Eccentricity of an ellipse

Percentage Accurate: 77.7% → 100.0%

Time: 3.3s

Alternatives: 4

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, 0.7× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (exp (* (log1p (- (pow (/ b a) 2.0))) 0.5)))

C

double code(double a, double b) {
	return exp((log1p(-pow((b / a), 2.0)) * 0.5));
}

Java

public static double code(double a, double b) {
	return Math.exp((Math.log1p(-Math.pow((b / a), 2.0)) * 0.5));
}

Python

def code(a, b):
	return math.exp((math.log1p(-math.pow((b / a), 2.0)) * 0.5))

Julia

function code(a, b)
	return exp(Float64(log1p(Float64(-(Float64(b / a) ^ 2.0))) * 0.5))
end

Wolfram

code[a_, b_] := N[Exp[N[(N[Log[1 + (-N[Power[N[(b / a), $MachinePrecision], 2.0], $MachinePrecision])], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.6%

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

   1. div-sub 76.6%

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

   2. *-inverses 76.6%

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

   3. times-frac 100.0%

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

3. Simplified 100.0%

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

   1. pow1/2 100.0%

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

   2. pow-to-exp 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. fabs-sqr 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. sub-neg 100.0%

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

   7. log1p-def 100.0%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(-\frac{b}{a} \cdot \frac{b}{a}\right)} \cdot 0.5} \]

   8. pow2 100.0%

      \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{{\left(\frac{b}{a}\right)}^{2}}\right) \cdot 0.5} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5}} \]

6. Final simplification 100.0%

   \[\leadsto e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5} \]

Alternative 2: 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 76.6%

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

   1. div-sub 76.6%

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

   2. *-inverses 76.6%

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

   3. times-frac 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in b around 0 76.6%

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

   1. unpow2 76.6%

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

   2. unpow2 76.6%

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

   3. times-frac 100.0%

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

   4. unpow2 100.0%

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

   5. rem-square-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. rem-square-sqrt 100.0%

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

6. Simplified 100.0%

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

   1. unpow2 98.4%

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

   2. clear-num 98.4%

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

   3. un-div-inv 98.4%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 3: 99.1% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. div-sub 76.6%

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

   2. *-inverses 76.6%

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

   3. times-frac 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in b around 0 76.6%

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

   1. unpow2 76.6%

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

   2. unpow2 76.6%

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

   3. times-frac 100.0%

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

   4. unpow2 100.0%

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

   5. rem-square-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. rem-square-sqrt 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in b around 0 75.9%

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

   1. unpow2 75.9%

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

   2. unpow2 75.9%

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

   3. times-frac 98.4%

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

   4. unpow2 98.4%

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

9. Simplified 98.4%

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

    1. unpow2 98.4%

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

    2. clear-num 98.4%

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

    3. un-div-inv 98.4%

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

11. Applied egg-rr 98.4%

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

12. Final simplification 98.4%

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

Alternative 4: 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 76.6%

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

   1. div-sub 76.6%

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

   2. *-inverses 76.6%

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

   3. times-frac 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in b around 0 76.6%

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

   1. unpow2 76.6%

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

   2. unpow2 76.6%

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

   3. times-frac 100.0%

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

   4. unpow2 100.0%

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

   5. rem-square-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. rem-square-sqrt 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in b around 0 97.1%

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

8. Final simplification 97.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023189 
(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)))))