Eccentricity of an ellipse

Percentage Accurate: 77.2% → 100.0%

Time: 4.1s

Alternatives: 3

Speedup: 42.0×

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: 77.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in a around 0

   \[\leadsto \sqrt{\color{blue}{\left|\frac{{a}^{2} - {b}^{2}}{{a}^{2}}\right|}} \]
4. Step-by-step derivation

   1. fabs-neg N/A

      \[\leadsto \sqrt{\color{blue}{\left|\mathsf{neg}\left(\frac{{a}^{2} - {b}^{2}}{{a}^{2}}\right)\right|}} \]

   2. lower-fabs.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left|\mathsf{neg}\left(\frac{{a}^{2} - {b}^{2}}{{a}^{2}}\right)\right|}} \]

   3. div-sub N/A

      \[\leadsto \sqrt{\left|\mathsf{neg}\left(\color{blue}{\left(\frac{{a}^{2}}{{a}^{2}} - \frac{{b}^{2}}{{a}^{2}}\right)}\right)\right|} \]

   4. *-inverses N/A

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

   5. *-lft-identity N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sign-sub-inv N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. distribute-lft-neg-in N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

   14. unpow2 N/A

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

   15. associate-/l* N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5 \cdot b, \frac{\frac{b}{a}}{a}, 1\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma (* -0.5 b) (/ (/ b a) a) 1.0))

C

double code(double a, double b) {
	return fma((-0.5 * b), ((b / a) / a), 1.0);
}

Julia

function code(a, b)
	return fma(Float64(-0.5 * b), Float64(Float64(b / a) / a), 1.0)
end

Wolfram

code[a_, b_] := N[(N[(-0.5 * b), $MachinePrecision] * N[(N[(b / a), $MachinePrecision] / a), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-prod N/A

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

   3. rem-sqrt-square-rev N/A

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

   4. lift-/.f64 N/A

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

   5. fabs-div N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. difference-of-squares N/A

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

   10. fabs-mul N/A

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

   11. rem-sqrt-square-rev N/A

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

   12. sqrt-prod N/A

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

   13. rem-square-sqrt N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-fabs.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. fabs-sub N/A

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

   21. lower-fabs.f64 N/A

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

   22. lower--.f64 82.8

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

4. Applied rewrites 82.8%

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

   1. lift-fabs.f64 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 82.8

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

6. Applied rewrites 82.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. fabs-div N/A

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

   7. fabs-mul N/A

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

   8. lift-fabs.f64 N/A

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

   9. fabs-fabs N/A

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

   10. lift--.f64 N/A

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

   11. fabs-sub N/A

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

   12. fabs-mul N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. difference-of-squares N/A

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

   16. fabs-div N/A

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

   17. lift-*.f64 N/A

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

8. Applied rewrites 83.1%

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

9. Taylor expanded in b around 0

   \[\leadsto \color{blue}{1 + b \cdot \left(\frac{-1}{2} \cdot \frac{b \cdot \left(1 + \frac{1}{4} \cdot \frac{{\left(a + -1 \cdot a\right)}^{2}}{{a}^{2}}\right)}{{a}^{2}} + \frac{1}{2} \cdot \frac{a + -1 \cdot a}{{a}^{2}}\right)} \]

11. Applied rewrites 100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot b, \frac{\frac{b}{a}}{a}, 1\right)} \]
12. Add Preprocessing

Alternative 3: 97.9% accurate, 42.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 1.0)

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return 1.0;
}

Python

def code(a, b):
	return 1.0

Julia

function code(a, b)
	return 1.0
end

MATLAB

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

Wolfram

code[a_, b_] := 1.0

Derivation

1. Initial program 83.4%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-prod N/A

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

   3. rem-sqrt-square-rev N/A

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

   4. lift-/.f64 N/A

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

   5. fabs-div N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. difference-of-squares N/A

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

   10. fabs-mul N/A

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

   11. rem-sqrt-square-rev N/A

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

   12. sqrt-prod N/A

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

   13. rem-square-sqrt N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-fabs.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. fabs-sub N/A

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

   21. lower-fabs.f64 N/A

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

   22. lower--.f64 82.8

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

4. Applied rewrites 82.8%

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

   1. lift-fabs.f64 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 82.8

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

6. Applied rewrites 82.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. fabs-div N/A

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

   7. fabs-mul N/A

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

   8. lift-fabs.f64 N/A

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

   9. fabs-fabs N/A

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

   10. lift--.f64 N/A

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

   11. fabs-sub N/A

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

   12. fabs-mul N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. difference-of-squares N/A

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

   16. fabs-div N/A

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

   17. lift-*.f64 N/A

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

8. Applied rewrites 83.1%

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

9. Taylor expanded in a around inf

   \[\leadsto \color{blue}{1} \]
10. Step-by-step derivation

11. Applied rewrites 99.3%

    \[\leadsto \color{blue}{1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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)))))