Eccentricity of an ellipse

Percentage Accurate: 78.6% → 100.0%

Time: 7.9s

Alternatives: 5

Speedup: 3.2×

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: 78.6% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|\frac{\mathsf{fma}\left(-b, \frac{b}{a}, a\right)}{a}\right|} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (sqrt (fabs (/ (fma (- b) (/ b a) a) a))))

C

double code(double a, double b) {
	return sqrt(fabs((fma(-b, (b / a), a) / a)));
}

Julia

function code(a, b)
	return sqrt(abs(Float64(fma(Float64(-b), Float64(b / a), a) / a)))
end

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. *-inverses N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. *-inverses N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-times N/A

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

   10. div-sub N/A

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

   11. difference-of-squares N/A

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

   12. lift-+.f64 N/A

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

   13. lift--.f64 N/A

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

   14. associate-*l/ N/A

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

   15. associate-/r* N/A

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

   16. lift-/.f64 N/A

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

   17. associate-*l/ N/A

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

   18. lower-/.f64 N/A

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

   19. lower-*.f64 100.0

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

   20. lift-+.f64 N/A

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

   21. +-commutative N/A

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

   22. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

      \[\leadsto \sqrt{\left|\frac{\color{blue}{\left(\mathsf{neg}\left(\frac{{b}^{2}}{a}\right)\right)} + a}{a}\right|} \]

   3. unpow2 N/A

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

   4. associate-*r/ N/A

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

   5. distribute-lft-neg-in N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 100.0

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

9. Applied rewrites 100.0%

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

Alternative 2: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 0:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|1 - b \cdot \frac{b}{a \cdot a}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 0.0)
   (sqrt (fabs 1.0))
   (sqrt (fabs (- 1.0 (* b (/ b (* a a))))))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 0.0) {
		tmp = sqrt(fabs(1.0));
	} else {
		tmp = sqrt(fabs((1.0 - (b * (b / (a * a))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * a) <= 0.0d0) then
        tmp = sqrt(abs(1.0d0))
    else
        tmp = sqrt(abs((1.0d0 - (b * (b / (a * a))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a * a) <= 0.0) {
		tmp = Math.sqrt(Math.abs(1.0));
	} else {
		tmp = Math.sqrt(Math.abs((1.0 - (b * (b / (a * a))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 0.0:
		tmp = math.sqrt(math.fabs(1.0))
	else:
		tmp = math.sqrt(math.fabs((1.0 - (b * (b / (a * a))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 0.0)
		tmp = sqrt(abs(1.0));
	else
		tmp = sqrt(abs(Float64(1.0 - Float64(b * Float64(b / Float64(a * a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 0.0)
		tmp = sqrt(abs(1.0));
	else
		tmp = sqrt(abs((1.0 - (b * (b / (a * a))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 0.0], N[Sqrt[N[Abs[1.0], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(1.0 - N[(b * N[(b / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 0.0

   1. Initial program 0.0%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 97.4%

      \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]

   if 0.0 < (*.f64 a a)

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \sqrt{\left|\color{blue}{1 - b \cdot \frac{b}{a \cdot a}}\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 0:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 0.0)
   (sqrt (fabs 1.0))
   (sqrt (fabs (fma b (/ b (* a a)) -1.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 0.0) {
		tmp = sqrt(fabs(1.0));
	} else {
		tmp = sqrt(fabs(fma(b, (b / (a * a)), -1.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 0.0)
		tmp = sqrt(abs(1.0));
	else
		tmp = sqrt(abs(fma(b, Float64(b / Float64(a * a)), -1.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 0.0], N[Sqrt[N[Abs[1.0], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(b * N[(b / N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 0.0

   1. Initial program 0.0%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 97.4%

      \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]

   if 0.0 < (*.f64 a a)

   1. Initial program 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. fabs-sub N/A

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

      7. lower-fabs.f64 N/A

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

      8. sub-neg N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \sqrt{\color{blue}{\left|\mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|\mathsf{fma}\left(\frac{b}{a}, \frac{b}{a}, -1\right)\right|} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return sqrt(fabs(fma((b / a), (b / a), -1.0)));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. *-inverses N/A

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

   6. fabs-sub N/A

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

   7. lower-fabs.f64 N/A

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

   8. sub-neg N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 79.0

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

4. Applied rewrites 79.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 5: 98.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|1\right|} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (sqrt (fabs 1.0)))

C

double code(double a, double b) {
	return sqrt(fabs(1.0));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(abs(1.0d0))
end function

Java

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

Python

def code(a, b):
	return math.sqrt(math.fabs(1.0))

Julia

function code(a, b)
	return sqrt(abs(1.0))
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt(abs(1.0));
end

Wolfram

code[a_, b_] := N[Sqrt[N[Abs[1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Taylor expanded in a around inf

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

5. Applied rewrites 97.5%

   \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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)))))