Eccentricity of an ellipse

Percentage Accurate: 77.2% → 100.0%

Time: 5.8s

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: 77.2% 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

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

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 81.6%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-/.f64 82.2

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

4. Applied rewrites 82.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot a - b \cdot b}{a \cdot a} \leq 0.9999999999959164:\\ \;\;\;\;\sqrt{\left|\left(a + b\right) \cdot \frac{a - b}{a \cdot a}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (- (* a a) (* b b)) (* a a)) 0.9999999999959164)
   (sqrt (fabs (* (+ a b) (/ (- a b) (* a a)))))
   (sqrt (fabs 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999959164) {
		tmp = sqrt(fabs(((a + b) * ((a - b) / (a * a)))));
	} else {
		tmp = sqrt(fabs(1.0));
	}
	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) - (b * b)) / (a * a)) <= 0.9999999999959164d0) then
        tmp = sqrt(abs(((a + b) * ((a - b) / (a * a)))))
    else
        tmp = sqrt(abs(1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999959164) {
		tmp = Math.sqrt(Math.abs(((a + b) * ((a - b) / (a * a)))));
	} else {
		tmp = Math.sqrt(Math.abs(1.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (((a * a) - (b * b)) / (a * a)) <= 0.9999999999959164:
		tmp = math.sqrt(math.fabs(((a + b) * ((a - b) / (a * a)))))
	else:
		tmp = math.sqrt(math.fabs(1.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a)) <= 0.9999999999959164)
		tmp = sqrt(abs(Float64(Float64(a + b) * Float64(Float64(a - b) / Float64(a * a)))));
	else
		tmp = sqrt(abs(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999959164)
		tmp = sqrt(abs(((a + b) * ((a - b) / (a * a)))));
	else
		tmp = sqrt(abs(1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision], 0.9999999999959164], N[Sqrt[N[Abs[N[(N[(a + b), $MachinePrecision] * N[(N[(a - b), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[1.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 a a) (*.f64 b b)) (*.f64 a a)) < 0.99999999999591638

   1. Initial program 99.7%

      \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 0.99999999999591638 < (/.f64 (-.f64 (*.f64 a a) (*.f64 b b)) (*.f64 a a))

   1. Initial program 80.9%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 99.0%

      \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot a - b \cdot b}{a \cdot a} \leq 0.9999999999959164:\\ \;\;\;\;\sqrt{\left|\left(a + b\right) \cdot \frac{a - b}{a \cdot a}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot a - b \cdot b}{a \cdot a} \leq 0.9999999999959164:\\ \;\;\;\;\sqrt{\left|\frac{b}{a \cdot a} \cdot b - 1\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (- (* a a) (* b b)) (* a a)) 0.9999999999959164)
   (sqrt (fabs (- (* (/ b (* a a)) b) 1.0)))
   (sqrt (fabs 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999959164) {
		tmp = sqrt(fabs((((b / (a * a)) * b) - 1.0)));
	} else {
		tmp = sqrt(fabs(1.0));
	}
	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) - (b * b)) / (a * a)) <= 0.9999999999959164d0) then
        tmp = sqrt(abs((((b / (a * a)) * b) - 1.0d0)))
    else
        tmp = sqrt(abs(1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999959164) {
		tmp = Math.sqrt(Math.abs((((b / (a * a)) * b) - 1.0)));
	} else {
		tmp = Math.sqrt(Math.abs(1.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (((a * a) - (b * b)) / (a * a)) <= 0.9999999999959164:
		tmp = math.sqrt(math.fabs((((b / (a * a)) * b) - 1.0)))
	else:
		tmp = math.sqrt(math.fabs(1.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a)) <= 0.9999999999959164)
		tmp = sqrt(abs(Float64(Float64(Float64(b / Float64(a * a)) * b) - 1.0)));
	else
		tmp = sqrt(abs(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999959164)
		tmp = sqrt(abs((((b / (a * a)) * b) - 1.0)));
	else
		tmp = sqrt(abs(1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision], 0.9999999999959164], N[Sqrt[N[Abs[N[(N[(N[(b / N[(a * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[1.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 a a) (*.f64 b b)) (*.f64 a a)) < 0.99999999999591638

   1. Initial program 99.7%

      \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   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. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if 0.99999999999591638 < (/.f64 (-.f64 (*.f64 a a) (*.f64 b b)) (*.f64 a a))

   1. Initial program 80.9%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 99.0%

      \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot a - b \cdot b}{a \cdot a} \leq 0.9999999999959164:\\ \;\;\;\;\sqrt{\left|\frac{b}{a \cdot a} \cdot b - 1\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.6%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 98.6%

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

   if 0.0 < (*.f64 b b)

   1. Initial program 100.0%

      \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   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. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \sqrt{\color{blue}{\left|\mathsf{fma}\left(\frac{b}{a \cdot a}, b, -1\right)\right|}} \]
3. Recombined 2 regimes into one program.
4. 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 81.6%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 97.0%

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

Reproduce

herbie shell --seed 2024235 
(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)))))