Eccentricity of an ellipse

Percentage Accurate: 77.7% → 100.0%

Time: 13.7s

Alternatives: 5

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: 77.7% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. add-sqr-sqrt 75.4%

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

   2. fabs-sqr 75.4%

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

   3. add-sqr-sqrt 75.4%

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

   4. div-sub 75.4%

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

   5. *-inverses 75.4%

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

   6. frac-times 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 99.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \frac{a + b \cdot \left(b \cdot \left(\left(\left(b \cdot \frac{b}{a}\right) \cdot \frac{1}{a}\right) \cdot \frac{-0.125}{a} + \frac{-0.5}{a}\right)\right)}{a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/
  (+ a (* b (* b (+ (* (* (* b (/ b a)) (/ 1.0 a)) (/ -0.125 a)) (/ -0.5 a)))))
  a))

C

double code(double a, double b) {
	return (a + (b * (b * ((((b * (b / a)) * (1.0 / a)) * (-0.125 / a)) + (-0.5 / a))))) / a;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a + (b * (b * ((((b * (b / a)) * (1.0d0 / a)) * ((-0.125d0) / a)) + ((-0.5d0) / a))))) / a
end function

Java

public static double code(double a, double b) {
	return (a + (b * (b * ((((b * (b / a)) * (1.0 / a)) * (-0.125 / a)) + (-0.5 / a))))) / a;
}

Python

def code(a, b):
	return (a + (b * (b * ((((b * (b / a)) * (1.0 / a)) * (-0.125 / a)) + (-0.5 / a))))) / a

Julia

function code(a, b)
	return Float64(Float64(a + Float64(b * Float64(b * Float64(Float64(Float64(Float64(b * Float64(b / a)) * Float64(1.0 / a)) * Float64(-0.125 / a)) + Float64(-0.5 / a))))) / a)
end

MATLAB

function tmp = code(a, b)
	tmp = (a + (b * (b * ((((b * (b / a)) * (1.0 / a)) * (-0.125 / a)) + (-0.5 / a))))) / a;
end

Wolfram

code[a_, b_] := N[(N[(a + N[(b * N[(b * N[(N[(N[(N[(b * N[(b / a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 75.4%

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

   1. add-sqr-sqrt 75.4%

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

   2. fabs-sqr 75.4%

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

   3. add-sqr-sqrt 75.4%

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

   4. sqrt-div 75.4%

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

   5. sqrt-prod 75.5%

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

   6. add-sqr-sqrt 76.1%

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

   7. pow1/2 76.1%

      \[\leadsto \frac{\color{blue}{{\left(a \cdot a - b \cdot b\right)}^{0.5}}}{a} \]

4. Applied egg-rr 76.1%

   \[\leadsto \color{blue}{\frac{{\left(a \cdot a - b \cdot b\right)}^{0.5}}{a}} \]

5. Taylor expanded in b around 0 55.8%

   \[\leadsto \frac{\color{blue}{a + {b}^{2} \cdot \left(-0.125 \cdot \frac{{b}^{2}}{{a}^{3}} - 0.5 \cdot \frac{1}{a}\right)}}{a} \]
6. Step-by-step derivation

   1. unpow2 55.8%

      \[\leadsto \frac{a + \color{blue}{\left(b \cdot b\right)} \cdot \left(-0.125 \cdot \frac{{b}^{2}}{{a}^{3}} - 0.5 \cdot \frac{1}{a}\right)}{a} \]

   2. sub-neg 55.8%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \color{blue}{\left(-0.125 \cdot \frac{{b}^{2}}{{a}^{3}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}}{a} \]

   3. associate-*r/ 55.8%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\color{blue}{\frac{-0.125 \cdot {b}^{2}}{{a}^{3}}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}{a} \]

   4. cube-unmult 55.8%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125 \cdot {b}^{2}}{\color{blue}{a \cdot \left(a \cdot a\right)}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}{a} \]

   5. unpow2 55.8%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125 \cdot {b}^{2}}{a \cdot \color{blue}{{a}^{2}}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}{a} \]

   6. times-frac 75.3%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\color{blue}{\frac{-0.125}{a} \cdot \frac{{b}^{2}}{{a}^{2}}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}{a} \]

   7. unpow2 75.3%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{\color{blue}{b \cdot b}}{{a}^{2}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}{a} \]

   8. unpow2 75.3%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{\color{blue}{a \cdot a}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}{a} \]

   9. associate-*r/ 75.3%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{a \cdot a} + \left(-\color{blue}{\frac{0.5 \cdot 1}{a}}\right)\right)}{a} \]

   10. metadata-eval 75.3%

       \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{a \cdot a} + \left(-\frac{\color{blue}{0.5}}{a}\right)\right)}{a} \]

   11. distribute-neg-frac 75.3%

       \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{a \cdot a} + \color{blue}{\frac{-0.5}{a}}\right)}{a} \]

   12. metadata-eval 75.3%

       \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{a \cdot a} + \frac{\color{blue}{-0.5}}{a}\right)}{a} \]

7. Simplified 75.3%

   \[\leadsto \frac{\color{blue}{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{a \cdot a} + \frac{-0.5}{a}\right)}}{a} \]
8. Step-by-step derivation

   1. associate-*l* 75.3%

      \[\leadsto \frac{a + \color{blue}{b \cdot \left(b \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{a \cdot a} + \frac{-0.5}{a}\right)\right)}}{a} \]

   2. *-commutative 75.3%

      \[\leadsto \frac{a + b \cdot \left(b \cdot \left(\color{blue}{\frac{b \cdot b}{a \cdot a} \cdot \frac{-0.125}{a}} + \frac{-0.5}{a}\right)\right)}{a} \]

   3. associate-/l* 75.5%

      \[\leadsto \frac{a + b \cdot \left(b \cdot \left(\color{blue}{\left(b \cdot \frac{b}{a \cdot a}\right)} \cdot \frac{-0.125}{a} + \frac{-0.5}{a}\right)\right)}{a} \]

   4. associate-/l/ 98.5%

      \[\leadsto \frac{a + b \cdot \left(b \cdot \left(\left(b \cdot \color{blue}{\frac{\frac{b}{a}}{a}}\right) \cdot \frac{-0.125}{a} + \frac{-0.5}{a}\right)\right)}{a} \]

9. Applied egg-rr 98.5%

   \[\leadsto \frac{a + \color{blue}{b \cdot \left(b \cdot \left(\left(b \cdot \frac{\frac{b}{a}}{a}\right) \cdot \frac{-0.125}{a} + \frac{-0.5}{a}\right)\right)}}{a} \]
10. Step-by-step derivation

    1. associate-*r/ 98.5%

       \[\leadsto \frac{a + b \cdot \left(b \cdot \left(\color{blue}{\frac{b \cdot \frac{b}{a}}{a}} \cdot \frac{-0.125}{a} + \frac{-0.5}{a}\right)\right)}{a} \]

    2. div-inv 98.5%

       \[\leadsto \frac{a + b \cdot \left(b \cdot \left(\color{blue}{\left(\left(b \cdot \frac{b}{a}\right) \cdot \frac{1}{a}\right)} \cdot \frac{-0.125}{a} + \frac{-0.5}{a}\right)\right)}{a} \]

11. Applied egg-rr 98.5%

    \[\leadsto \frac{a + b \cdot \left(b \cdot \left(\color{blue}{\left(\left(b \cdot \frac{b}{a}\right) \cdot \frac{1}{a}\right)} \cdot \frac{-0.125}{a} + \frac{-0.5}{a}\right)\right)}{a} \]
12. Add Preprocessing

Alternative 3: 99.4% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \frac{a + b \cdot \left(b \cdot \left(\frac{-0.5}{a} + \frac{-0.125}{a} \cdot \left(b \cdot \frac{\frac{b}{a}}{a}\right)\right)\right)}{a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/ (+ a (* b (* b (+ (/ -0.5 a) (* (/ -0.125 a) (* b (/ (/ b a) a))))))) a))

C

double code(double a, double b) {
	return (a + (b * (b * ((-0.5 / a) + ((-0.125 / a) * (b * ((b / a) / a))))))) / a;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a + (b * (b * (((-0.5d0) / a) + (((-0.125d0) / a) * (b * ((b / a) / a))))))) / a
end function

Java

public static double code(double a, double b) {
	return (a + (b * (b * ((-0.5 / a) + ((-0.125 / a) * (b * ((b / a) / a))))))) / a;
}

Python

def code(a, b):
	return (a + (b * (b * ((-0.5 / a) + ((-0.125 / a) * (b * ((b / a) / a))))))) / a

Julia

function code(a, b)
	return Float64(Float64(a + Float64(b * Float64(b * Float64(Float64(-0.5 / a) + Float64(Float64(-0.125 / a) * Float64(b * Float64(Float64(b / a) / a))))))) / a)
end

MATLAB

function tmp = code(a, b)
	tmp = (a + (b * (b * ((-0.5 / a) + ((-0.125 / a) * (b * ((b / a) / a))))))) / a;
end

Wolfram

code[a_, b_] := N[(N[(a + N[(b * N[(b * N[(N[(-0.5 / a), $MachinePrecision] + N[(N[(-0.125 / a), $MachinePrecision] * N[(b * N[(N[(b / a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 75.4%

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

   1. add-sqr-sqrt 75.4%

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

   2. fabs-sqr 75.4%

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

   3. add-sqr-sqrt 75.4%

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

   4. sqrt-div 75.4%

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

   5. sqrt-prod 75.5%

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

   6. add-sqr-sqrt 76.1%

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

   7. pow1/2 76.1%

      \[\leadsto \frac{\color{blue}{{\left(a \cdot a - b \cdot b\right)}^{0.5}}}{a} \]

4. Applied egg-rr 76.1%

   \[\leadsto \color{blue}{\frac{{\left(a \cdot a - b \cdot b\right)}^{0.5}}{a}} \]

5. Taylor expanded in b around 0 55.8%

   \[\leadsto \frac{\color{blue}{a + {b}^{2} \cdot \left(-0.125 \cdot \frac{{b}^{2}}{{a}^{3}} - 0.5 \cdot \frac{1}{a}\right)}}{a} \]
6. Step-by-step derivation

   1. unpow2 55.8%

      \[\leadsto \frac{a + \color{blue}{\left(b \cdot b\right)} \cdot \left(-0.125 \cdot \frac{{b}^{2}}{{a}^{3}} - 0.5 \cdot \frac{1}{a}\right)}{a} \]

   2. sub-neg 55.8%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \color{blue}{\left(-0.125 \cdot \frac{{b}^{2}}{{a}^{3}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}}{a} \]

   3. associate-*r/ 55.8%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\color{blue}{\frac{-0.125 \cdot {b}^{2}}{{a}^{3}}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}{a} \]

   4. cube-unmult 55.8%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125 \cdot {b}^{2}}{\color{blue}{a \cdot \left(a \cdot a\right)}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}{a} \]

   5. unpow2 55.8%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125 \cdot {b}^{2}}{a \cdot \color{blue}{{a}^{2}}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}{a} \]

   6. times-frac 75.3%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\color{blue}{\frac{-0.125}{a} \cdot \frac{{b}^{2}}{{a}^{2}}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}{a} \]

   7. unpow2 75.3%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{\color{blue}{b \cdot b}}{{a}^{2}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}{a} \]

   8. unpow2 75.3%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{\color{blue}{a \cdot a}} + \left(-0.5 \cdot \frac{1}{a}\right)\right)}{a} \]

   9. associate-*r/ 75.3%

      \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{a \cdot a} + \left(-\color{blue}{\frac{0.5 \cdot 1}{a}}\right)\right)}{a} \]

   10. metadata-eval 75.3%

       \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{a \cdot a} + \left(-\frac{\color{blue}{0.5}}{a}\right)\right)}{a} \]

   11. distribute-neg-frac 75.3%

       \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{a \cdot a} + \color{blue}{\frac{-0.5}{a}}\right)}{a} \]

   12. metadata-eval 75.3%

       \[\leadsto \frac{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{a \cdot a} + \frac{\color{blue}{-0.5}}{a}\right)}{a} \]

7. Simplified 75.3%

   \[\leadsto \frac{\color{blue}{a + \left(b \cdot b\right) \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{a \cdot a} + \frac{-0.5}{a}\right)}}{a} \]
8. Step-by-step derivation

   1. associate-*l* 75.3%

      \[\leadsto \frac{a + \color{blue}{b \cdot \left(b \cdot \left(\frac{-0.125}{a} \cdot \frac{b \cdot b}{a \cdot a} + \frac{-0.5}{a}\right)\right)}}{a} \]

   2. *-commutative 75.3%

      \[\leadsto \frac{a + b \cdot \left(b \cdot \left(\color{blue}{\frac{b \cdot b}{a \cdot a} \cdot \frac{-0.125}{a}} + \frac{-0.5}{a}\right)\right)}{a} \]

   3. associate-/l* 75.5%

      \[\leadsto \frac{a + b \cdot \left(b \cdot \left(\color{blue}{\left(b \cdot \frac{b}{a \cdot a}\right)} \cdot \frac{-0.125}{a} + \frac{-0.5}{a}\right)\right)}{a} \]

   4. associate-/l/ 98.5%

      \[\leadsto \frac{a + b \cdot \left(b \cdot \left(\left(b \cdot \color{blue}{\frac{\frac{b}{a}}{a}}\right) \cdot \frac{-0.125}{a} + \frac{-0.5}{a}\right)\right)}{a} \]

9. Applied egg-rr 98.5%

   \[\leadsto \frac{a + \color{blue}{b \cdot \left(b \cdot \left(\left(b \cdot \frac{\frac{b}{a}}{a}\right) \cdot \frac{-0.125}{a} + \frac{-0.5}{a}\right)\right)}}{a} \]

10. Final simplification 98.5%

    \[\leadsto \frac{a + b \cdot \left(b \cdot \left(\frac{-0.5}{a} + \frac{-0.125}{a} \cdot \left(b \cdot \frac{\frac{b}{a}}{a}\right)\right)\right)}{a} \]
11. Add Preprocessing

Alternative 4: 99.1% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 + ((-0.5d0) * (b * ((b / a) / a)))
end function

Java

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

Python

def code(a, b):
	return 1.0 + (-0.5 * (b * ((b / a) / a)))

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = 1.0 + (-0.5 * (b * ((b / a) / a)));
end

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. add-sqr-sqrt 75.4%

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

   2. fabs-sqr 75.4%

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

   3. add-sqr-sqrt 75.4%

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

   4. div-sub 75.4%

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

   5. *-inverses 75.4%

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

   6. frac-times 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in b around 0 75.1%

   \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{b}^{2}}{{a}^{2}}} \]
6. Step-by-step derivation

   1. unpow2 75.1%

      \[\leadsto 1 + -0.5 \cdot \frac{\color{blue}{b \cdot b}}{{a}^{2}} \]

   2. unpow2 75.1%

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

7. Simplified 75.1%

   \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{b \cdot b}{a \cdot a}} \]
8. Step-by-step derivation

   1. *-commutative 75.1%

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

   2. associate-/l* 75.2%

      \[\leadsto 1 + \color{blue}{\left(b \cdot \frac{b}{a \cdot a}\right)} \cdot -0.5 \]

   3. associate-/l/ 97.8%

      \[\leadsto 1 + \left(b \cdot \color{blue}{\frac{\frac{b}{a}}{a}}\right) \cdot -0.5 \]

9. Applied egg-rr 97.8%

   \[\leadsto 1 + \color{blue}{\left(b \cdot \frac{\frac{b}{a}}{a}\right) \cdot -0.5} \]

10. Final simplification 97.8%

    \[\leadsto 1 + -0.5 \cdot \left(b \cdot \frac{\frac{b}{a}}{a}\right) \]
11. Add Preprocessing

Alternative 5: 98.0% 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 75.4%

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

   1. add-sqr-sqrt 75.4%

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

   2. fabs-sqr 75.4%

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

   3. add-sqr-sqrt 75.4%

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

   4. div-sub 75.4%

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

   5. *-inverses 75.4%

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

   6. frac-times 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in b around 0 95.9%

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

Reproduce

herbie shell --seed 2024098 
(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)))))