Eccentricity of an ellipse

Percentage Accurate: 77.9% → 100.0%

Time: 4.4s

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: 77.9% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

   1. sqr-neg 76.2%

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

   2. fabs-div 76.2%

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

   3. sqr-neg 76.2%

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

   4. fabs-sub 76.2%

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

   5. sqr-neg 76.2%

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

   6. distribute-rgt-neg-out 76.2%

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

   7. fabs-neg 76.2%

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

   8. fabs-div 76.2%

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

   9. cancel-sign-sub-inv 76.2%

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

   10. +-commutative 76.2%

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

   11. sqr-neg 76.2%

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

   12. cancel-sign-sub-inv 76.2%

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

3. Simplified 76.9%

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

   1. associate-*r/ 76.2%

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

   2. frac-times 100.0%

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

   3. clear-num 100.0%

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

   4. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \sqrt{\left|1 - \frac{\frac{b}{a}}{\frac{a}{b}}\right|} \]
8. Add Preprocessing

Alternative 2: 97.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(1, \frac{b}{a}\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (hypot 1.0 (/ b a)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[Sqrt[1.0 ^ 2 + N[(b / a), $MachinePrecision] ^ 2], $MachinePrecision]

Derivation

1. Initial program 76.2%

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

   1. add-sqr-sqrt 76.2%

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

   2. fabs-sqr 76.2%

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

   3. add-sqr-sqrt 76.2%

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

   4. div-sub 76.2%

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

   5. *-inverses 76.2%

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

   6. associate-*r/ 76.9%

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

   7. fabs-sub 76.9%

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

   8. add-sqr-sqrt 76.9%

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

   9. fma-neg 76.9%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in b around 0 100.0%

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

7. Simplified 98.9%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \frac{b}{a}\right)} \]

8. Final simplification 98.9%

   \[\leadsto \mathsf{hypot}\left(1, \frac{b}{a}\right) \]
9. Add Preprocessing

Alternative 3: 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 76.2%

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

3. Taylor expanded in a around inf 99.0%

   \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]

4. Final simplification 99.0%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(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)))))