Eccentricity of an ellipse

Percentage Accurate: 77.0% → 100.0%

Time: 5.8s

Alternatives: 3

Speedup: 211.0×

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: 77.0% 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, 1.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 - \frac{\frac{b}{a}}{\frac{a}{b}}} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return sqrt((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((1.0d0 - ((b / a) / (a / b))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. sqr-neg 80.1%

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

   2. associate-/r* 79.6%

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

   3. sqr-neg 79.6%

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

   4. associate-/r* 80.1%

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

   5. div-sub 80.1%

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

   6. fabs-sub 80.1%

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

   7. times-frac 80.1%

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

   8. *-inverses 100.0%

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

   9. difference-of-sqr-1 100.0%

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

   10. difference-of-sqr--1 100.0%

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

   11. fma-define 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. associate-/l* 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. sqrt-prod 80.7%

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

   5. sqrt-div 80.7%

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

   6. sqrt-prod 80.7%

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

   7. add-sqr-sqrt 80.7%

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

   8. associate-/l* 80.7%

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

   9. add-sqr-sqrt 80.7%

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

   10. sqrt-prod 80.7%

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

   11. sqrt-div 80.7%

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

   12. sqrt-prod 80.7%

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

   13. metadata-eval 80.7%

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

   14. fma-neg 80.7%

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

   15. add-sqr-sqrt 80.7%

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

   16. fabs-sub 80.7%

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

   17. add-sqr-sqrt 80.1%

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

6. Applied egg-rr 100.0%

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

   1. unpow2 100.0%

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

   2. clear-num 100.0%

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

   3. un-div-inv 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}}} \]
9. Add Preprocessing

Alternative 2: 97.9% accurate, 211.0× 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 80.1%

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

   1. sqr-neg 80.1%

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

   2. associate-/r* 79.6%

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

   3. sqr-neg 79.6%

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

   4. associate-/r* 80.1%

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

   5. div-sub 80.1%

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

   6. fabs-sub 80.1%

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

   7. times-frac 80.1%

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

   8. *-inverses 100.0%

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

   9. difference-of-sqr-1 100.0%

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

   10. difference-of-sqr--1 100.0%

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

   11. fma-define 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. associate-/l* 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. sqrt-prod 80.7%

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

   5. sqrt-div 80.7%

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

   6. sqrt-prod 80.7%

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

   7. add-sqr-sqrt 80.7%

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

   8. associate-/l* 80.7%

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

   9. add-sqr-sqrt 80.7%

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

   10. sqrt-prod 80.7%

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

   11. sqrt-div 80.7%

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

   12. sqrt-prod 80.7%

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

   13. metadata-eval 80.7%

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

   14. fma-neg 80.7%

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

   15. add-sqr-sqrt 80.7%

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

   16. fabs-sub 80.7%

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

   17. add-sqr-sqrt 80.1%

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

6. Applied egg-rr 100.0%

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

   1. unpow2 100.0%

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

   2. clear-num 100.0%

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

   3. un-div-inv 100.0%

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

8. Applied egg-rr 100.0%

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

9. Taylor expanded in b around 0 98.6%

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

Alternative 3: 1.6% accurate, 211.0× 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 80.1%

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

   1. sqr-neg 80.1%

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

   2. associate-/r* 79.6%

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

   3. sqr-neg 79.6%

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

   4. associate-/r* 80.1%

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

   5. div-sub 80.1%

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

   6. fabs-sub 80.1%

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

   7. times-frac 80.1%

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

   8. *-inverses 100.0%

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

   9. difference-of-sqr-1 100.0%

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

   10. difference-of-sqr--1 100.0%

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

   11. fma-define 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. associate-/l* 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. sqrt-prod 80.7%

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

   5. sqrt-div 80.7%

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

   6. sqrt-prod 80.7%

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

   7. add-sqr-sqrt 80.7%

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

   8. associate-/l* 80.7%

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

   9. add-sqr-sqrt 80.7%

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

   10. sqrt-prod 80.7%

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

   11. sqrt-div 80.7%

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

   12. sqrt-prod 80.7%

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

   13. metadata-eval 80.7%

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

   14. fma-neg 80.7%

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

   15. add-sqr-sqrt 80.7%

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

   16. fabs-sub 80.7%

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

   17. *-inverses 80.1%

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

   18. associate-*r/ 80.1%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in a around -inf 0.0%

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

   1. unpow2 0.0%

      \[\leadsto \color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]

   2. rem-square-sqrt 1.6%

      \[\leadsto \color{blue}{-1} \]

9. Simplified 1.6%

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

Reproduce

herbie shell --seed 2024135 
(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)))))