	\[ \sqrt{1 - \sqrt{\color{blue}{\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot \left(-{\left(\frac{b}{a}\right)}^{2}\right)}}} \]
   sqrt-unprod [<=] 19.9	\[ \sqrt{1 - \color{blue}{\sqrt{-{\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{-{\left(\frac{b}{a}\right)}^{2}}}} \]
   add-sqr-sqrt [<=] 98.4	\[ \sqrt{1 - \color{blue}{\left(-{\left(\frac{b}{a}\right)}^{2}\right)}} \]
   expm1-log1p-u [=>] 98.4	\[ \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right)\right)}} \]
   expm1-udef [=>] 98.4	\[ \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right)} - 1\right)}} \]
   log1p-udef [=>] 98.4	\[ \sqrt{1 - \left(e^{\color{blue}{\log \left(1 + \left(-{\left(\frac{b}{a}\right)}^{2}\right)\right)}} - 1\right)} \]
   add-exp-log [<=] 98.4	\[ \sqrt{1 - \left(\color{blue}{\left(1 + \left(-{\left(\frac{b}{a}\right)}^{2}\right)\right)} - 1\right)} \]
   +-commutative [=>] 98.4	\[ \sqrt{1 - \left(\color{blue}{\left(\left(-{\left(\frac{b}{a}\right)}^{2}\right) + 1\right)} - 1\right)} \]
   add-sqr-sqrt [=>] 19.9	\[ \sqrt{1 - \left(\left(\color{blue}{\sqrt{-{\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{-{\left(\frac{b}{a}\right)}^{2}}} + 1\right) - 1\right)} \]
   sqrt-unprod [=>] 100.0	\[ \sqrt{1 - \left(\left(\color{blue}{\sqrt{\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot \left(-{\left(\frac{b}{a}\right)}^{2}\right)}} + 1\right) - 1\right)} \]
   sqr-neg [=>] 100.0	\[ \sqrt{1 - \left(\left(\sqrt{\color{blue}{{\left(\frac{b}{a}\right)}^{2} \cdot {\left(\frac{b}{a}\right)}^{2}}} + 1\right) - 1\right)} \]
   sqrt-unprod [<=] 100.0	\[ \sqrt{1 - \left(\left(\color{blue}{\sqrt{{\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{{\left(\frac{b}{a}\right)}^{2}}} + 1\right) - 1\right)} \]
   add-sqr-sqrt [<=] 100.0	\[ \sqrt{1 - \left(\left(\color{blue}{{\left(\frac{b}{a}\right)}^{2}} + 1\right) - 1\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto \sqrt{1 - \left(\left(\color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}} + 1\right) - 1\right)} \]
   Step-by-step derivation

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

7. Final simplification 100.0%

   \[\leadsto \sqrt{1 + \left(1 + \left(-1 - \frac{\frac{b}{a}}{\frac{a}{b}}\right)\right)} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6976

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

Alternative 2
Accuracy	98.9%
Cost	704

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

Alternative 3
Accuracy	97.9%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023157 
(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)))))