Eccentricity of an ellipse

Percentage Accurate: 76.8% → 100.0%

Time: 7.4s

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: 76.8% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. sqr-neg 76.9%

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

   2. fabs-div 76.9%

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

   3. sqr-neg 76.9%

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

   4. fabs-sub 76.9%

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

   5. sqr-neg 76.9%

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

   6. distribute-rgt-neg-out 76.9%

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

   7. fabs-neg 76.9%

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

   8. fabs-div 76.9%

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

   9. cancel-sign-sub-inv 76.9%

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

   10. +-commutative 76.9%

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

   11. sqr-neg 76.9%

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

   12. cancel-sign-sub-inv 76.9%

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

3. Simplified 77.7%

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

   1. pow1/2 77.7%

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

   2. pow-to-exp 77.7%

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

   3. add-sqr-sqrt 76.9%

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

   4. fabs-sqr 76.9%

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

   5. add-sqr-sqrt 76.9%

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

   6. sub-neg 76.9%

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

   7. log1p-define 77.0%

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

   8. associate-*r/ 77.0%

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

   9. frac-times 100.0%

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

   10. pow2 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5}} \]
7. Step-by-step derivation

   1. unpow2 100.0%

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

   2. clear-num 100.0%

      \[\leadsto e^{\mathsf{log1p}\left(-\frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right) \cdot 0.5} \]

   3. un-div-inv 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

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(b / Float64(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[(b / N[(a * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.9%

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

   1. sqr-neg 76.9%

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

   2. fabs-div 76.9%

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

   3. sqr-neg 76.9%

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

   4. fabs-sub 76.9%

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

   5. sqr-neg 76.9%

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

   6. distribute-rgt-neg-out 76.9%

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

   7. fabs-neg 76.9%

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

   8. fabs-div 76.9%

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

   9. cancel-sign-sub-inv 76.9%

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

   10. +-commutative 76.9%

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

   11. sqr-neg 76.9%

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

   12. cancel-sign-sub-inv 76.9%

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

3. Simplified 77.7%

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

   1. associate-*r/ 76.9%

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

   2. frac-times 99.9%

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

   3. clear-num 100.0%

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

   4. frac-times 100.0%

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

   5. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 3: 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 76.9%

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

   1. sqr-neg 76.9%

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

   2. fabs-div 76.9%

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

   3. sqr-neg 76.9%

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

   4. fabs-sub 76.9%

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

   5. sqr-neg 76.9%

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

   6. distribute-rgt-neg-out 76.9%

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

   7. fabs-neg 76.9%

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

   8. fabs-div 76.9%

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

   9. cancel-sign-sub-inv 76.9%

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

   10. +-commutative 76.9%

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

   11. sqr-neg 76.9%

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

   12. cancel-sign-sub-inv 76.9%

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

3. Simplified 77.7%

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

   1. fabs-sub 77.7%

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

   2. sub-neg 77.7%

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

   3. metadata-eval 77.7%

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

   4. associate-*r/ 76.9%

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

   5. frac-times 99.9%

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

   6. fma-undefine 99.9%

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

   7. add-exp-log 99.9%

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

   8. add-sqr-sqrt 99.9%

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

   9. log-prod 99.9%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{0.5 \cdot \mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right)}} \]
7. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. log1p-undefine 99.9%

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

   3. +-commutative 99.9%

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

   4. metadata-eval 99.9%

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

   5. distribute-neg-in 99.9%

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

   6. exp-to-pow 99.9%

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

   7. unpow1/2 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. metadata-eval 99.9%

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

   10. +-commutative 99.9%

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

   11. sub-neg 99.9%

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

8. Simplified 99.9%

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

   1. unpow2 100.0%

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

   2. clear-num 100.0%

      \[\leadsto e^{\mathsf{log1p}\left(-\frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right) \cdot 0.5} \]

   3. un-div-inv 100.0%

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

10. Applied egg-rr 100.0%

    \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}}} \]
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 76.9%

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

   1. sqr-neg 76.9%

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

   2. fabs-div 76.9%

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

   3. sqr-neg 76.9%

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

   4. fabs-sub 76.9%

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

   5. sqr-neg 76.9%

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

   6. distribute-rgt-neg-out 76.9%

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

   7. fabs-neg 76.9%

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

   8. fabs-div 76.9%

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

   9. cancel-sign-sub-inv 76.9%

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

   10. +-commutative 76.9%

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

   11. sqr-neg 76.9%

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

   12. cancel-sign-sub-inv 76.9%

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

3. Simplified 77.7%

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

   1. fabs-sub 77.7%

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

   2. sub-neg 77.7%

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

   3. metadata-eval 77.7%

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

   4. associate-*r/ 76.9%

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

   5. frac-times 99.9%

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

   6. fma-undefine 99.9%

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

   7. add-exp-log 99.9%

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

   8. add-sqr-sqrt 99.9%

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

   9. log-prod 99.9%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{0.5 \cdot \mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right)}} \]
7. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. log1p-undefine 99.9%

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

   3. +-commutative 99.9%

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

   4. metadata-eval 99.9%

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

   5. distribute-neg-in 99.9%

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

   6. exp-to-pow 99.9%

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

   7. unpow1/2 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. metadata-eval 99.9%

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

   10. +-commutative 99.9%

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

   11. sub-neg 99.9%

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

8. Simplified 99.9%

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

9. Taylor expanded in b around 0 76.0%

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

    1. +-commutative 76.0%

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

    2. unpow2 76.0%

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

    3. unpow2 76.0%

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

    4. times-frac 97.7%

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

    5. unpow2 97.7%

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

11. Simplified 97.7%

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

    1. unpow2 97.7%

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

    2. clear-num 97.7%

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

    3. frac-times 97.7%

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

    4. *-un-lft-identity 97.7%

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

    5. clear-num 97.7%

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

    6. associate-/r/ 97.7%

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

    7. associate-/r* 97.7%

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

    8. clear-num 97.7%

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

13. Applied egg-rr 97.7%

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

14. Final simplification 97.7%

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

Alternative 5: 98.1% 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 76.9%

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

   1. sqr-neg 76.9%

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

   2. fabs-div 76.9%

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

   3. sqr-neg 76.9%

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

   4. fabs-sub 76.9%

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

   5. sqr-neg 76.9%

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

   6. distribute-rgt-neg-out 76.9%

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

   7. fabs-neg 76.9%

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

   8. fabs-div 76.9%

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

   9. cancel-sign-sub-inv 76.9%

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

   10. +-commutative 76.9%

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

   11. sqr-neg 76.9%

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

   12. cancel-sign-sub-inv 76.9%

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

3. Simplified 77.7%

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

   1. fabs-sub 77.7%

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

   2. sub-neg 77.7%

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

   3. metadata-eval 77.7%

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

   4. associate-*r/ 76.9%

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

   5. frac-times 99.9%

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

   6. fma-undefine 99.9%

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

   7. add-exp-log 99.9%

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

   8. add-sqr-sqrt 99.9%

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

   9. log-prod 99.9%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{0.5 \cdot \mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right)}} \]
7. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. log1p-undefine 99.9%

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

   3. +-commutative 99.9%

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

   4. metadata-eval 99.9%

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

   5. distribute-neg-in 99.9%

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

   6. exp-to-pow 99.9%

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

   7. unpow1/2 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. metadata-eval 99.9%

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

   10. +-commutative 99.9%

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

   11. sub-neg 99.9%

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

8. Simplified 99.9%

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

9. Taylor expanded in b around 0 96.2%

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

Reproduce

herbie shell --seed 2024083 
(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)))))