Eccentricity of an ellipse

Percentage Accurate: 77.3% → 100.0%

Time: 3.8s

Alternatives: 4

Speedup: 42.0×

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: 77.3% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in a around 0

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

   1. fabs-neg N/A

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

   2. lower-fabs.f64 N/A

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

   3. div-sub N/A

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

   4. *-inverses N/A

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

   5. unsub-neg N/A

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

   6. mul-1-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. mul-1-neg N/A

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

   10. remove-double-neg N/A

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

   11. unpow2 N/A

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

   12. associate-/l* N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. unpow2 N/A

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

   17. associate-/r* N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 3: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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 99.0%

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

   1. lift-fabs.f64 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 99.0

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in a around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b \cdot b}{\color{blue}{a \cdot a}}, 1\right) \]

   7. lower-*.f64 78.9

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

10. Applied rewrites 78.9%

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

12. Applied rewrites 99.8%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{\frac{b}{a} \cdot b}{\color{blue}{a}}, 1\right) \]
13. Add Preprocessing

Alternative 4: 97.8% accurate, 42.0× 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 78.9%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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 99.0%

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

   1. lift-fabs.f64 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 99.0

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in a around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b \cdot b}{\color{blue}{a \cdot a}}, 1\right) \]

   7. lower-*.f64 78.9

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

10. Applied rewrites 78.9%

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

11. Taylor expanded in a around inf

    \[\leadsto \color{blue}{1} \]
12. Step-by-step derivation

13. Applied rewrites 99.0%

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

Reproduce

herbie shell --seed 2024312 
(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)))))