Eccentricity of an ellipse

Percentage Accurate: 78.2% → 100.0%

Time: 5.7s

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: 78.2% 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, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. add-sqr-sqrt 79.3%

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

   2. fabs-sqr 79.3%

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

   3. add-sqr-sqrt 79.3%

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

   4. div-sub 79.3%

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

   5. *-inverses 79.3%

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

   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. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{a} \cdot \frac{b}{a}\right)\right)}} \]

   2. expm1-undefine 100.0%

      \[\leadsto \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{b}{a} \cdot \frac{b}{a}\right)} - 1\right)}} \]

   3. log1p-undefine 100.0%

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

   4. add-exp-log 100.0%

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

   5. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 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 79.3%

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

   1. add-sqr-sqrt 79.3%

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

   2. fabs-sqr 79.3%

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

   3. add-sqr-sqrt 79.3%

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

   4. div-sub 79.3%

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

   5. *-inverses 79.3%

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

   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 3: 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 79.3%

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

   1. add-cbrt-cube 79.3%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \cdot \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\right) \cdot \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}}} \]

   2. add-sqr-sqrt 79.3%

      \[\leadsto \sqrt[3]{\color{blue}{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \cdot \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}} \]

   3. pow1 79.3%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|\right)}^{1}} \cdot \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}} \]

   4. pow1/2 79.3%

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

   5. pow-prod-up 79.3%

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

   6. add-sqr-sqrt 79.3%

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

   7. fabs-sqr 79.3%

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

   8. add-sqr-sqrt 79.3%

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

   9. div-sub 79.3%

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

   10. *-inverses 79.3%

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

   11. frac-times 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)}^{1.5}}} \]

5. Taylor expanded in b around 0 98.2%

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

Reproduce

herbie shell --seed 2024097 
(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)))))