Eccentricity of an ellipse

Percentage Accurate: 77.5% → 100.0%

Time: 3.2s

Alternatives: 2

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Initial Program: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|\frac{b}{a} \cdot \frac{b}{a} + -1\right|} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return Math.sqrt(Math.abs((((b / a) * (b / a)) + -1.0)));
}

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = sqrt(abs((((b / a) * (b / a)) + -1.0)));
end

Wolfram

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

Derivation

1. Initial program 78.9%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
2. Step-by-step derivation

   1. sqr-neg 78.9%

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

   2. associate-/r* 79.2%

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

   3. sqr-neg 79.2%

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

   4. associate-/r* 78.9%

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

   5. div-sub 78.9%

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

   6. fabs-sub 78.9%

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

   7. times-frac 78.9%

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

   8. *-inverses 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 0.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\left(\frac{a}{b}\right)}^{-2} + -1} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (sqrt (+ (pow (/ a b) -2.0) -1.0)))

C

double code(double a, double b) {
	return sqrt((pow((a / b), -2.0) + -1.0));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt((((a / b) ** (-2.0d0)) + (-1.0d0)))
end function

Java

public static double code(double a, double b) {
	return Math.sqrt((Math.pow((a / b), -2.0) + -1.0));
}

Python

def code(a, b):
	return math.sqrt((math.pow((a / b), -2.0) + -1.0))

Julia

function code(a, b)
	return sqrt(Float64((Float64(a / b) ^ -2.0) + -1.0))
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt((((a / b) ^ -2.0) + -1.0));
end

Wolfram

code[a_, b_] := N[Sqrt[N[(N[Power[N[(a / b), $MachinePrecision], -2.0], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.9%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
2. Step-by-step derivation

   1. sqr-neg 78.9%

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

   2. associate-/r* 79.2%

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

   3. sqr-neg 79.2%

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

   4. associate-/r* 78.9%

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

   5. div-sub 78.9%

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

   6. fabs-sub 78.9%

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

   7. times-frac 78.9%

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

   8. *-inverses 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \color{blue}{\log \left(e^{\sqrt{\left|\frac{b}{a} \cdot \frac{b}{a} - 1\right|}}\right)} \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{\left|\frac{b}{a} \cdot \frac{b}{a} - 1\right|}}\right)} \]

   3. log-prod 100.0%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{\left|\frac{b}{a} \cdot \frac{b}{a} - 1\right|}}\right)} \]

   4. metadata-eval 100.0%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{\left|\frac{b}{a} \cdot \frac{b}{a} - 1\right|}}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\sqrt{\left|\frac{b}{a} \cdot \frac{b}{a} - 1\right|}} \]

   6. add-sqr-sqrt 0.0%

      \[\leadsto 0 + \sqrt{\left|\color{blue}{\sqrt{\frac{b}{a} \cdot \frac{b}{a} - 1} \cdot \sqrt{\frac{b}{a} \cdot \frac{b}{a} - 1}}\right|} \]

   7. fabs-sqr 0.0%

      \[\leadsto 0 + \sqrt{\color{blue}{\sqrt{\frac{b}{a} \cdot \frac{b}{a} - 1} \cdot \sqrt{\frac{b}{a} \cdot \frac{b}{a} - 1}}} \]

   8. add-sqr-sqrt 0.0%

      \[\leadsto 0 + \sqrt{\color{blue}{\frac{b}{a} \cdot \frac{b}{a} - 1}} \]

   9. sub-neg 0.0%

      \[\leadsto 0 + \sqrt{\color{blue}{\frac{b}{a} \cdot \frac{b}{a} + \left(-1\right)}} \]

   10. pow2 0.0%

       \[\leadsto 0 + \sqrt{\color{blue}{{\left(\frac{b}{a}\right)}^{2}} + \left(-1\right)} \]

   11. clear-num 0.0%

       \[\leadsto 0 + \sqrt{{\color{blue}{\left(\frac{1}{\frac{a}{b}}\right)}}^{2} + \left(-1\right)} \]

   12. inv-pow 0.0%

       \[\leadsto 0 + \sqrt{{\color{blue}{\left({\left(\frac{a}{b}\right)}^{-1}\right)}}^{2} + \left(-1\right)} \]

   13. metadata-eval 0.0%

       \[\leadsto 0 + \sqrt{{\left({\left(\frac{a}{b}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{2} + \left(-1\right)} \]

   14. pow-pow 0.0%

       \[\leadsto 0 + \sqrt{\color{blue}{{\left(\frac{a}{b}\right)}^{\left(\left(-1\right) \cdot 2\right)}} + \left(-1\right)} \]

   15. metadata-eval 0.0%

       \[\leadsto 0 + \sqrt{{\left(\frac{a}{b}\right)}^{\left(\color{blue}{-1} \cdot 2\right)} + \left(-1\right)} \]

   16. metadata-eval 0.0%

       \[\leadsto 0 + \sqrt{{\left(\frac{a}{b}\right)}^{\color{blue}{-2}} + \left(-1\right)} \]

   17. metadata-eval 0.0%

       \[\leadsto 0 + \sqrt{{\left(\frac{a}{b}\right)}^{-2} + \color{blue}{-1}} \]

5. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{0 + \sqrt{{\left(\frac{a}{b}\right)}^{-2} + -1}} \]
6. Step-by-step derivation

   1. +-lft-identity 0.0%

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

7. Simplified 0.0%

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

8. Final simplification 0.0%

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

Reproduce

herbie shell --seed 2023287 
(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)))))