Eccentricity of an ellipse

Percentage Accurate: 77.6% → 100.0%

Time: 6.0s

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: 77.6% 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 79.3%

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

   1. sqr-neg 79.3%

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

   2. fabs-div 79.3%

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

   3. sqr-neg 79.3%

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

   4. fabs-sub 79.3%

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

   5. sqr-neg 79.3%

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

   6. distribute-rgt-neg-out 79.3%

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

   7. fabs-neg 79.3%

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

   8. fabs-div 79.3%

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

   9. cancel-sign-sub-inv 79.3%

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

   10. +-commutative 79.3%

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

   11. sqr-neg 79.3%

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

   12. cancel-sign-sub-inv 79.3%

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

   13. div-sub 79.3%

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

3. Simplified 79.9%

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

   1. pow1/2 79.9%

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

   2. pow-to-exp 79.9%

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

   3. add-sqr-sqrt 79.3%

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

   4. fabs-sqr 79.3%

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

   5. add-sqr-sqrt 79.3%

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

   6. sub-neg 79.3%

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

   7. log1p-define 79.3%

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

   8. associate-*r/ 79.3%

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

   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. Add Preprocessing

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[Sqrt[N[(N[(a - b), $MachinePrecision] / N[(a * N[(a / N[(b + a), $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. Step-by-step derivation

   1. sqr-neg 79.3%

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

   2. associate-/r* 78.7%

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

   3. sqr-neg 78.7%

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

   4. associate-/r* 79.3%

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

   5. div-sub 79.3%

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

   6. fabs-sub 79.3%

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

   7. times-frac 79.3%

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

   8. *-inverses 100.0%

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

   9. difference-of-sqr-1 99.9%

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

   10. difference-of-sqr--1 100.0%

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

   11. fma-define 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. associate-/l* 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. sqrt-prod 79.9%

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

   5. sqrt-div 79.9%

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

   6. sqrt-prod 79.9%

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

   7. add-sqr-sqrt 79.9%

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

   8. associate-/l* 79.9%

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

   9. add-sqr-sqrt 79.9%

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

   10. sqrt-prod 79.9%

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

   11. sqrt-div 79.9%

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

   12. sqrt-prod 79.9%

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

   13. metadata-eval 79.9%

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

   14. fmm-def 79.9%

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

   15. add-sqr-sqrt 79.9%

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

   16. fabs-sub 79.9%

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

   17. *-inverses 79.3%

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

   18. associate-*r/ 79.3%

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

6. Applied egg-rr 99.9%

   \[\leadsto \sqrt{\color{blue}{\frac{b + a}{a} \cdot \frac{a - b}{a}}} \]
7. Step-by-step derivation

   1. clear-num 100.0%

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

   2. frac-times 100.0%

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

   3. *-un-lft-identity 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \sqrt{\frac{a - b}{a \cdot \frac{a}{b + a}}} \]
10. Add Preprocessing

Alternative 3: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[Sqrt[N[(N[(b + a), $MachinePrecision] / N[(a * N[(a / N[(a - b), $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. Step-by-step derivation

   1. sqr-neg 79.3%

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

   2. associate-/r* 78.7%

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

   3. sqr-neg 78.7%

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

   4. associate-/r* 79.3%

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

   5. div-sub 79.3%

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

   6. fabs-sub 79.3%

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

   7. times-frac 79.3%

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

   8. *-inverses 100.0%

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

   9. difference-of-sqr-1 99.9%

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

   10. difference-of-sqr--1 100.0%

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

   11. fma-define 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. associate-/l* 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. sqrt-prod 79.9%

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

   5. sqrt-div 79.9%

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

   6. sqrt-prod 79.9%

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

   7. add-sqr-sqrt 79.9%

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

   8. associate-/l* 79.9%

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

   9. add-sqr-sqrt 79.9%

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

   10. sqrt-prod 79.9%

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

   11. sqrt-div 79.9%

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

   12. sqrt-prod 79.9%

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

   13. metadata-eval 79.9%

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

   14. fmm-def 79.9%

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

   15. add-sqr-sqrt 79.9%

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

   16. fabs-sub 79.9%

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

   17. *-inverses 79.3%

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

   18. associate-*r/ 79.3%

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

6. Applied egg-rr 99.9%

   \[\leadsto \sqrt{\color{blue}{\frac{b + a}{a} \cdot \frac{a - b}{a}}} \]
7. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. clear-num 100.0%

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

   3. frac-times 100.0%

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

   4. *-un-lft-identity 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \sqrt{\frac{b + a}{a \cdot \frac{a}{a - b}}} \]
10. Add Preprocessing

Alternative 4: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. sqr-neg 79.3%

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

   2. associate-/r* 78.7%

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

   3. sqr-neg 78.7%

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

   4. associate-/r* 79.3%

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

   5. div-sub 79.3%

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

   6. fabs-sub 79.3%

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

   7. times-frac 79.3%

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

   8. *-inverses 100.0%

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

   9. difference-of-sqr-1 99.9%

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

   10. difference-of-sqr--1 100.0%

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

   11. fma-define 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. associate-/l* 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. sqrt-prod 79.9%

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

   5. sqrt-div 79.9%

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

   6. sqrt-prod 79.9%

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

   7. add-sqr-sqrt 79.9%

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

   8. associate-/l* 79.9%

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

   9. add-sqr-sqrt 79.9%

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

   10. sqrt-prod 79.9%

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

   11. sqrt-div 79.9%

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

   12. sqrt-prod 79.9%

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

   13. metadata-eval 79.9%

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

   14. fmm-def 79.9%

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

   15. add-sqr-sqrt 79.9%

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

   16. fabs-sub 79.9%

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

   17. *-inverses 79.3%

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

   18. associate-*r/ 79.3%

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

6. Applied egg-rr 99.9%

   \[\leadsto \sqrt{\color{blue}{\frac{b + a}{a} \cdot \frac{a - b}{a}}} \]
7. 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 79.3%

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

   1. sqr-neg 79.3%

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

   2. fabs-div 79.3%

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

   3. sqr-neg 79.3%

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

   4. fabs-sub 79.3%

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

   5. sqr-neg 79.3%

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

   6. distribute-rgt-neg-out 79.3%

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

   7. fabs-neg 79.3%

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

   8. fabs-div 79.3%

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

   9. cancel-sign-sub-inv 79.3%

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

   10. +-commutative 79.3%

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

   11. sqr-neg 79.3%

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

   12. cancel-sign-sub-inv 79.3%

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

   13. div-sub 79.3%

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

3. Simplified 79.9%

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

   1. pow1/2 79.9%

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

   2. pow-to-exp 79.9%

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

   3. add-sqr-sqrt 79.3%

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

   4. fabs-sqr 79.3%

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

   5. add-sqr-sqrt 79.3%

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

   6. sub-neg 79.3%

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

   7. log1p-define 79.3%

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

   8. associate-*r/ 79.3%

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

   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. Taylor expanded in b around 0 98.0%

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

Reproduce

herbie shell --seed 2024186 
(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)))))