Eccentricity of an ellipse

Percentage Accurate: 76.5% → 100.0%

Time: 7.6s

Alternatives: 3

Speedup: 3.2×

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: 76.5% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|\mathsf{fma}\left(b \cdot \frac{b}{a}, \frac{1}{a}, -1\right)\right|} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. *-inverses N/A

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

   6. fabs-sub N/A

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

   7. lower-fabs.f64 N/A

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

   8. sub-neg N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 74.6

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

4. Applied rewrites 74.6%

   \[\leadsto \sqrt{\color{blue}{\left|\mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right|}} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. frac-times N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 73.8%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. *-inverses N/A

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

   6. fabs-sub N/A

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

   7. lower-fabs.f64 N/A

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

   8. sub-neg N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 74.6

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

4. Applied rewrites 74.6%

   \[\leadsto \sqrt{\color{blue}{\left|\mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right|}} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. frac-times N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lower-fma.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 3: 97.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|1\right|} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (sqrt (fabs 1.0)))

C

double code(double a, double b) {
	return sqrt(fabs(1.0));
}

Fortran

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

Java

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

Python

def code(a, b):
	return math.sqrt(math.fabs(1.0))

Julia

function code(a, b)
	return sqrt(abs(1.0))
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt(abs(1.0));
end

Wolfram

code[a_, b_] := N[Sqrt[N[Abs[1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
2. Add Preprocessing

3. Taylor expanded in a around inf

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

5. Applied rewrites 98.1%

   \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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)))))