Eccentricity of an ellipse

Percentage Accurate: 78.3% → 100.0%

Time: 7.1s

Alternatives: 6

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: 78.3% 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 73.3%

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

   1. sqr-neg 73.3%

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

   2. fabs-div 73.3%

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

   3. sqr-neg 73.3%

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

   4. fabs-sub 73.3%

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

   5. sqr-neg 73.3%

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

   6. distribute-rgt-neg-out 73.3%

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

   7. fabs-neg 73.3%

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

   8. fabs-div 73.3%

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

   9. cancel-sign-sub-inv 73.3%

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

   10. +-commutative 73.3%

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

   11. sqr-neg 73.3%

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

   12. cancel-sign-sub-inv 73.3%

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

3. Simplified 74.3%

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

   1. pow1/2 74.3%

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

   2. pow-to-exp 74.3%

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

   3. add-sqr-sqrt 73.4%

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

   4. fabs-sqr 73.4%

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

   5. add-sqr-sqrt 73.4%

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

   6. sub-neg 73.4%

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

   7. log1p-define 73.4%

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

   8. associate-*r/ 73.3%

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

   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, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (sqrt (- 1.0 (pow (/ b a) 2.0))))

C

double code(double a, double b) {
	return sqrt((1.0 - pow((b / a), 2.0)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt((1.0d0 - ((b / a) ** 2.0d0)))
end function

Java

public static double code(double a, double b) {
	return Math.sqrt((1.0 - Math.pow((b / a), 2.0)));
}

Python

def code(a, b):
	return math.sqrt((1.0 - math.pow((b / a), 2.0)))

Julia

function code(a, b)
	return sqrt(Float64(1.0 - (Float64(b / a) ^ 2.0)))
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt((1.0 - ((b / a) ^ 2.0)));
end

Wolfram

code[a_, b_] := N[Sqrt[N[(1.0 - N[Power[N[(b / a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.3%

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

   1. sqr-neg 73.3%

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

   2. associate-/r* 73.9%

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

   3. sqr-neg 73.9%

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

   4. associate-/r* 73.3%

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

   5. div-sub 73.3%

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

   6. fabs-sub 73.3%

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

   7. times-frac 73.4%

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

   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 74.3%

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

   5. sqrt-div 74.3%

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

   6. sqrt-prod 74.3%

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

   7. add-sqr-sqrt 74.3%

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

   8. associate-/l* 74.3%

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

   9. add-sqr-sqrt 74.3%

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

   10. sqrt-prod 74.3%

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

   11. sqrt-div 74.3%

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

   12. sqrt-prod 74.3%

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

   13. metadata-eval 74.3%

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

   14. fma-neg 74.3%

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

   15. add-sqr-sqrt 74.3%

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

   16. fabs-sub 74.3%

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

   17. add-sqr-sqrt 73.4%

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

6. Applied egg-rr 100.0%

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

Alternative 3: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. sqr-neg 73.3%

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

   2. associate-/r* 73.9%

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

   3. sqr-neg 73.9%

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

   4. associate-/r* 73.3%

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

   5. div-sub 73.3%

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

   6. fabs-sub 73.3%

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

   7. times-frac 73.4%

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

   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 74.3%

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

   5. sqrt-div 74.3%

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

   6. sqrt-prod 74.3%

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

   7. add-sqr-sqrt 74.3%

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

   8. associate-/l* 74.3%

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

   9. add-sqr-sqrt 74.3%

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

   10. sqrt-prod 74.3%

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

   11. sqrt-div 74.3%

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

   12. sqrt-prod 74.3%

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

   13. metadata-eval 74.3%

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

   14. fma-neg 74.3%

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

   15. add-sqr-sqrt 74.3%

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

   16. fabs-sub 74.3%

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

   17. *-inverses 73.4%

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

   18. associate-*r/ 73.3%

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

6. Applied egg-rr 73.7%

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

   1. associate-*r/ 73.3%

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

   2. unpow2 73.3%

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

   3. frac-times 99.9%

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

   4. *-commutative 99.9%

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

   5. clear-num 100.0%

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

   6. un-div-inv 100.0%

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

   7. div-sub 100.0%

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

   8. *-inverses 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \sqrt{\color{blue}{\frac{1 - \frac{b}{a}}{\frac{a}{b + a}}}} \]
9. Add Preprocessing

Alternative 4: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. sqr-neg 73.3%

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

   2. associate-/r* 73.9%

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

   3. sqr-neg 73.9%

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

   4. associate-/r* 73.3%

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

   5. div-sub 73.3%

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

   6. fabs-sub 73.3%

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

   7. times-frac 73.4%

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

   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 74.3%

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

   5. sqrt-div 74.3%

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

   6. sqrt-prod 74.3%

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

   7. add-sqr-sqrt 74.3%

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

   8. associate-/l* 74.3%

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

   9. add-sqr-sqrt 74.3%

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

   10. sqrt-prod 74.3%

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

   11. sqrt-div 74.3%

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

   12. sqrt-prod 74.3%

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

   13. metadata-eval 74.3%

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

   14. fma-neg 74.3%

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

   15. add-sqr-sqrt 74.3%

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

   16. fabs-sub 74.3%

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

   17. *-inverses 73.4%

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

   18. associate-*r/ 73.3%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in b around 0 99.9%

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

8. Final simplification 99.9%

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

Alternative 5: 99.1% accurate, 19.2× speedup

Math

\[\begin{array}{l} \\ 1 - 0.5 \cdot \left(\frac{b}{a} \cdot \frac{b}{a}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. sqr-neg 73.3%

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

   2. associate-/r* 73.9%

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

   3. sqr-neg 73.9%

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

   4. associate-/r* 73.3%

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

   5. div-sub 73.3%

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

   6. fabs-sub 73.3%

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

   7. times-frac 73.4%

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

   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 74.3%

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

   5. sqrt-div 74.3%

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

   6. sqrt-prod 74.3%

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

   7. add-sqr-sqrt 74.3%

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

   8. associate-/l* 74.3%

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

   9. add-sqr-sqrt 74.3%

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

   10. sqrt-prod 74.3%

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

   11. sqrt-div 74.3%

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

   12. sqrt-prod 74.3%

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

   13. metadata-eval 74.3%

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

   14. fma-neg 74.3%

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

   15. add-sqr-sqrt 74.3%

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

   16. fabs-sub 74.3%

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

   17. *-inverses 73.4%

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

   18. associate-*r/ 73.3%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in b around 0 99.9%

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

8. Taylor expanded in a around inf 72.2%

   \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{b + -1 \cdot b}{a} + 0.5 \cdot \frac{-1 \cdot {b}^{2} - 0.25 \cdot {\left(b + -1 \cdot b\right)}^{2}}{{a}^{2}}\right)} \]

10. Simplified 98.7%

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

    1. unpow2 98.7%

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

    2. distribute-rgt-neg-in 98.7%

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

12. Applied egg-rr 98.7%

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

13. Final simplification 98.7%

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

Alternative 6: 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 73.3%

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

   1. sqr-neg 73.3%

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

   2. fabs-div 73.3%

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

   3. sqr-neg 73.3%

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

   4. fabs-sub 73.3%

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

   5. sqr-neg 73.3%

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

   6. distribute-rgt-neg-out 73.3%

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

   7. fabs-neg 73.3%

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

   8. fabs-div 73.3%

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

   9. cancel-sign-sub-inv 73.3%

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

   10. +-commutative 73.3%

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

   11. sqr-neg 73.3%

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

   12. cancel-sign-sub-inv 73.3%

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

3. Simplified 74.3%

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

   1. pow1/2 74.3%

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

   2. pow-to-exp 74.3%

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

   3. add-sqr-sqrt 73.4%

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

   4. fabs-sqr 73.4%

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

   5. add-sqr-sqrt 73.4%

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

   6. sub-neg 73.4%

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

   7. log1p-define 73.4%

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

   8. associate-*r/ 73.3%

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

   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 96.5%

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

Reproduce

herbie shell --seed 2024137 
(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)))))