Quadratic roots, medium range

Percentage Accurate: 31.6% → 99.8%

Time: 17.7s

Alternatives: 10

Speedup: 23.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 31.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -2}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c -2.0) (+ b (sqrt (+ (* b b) (* -4.0 (* c a)))))))

C

double code(double a, double b, double c) {
	return (c * -2.0) / (b + sqrt(((b * b) + (-4.0 * (c * a)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c * (-2.0d0)) / (b + sqrt(((b * b) + ((-4.0d0) * (c * a)))))
end function

Java

public static double code(double a, double b, double c) {
	return (c * -2.0) / (b + Math.sqrt(((b * b) + (-4.0 * (c * a)))));
}

Python

def code(a, b, c):
	return (c * -2.0) / (b + math.sqrt(((b * b) + (-4.0 * (c * a)))))

Julia

function code(a, b, c)
	return Float64(Float64(c * -2.0) / Float64(b + sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * -2.0) / (b + sqrt(((b * b) + (-4.0 * (c * a)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 32.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot 2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{a}}{2} \cdot \left(\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right) \]

   4. flip-- N/A

      \[\leadsto \frac{\frac{1}{a}}{2} \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b \cdot b}{\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} + b}} \]

   5. frac-times N/A

      \[\leadsto \frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b \cdot b\right)}{\color{blue}{2 \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} + b\right)}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b \cdot b\right)\right), \color{blue}{\left(2 \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} + b\right)\right)}\right) \]

6. Applied egg-rr 33.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{a} \cdot \left(b \cdot b + \left(a \cdot \left(c \cdot -4\right) - b \cdot b\right)\right)}{2 \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-4 \cdot c\right)}, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{-4 \cdot c}}{2 \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)} \]
10. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \frac{\frac{-4 \cdot c}{2}}{\color{blue}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-4 \cdot c}{2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{c \cdot -4}{2}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

    4. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-4}{2}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-4}{2}\right)\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

    8. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \left(\sqrt{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right)\right)\right) \]

    9. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right) \]

    10. rem-square-sqrt N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

    11. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

    13. associate-*r* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

    16. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

    17. *-lowering-*.f64 99.7%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

11. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\frac{c \cdot -2}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-2}{b + \sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (/ -2.0 (+ b (sqrt (+ (* b b) (* c (* -4.0 a))))))))

C

double code(double a, double b, double c) {
	return c * (-2.0 / (b + sqrt(((b * b) + (c * (-4.0 * a))))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * ((-2.0d0) / (b + sqrt(((b * b) + (c * ((-4.0d0) * a))))))
end function

Java

public static double code(double a, double b, double c) {
	return c * (-2.0 / (b + Math.sqrt(((b * b) + (c * (-4.0 * a))))));
}

Python

def code(a, b, c):
	return c * (-2.0 / (b + math.sqrt(((b * b) + (c * (-4.0 * a))))))

Julia

function code(a, b, c)
	return Float64(c * Float64(-2.0 / Float64(b + sqrt(Float64(Float64(b * b) + Float64(c * Float64(-4.0 * a)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (-2.0 / (b + sqrt(((b * b) + (c * (-4.0 * a))))));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(-2.0 / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 32.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot 2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{a}}{2} \cdot \left(\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right) \]

   4. flip-- N/A

      \[\leadsto \frac{\frac{1}{a}}{2} \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b \cdot b}{\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} + b}} \]

   5. frac-times N/A

      \[\leadsto \frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b \cdot b\right)}{\color{blue}{2 \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} + b\right)}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b \cdot b\right)\right), \color{blue}{\left(2 \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} + b\right)\right)}\right) \]

6. Applied egg-rr 33.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{a} \cdot \left(b \cdot b + \left(a \cdot \left(c \cdot -4\right) - b \cdot b\right)\right)}{2 \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-4 \cdot c\right)}, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{-4 \cdot c}}{2 \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)} \]
10. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \frac{\frac{-4 \cdot c}{2}}{\color{blue}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-4 \cdot c}{2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{c \cdot -4}{2}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

    4. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-4}{2}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-4}{2}\right)\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

    8. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \left(\sqrt{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right)\right)\right) \]

    9. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right) \]

    10. rem-square-sqrt N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

    11. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

    13. associate-*r* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

    16. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

    17. *-lowering-*.f64 99.7%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

11. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\frac{c \cdot -2}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]
12. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto c \cdot \color{blue}{\frac{-2}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

    2. *-commutative N/A

       \[\leadsto \frac{-2}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}} \cdot \color{blue}{c} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-2}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}\right), \color{blue}{c}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right)\right), c\right) \]

    5. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right)\right)\right), c\right) \]

    6. rem-square-sqrt N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \left(\sqrt{\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}\right)\right)\right), c\right) \]

    7. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right)\right)\right)\right), c\right) \]

    8. rem-square-sqrt N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)\right)\right)\right), c\right) \]

    9. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(-4 \cdot \left(c \cdot a\right)\right)\right)\right)\right)\right), c\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(c \cdot a\right)\right)\right)\right)\right)\right), c\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right)\right), c\right) \]

    12. associate-*l* N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right)\right), c\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right)\right), c\right) \]

    14. *-lowering-*.f64 99.5%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right) \]

13. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{\frac{-2}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}} \cdot c} \]

14. Final simplification 99.5%

    \[\leadsto c \cdot \frac{-2}{b + \sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)}} \]
15. Add Preprocessing

Alternative 3: 95.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \left(\frac{c \cdot \left(c \cdot \left(c \cdot -2\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 20\right)}{\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} \cdot \frac{-0.25}{b}\right) - \frac{\frac{c \cdot c}{b}}{b \cdot b}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (+
      (/ (* c (* c (* c -2.0))) (* (* b b) (* b (* b b))))
      (*
       (/ (* (* c (* c (* c c))) (* a 20.0)) (* (* b b) (* (* b b) (* b b))))
       (/ -0.25 b))))
    (/ (/ (* c c) b) (* b b))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((a * (((c * (c * (c * -2.0))) / ((b * b) * (b * (b * b)))) + ((((c * (c * (c * c))) * (a * 20.0)) / ((b * b) * ((b * b) * (b * b)))) * (-0.25 / b)))) - (((c * c) / b) / (b * b)))) - (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (a * ((a * (((c * (c * (c * (-2.0d0)))) / ((b * b) * (b * (b * b)))) + ((((c * (c * (c * c))) * (a * 20.0d0)) / ((b * b) * ((b * b) * (b * b)))) * ((-0.25d0) / b)))) - (((c * c) / b) / (b * b)))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((a * (((c * (c * (c * -2.0))) / ((b * b) * (b * (b * b)))) + ((((c * (c * (c * c))) * (a * 20.0)) / ((b * b) * ((b * b) * (b * b)))) * (-0.25 / b)))) - (((c * c) / b) / (b * b)))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((a * (((c * (c * (c * -2.0))) / ((b * b) * (b * (b * b)))) + ((((c * (c * (c * c))) * (a * 20.0)) / ((b * b) * ((b * b) * (b * b)))) * (-0.25 / b)))) - (((c * c) / b) / (b * b)))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(a * Float64(Float64(Float64(c * Float64(c * Float64(c * -2.0))) / Float64(Float64(b * b) * Float64(b * Float64(b * b)))) + Float64(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * 20.0)) / Float64(Float64(b * b) * Float64(Float64(b * b) * Float64(b * b)))) * Float64(-0.25 / b)))) - Float64(Float64(Float64(c * c) / b) / Float64(b * b)))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((a * (((c * (c * (c * -2.0))) / ((b * b) * (b * (b * b)))) + ((((c * (c * (c * c))) * (a * 20.0)) / ((b * b) * ((b * b) * (b * b)))) * (-0.25 / b)))) - (((c * c) / b) / (b * b)))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(N[(N[(c * N[(c * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 20.0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 32.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

7. Simplified 95.1%

   \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -2}{{b}^{5}} + \frac{-0.25 \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(20 \cdot a\right)\right)}{b}\right) - \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}\right) - \frac{c}{b}} \]

9. Applied egg-rr 95.1%

   \[\leadsto \color{blue}{\left(a \cdot \left(\frac{c \cdot \left(c \cdot \left(c \cdot -2\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 20\right)}{\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} \cdot \frac{-0.25}{b}\right) - \frac{\frac{c \cdot c}{b}}{b \cdot b}\right) \cdot a} - \frac{c}{b} \]

10. Final simplification 95.1%

    \[\leadsto a \cdot \left(a \cdot \left(\frac{c \cdot \left(c \cdot \left(c \cdot -2\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 20\right)}{\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} \cdot \frac{-0.25}{b}\right) - \frac{\frac{c \cdot c}{b}}{b \cdot b}\right) - \frac{c}{b} \]
11. Add Preprocessing

Alternative 4: 94.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -4}{b \cdot 4 + c \cdot \left(-4 \cdot \left(\frac{a}{b} + \frac{c \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* c -4.0)
  (+ (* b 4.0) (* c (* -4.0 (+ (/ a b) (/ (* c (* a a)) (* b (* b b)))))))))

C

double code(double a, double b, double c) {
	return (c * -4.0) / ((b * 4.0) + (c * (-4.0 * ((a / b) + ((c * (a * a)) / (b * (b * b)))))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c * (-4.0d0)) / ((b * 4.0d0) + (c * ((-4.0d0) * ((a / b) + ((c * (a * a)) / (b * (b * b)))))))
end function

Java

public static double code(double a, double b, double c) {
	return (c * -4.0) / ((b * 4.0) + (c * (-4.0 * ((a / b) + ((c * (a * a)) / (b * (b * b)))))));
}

Python

def code(a, b, c):
	return (c * -4.0) / ((b * 4.0) + (c * (-4.0 * ((a / b) + ((c * (a * a)) / (b * (b * b)))))))

Julia

function code(a, b, c)
	return Float64(Float64(c * -4.0) / Float64(Float64(b * 4.0) + Float64(c * Float64(-4.0 * Float64(Float64(a / b) + Float64(Float64(c * Float64(a * a)) / Float64(b * Float64(b * b))))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * -4.0) / ((b * 4.0) + (c * (-4.0 * ((a / b) + ((c * (a * a)) / (b * (b * b)))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(c * -4.0), $MachinePrecision] / N[(N[(b * 4.0), $MachinePrecision] + N[(c * N[(-4.0 * N[(N[(a / b), $MachinePrecision] + N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 32.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot 2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{a}}{2} \cdot \left(\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right) \]

   4. flip-- N/A

      \[\leadsto \frac{\frac{1}{a}}{2} \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b \cdot b}{\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} + b}} \]

   5. frac-times N/A

      \[\leadsto \frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b \cdot b\right)}{\color{blue}{2 \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} + b\right)}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b \cdot b\right)\right), \color{blue}{\left(2 \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} + b\right)\right)}\right) \]

6. Applied egg-rr 33.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{a} \cdot \left(b \cdot b + \left(a \cdot \left(c \cdot -4\right) - b \cdot b\right)\right)}{2 \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-4 \cdot c\right)}, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{-4 \cdot c}}{2 \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)} \]

10. Taylor expanded in c around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \color{blue}{\left(4 \cdot b + c \cdot \left(-4 \cdot \frac{a}{b} + -4 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right)}\right) \]
11. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\left(4 \cdot b\right), \color{blue}{\left(c \cdot \left(-4 \cdot \frac{a}{b} + -4 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right)}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\left(b \cdot 4\right), \left(\color{blue}{c} \cdot \left(-4 \cdot \frac{a}{b} + -4 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right)\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \left(\color{blue}{c} \cdot \left(-4 \cdot \frac{a}{b} + -4 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \color{blue}{\left(-4 \cdot \frac{a}{b} + -4 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}\right)\right)\right) \]

    5. distribute-lft-out N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \left(-4 \cdot \color{blue}{\left(\frac{a}{b} + \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}\right)\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \color{blue}{\left(\frac{a}{b} + \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}\right)\right)\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \mathsf{+.f64}\left(\left(\frac{a}{b}\right), \color{blue}{\left(\frac{{a}^{2} \cdot c}{{b}^{3}}\right)}\right)\right)\right)\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, b\right), \left(\frac{\color{blue}{{a}^{2} \cdot c}}{{b}^{3}}\right)\right)\right)\right)\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot c\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right)\right)\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(\left(c \cdot {a}^{2}\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right)\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left({a}^{2}\right)\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right)\right)\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot a\right)\right), \left({b}^{3}\right)\right)\right)\right)\right)\right)\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, a\right)\right), \left({b}^{3}\right)\right)\right)\right)\right)\right)\right) \]

    14. cube-mult N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, a\right)\right), \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right)\right)\right)\right) \]

    15. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, a\right)\right), \left(b \cdot {b}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right) \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

    17. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right)\right) \]

    18. *-lowering-*.f64 93.7%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-4, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right)\right) \]

12. Simplified 93.7%

    \[\leadsto \frac{-4 \cdot c}{\color{blue}{b \cdot 4 + c \cdot \left(-4 \cdot \left(\frac{a}{b} + \frac{c \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)\right)}} \]

13. Final simplification 93.7%

    \[\leadsto \frac{c \cdot -4}{b \cdot 4 + c \cdot \left(-4 \cdot \left(\frac{a}{b} + \frac{c \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)\right)} \]
14. Add Preprocessing

Alternative 5: 94.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 32.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot 2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{1}{a \cdot \color{blue}{\frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{a}\right), \color{blue}{\left(\frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \left(\frac{\color{blue}{2}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

   12. *-lowering-*.f64 32.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right) \]

6. Applied egg-rr 32.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

7. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-1 \cdot \frac{b}{a} + c \cdot \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{3}} + \frac{a}{{b}^{3}}\right)\right) + \frac{1}{b}\right)}{c}\right)}\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\left(-1 \cdot \frac{b}{a} + c \cdot \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{3}} + \frac{a}{{b}^{3}}\right)\right) + \frac{1}{b}\right)\right), \color{blue}{c}\right)\right) \]

9. Simplified 93.2%

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

    1. associate-/l/ N/A

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

    2. associate-/r* N/A

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

    3. clear-num N/A

       \[\leadsto \frac{\frac{c}{\frac{\mathsf{neg}\left(b\right)}{a} + c \cdot \left(-1 \cdot \left(c \cdot \left(-1 \cdot \frac{a}{b \cdot \left(b \cdot b\right)}\right)\right) + \frac{1}{b}\right)}}{a} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{c}{\frac{\mathsf{neg}\left(b\right)}{a} + c \cdot \left(-1 \cdot \left(c \cdot \left(-1 \cdot \frac{a}{b \cdot \left(b \cdot b\right)}\right)\right) + \frac{1}{b}\right)}\right), \color{blue}{a}\right) \]

11. Applied egg-rr 93.4%

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

    1. associate-/l/ N/A

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

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{\left(a \cdot \left(c \cdot \left(c \cdot \frac{a}{b \cdot \left(b \cdot b\right)} + \frac{1}{b}\right) - \frac{b}{a}\right)\right)}\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(a, \color{blue}{\left(c \cdot \left(c \cdot \frac{a}{b \cdot \left(b \cdot b\right)} + \frac{1}{b}\right) - \frac{b}{a}\right)}\right)\right) \]

    4. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\left(c \cdot \left(c \cdot \frac{a}{b \cdot \left(b \cdot b\right)} + \frac{1}{b}\right)\right), \color{blue}{\left(\frac{b}{a}\right)}\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \left(c \cdot \frac{a}{b \cdot \left(b \cdot b\right)} + \frac{1}{b}\right)\right), \left(\frac{\color{blue}{b}}{a}\right)\right)\right)\right) \]

    6. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(c \cdot \frac{a}{b \cdot \left(b \cdot b\right)}\right), \left(\frac{1}{b}\right)\right)\right), \left(\frac{b}{a}\right)\right)\right)\right) \]

    7. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(c \cdot \frac{1}{\frac{b \cdot \left(b \cdot b\right)}{a}}\right), \left(\frac{1}{b}\right)\right)\right), \left(\frac{b}{a}\right)\right)\right)\right) \]

    8. un-div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{c}{\frac{b \cdot \left(b \cdot b\right)}{a}}\right), \left(\frac{1}{b}\right)\right)\right), \left(\frac{b}{a}\right)\right)\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(c, \left(\frac{b \cdot \left(b \cdot b\right)}{a}\right)\right), \left(\frac{1}{b}\right)\right)\right), \left(\frac{b}{a}\right)\right)\right)\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(c, \mathsf{/.f64}\left(\left(b \cdot \left(b \cdot b\right)\right), a\right)\right), \left(\frac{1}{b}\right)\right)\right), \left(\frac{b}{a}\right)\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(c, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot b\right)\right), a\right)\right), \left(\frac{1}{b}\right)\right)\right), \left(\frac{b}{a}\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(c, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right), \left(\frac{1}{b}\right)\right)\right), \left(\frac{b}{a}\right)\right)\right)\right) \]

    13. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(c, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right), \mathsf{/.f64}\left(1, b\right)\right)\right), \left(\frac{b}{a}\right)\right)\right)\right) \]

    14. /-lowering-/.f64 93.5%

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(c, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), a\right)\right), \mathsf{/.f64}\left(1, b\right)\right)\right), \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right)\right)\right) \]

13. Applied egg-rr 93.5%

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

Alternative 6: 93.8% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b, c):
	return (c / (((c + ((a * (c * c)) / (b * b))) / b) - (b / a))) / a

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 32.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot 2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{1}{a \cdot \color{blue}{\frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{a}\right), \color{blue}{\left(\frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \left(\frac{\color{blue}{2}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

   12. *-lowering-*.f64 32.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right) \]

6. Applied egg-rr 32.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

7. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-1 \cdot \frac{b}{a} + c \cdot \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{3}} + \frac{a}{{b}^{3}}\right)\right) + \frac{1}{b}\right)}{c}\right)}\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\left(-1 \cdot \frac{b}{a} + c \cdot \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{3}} + \frac{a}{{b}^{3}}\right)\right) + \frac{1}{b}\right)\right), \color{blue}{c}\right)\right) \]

9. Simplified 93.2%

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

    1. associate-/l/ N/A

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

    2. associate-/r* N/A

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

    3. clear-num N/A

       \[\leadsto \frac{\frac{c}{\frac{\mathsf{neg}\left(b\right)}{a} + c \cdot \left(-1 \cdot \left(c \cdot \left(-1 \cdot \frac{a}{b \cdot \left(b \cdot b\right)}\right)\right) + \frac{1}{b}\right)}}{a} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{c}{\frac{\mathsf{neg}\left(b\right)}{a} + c \cdot \left(-1 \cdot \left(c \cdot \left(-1 \cdot \frac{a}{b \cdot \left(b \cdot b\right)}\right)\right) + \frac{1}{b}\right)}\right), \color{blue}{a}\right) \]

11. Applied egg-rr 93.4%

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

12. Taylor expanded in b around inf

    \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\right)}, \mathsf{/.f64}\left(b, a\right)\right)\right), a\right) \]
13. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right), b\right), \mathsf{/.f64}\left(b, a\right)\right)\right), a\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(c, \left(\frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right), \mathsf{/.f64}\left(b, a\right)\right)\right), a\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(c, \mathsf{/.f64}\left(\left(a \cdot {c}^{2}\right), \left({b}^{2}\right)\right)\right), b\right), \mathsf{/.f64}\left(b, a\right)\right)\right), a\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(c, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({c}^{2}\right)\right), \left({b}^{2}\right)\right)\right), b\right), \mathsf{/.f64}\left(b, a\right)\right)\right), a\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(c, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot c\right)\right), \left({b}^{2}\right)\right)\right), b\right), \mathsf{/.f64}\left(b, a\right)\right)\right), a\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(c, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right), \left({b}^{2}\right)\right)\right), b\right), \mathsf{/.f64}\left(b, a\right)\right)\right), a\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(c, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right), \left(b \cdot b\right)\right)\right), b\right), \mathsf{/.f64}\left(b, a\right)\right)\right), a\right) \]

    8. *-lowering-*.f64 93.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(c, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), b\right), \mathsf{/.f64}\left(b, a\right)\right)\right), a\right) \]

14. Simplified 93.4%

    \[\leadsto \frac{\frac{c}{\color{blue}{\frac{c + \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}} - \frac{b}{a}}}{a} \]
15. Add Preprocessing

Alternative 7: 91.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -4}{b \cdot 4 + -4 \cdot \frac{c \cdot a}{b}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c -4.0) (+ (* b 4.0) (* -4.0 (/ (* c a) b)))))

C

double code(double a, double b, double c) {
	return (c * -4.0) / ((b * 4.0) + (-4.0 * ((c * a) / b)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c * (-4.0d0)) / ((b * 4.0d0) + ((-4.0d0) * ((c * a) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return (c * -4.0) / ((b * 4.0) + (-4.0 * ((c * a) / b)));
}

Python

def code(a, b, c):
	return (c * -4.0) / ((b * 4.0) + (-4.0 * ((c * a) / b)))

Julia

function code(a, b, c)
	return Float64(Float64(c * -4.0) / Float64(Float64(b * 4.0) + Float64(-4.0 * Float64(Float64(c * a) / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * -4.0) / ((b * 4.0) + (-4.0 * ((c * a) / b)));
end

Wolfram

code[a_, b_, c_] := N[(N[(c * -4.0), $MachinePrecision] / N[(N[(b * 4.0), $MachinePrecision] + N[(-4.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 32.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot 2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{a}}{2} \cdot \left(\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right) \]

   4. flip-- N/A

      \[\leadsto \frac{\frac{1}{a}}{2} \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b \cdot b}{\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} + b}} \]

   5. frac-times N/A

      \[\leadsto \frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b \cdot b\right)}{\color{blue}{2 \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} + b\right)}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b \cdot b\right)\right), \color{blue}{\left(2 \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} + b\right)\right)}\right) \]

6. Applied egg-rr 33.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{a} \cdot \left(b \cdot b + \left(a \cdot \left(c \cdot -4\right) - b \cdot b\right)\right)}{2 \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-4 \cdot c\right)}, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{-4 \cdot c}}{2 \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)} \]

10. Taylor expanded in a around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \color{blue}{\left(-4 \cdot \frac{a \cdot c}{b} + 4 \cdot b\right)}\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \left(4 \cdot b + \color{blue}{-4 \cdot \frac{a \cdot c}{b}}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\left(4 \cdot b\right), \color{blue}{\left(-4 \cdot \frac{a \cdot c}{b}\right)}\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\left(b \cdot 4\right), \left(\color{blue}{-4} \cdot \frac{a \cdot c}{b}\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \left(\color{blue}{-4} \cdot \frac{a \cdot c}{b}\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(-4, \color{blue}{\left(\frac{a \cdot c}{b}\right)}\right)\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(a \cdot c\right), \color{blue}{b}\right)\right)\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(c \cdot a\right), b\right)\right)\right)\right) \]

    8. *-lowering-*.f64 90.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, c\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\right)\right)\right)\right) \]

12. Simplified 90.2%

    \[\leadsto \frac{-4 \cdot c}{\color{blue}{b \cdot 4 + -4 \cdot \frac{c \cdot a}{b}}} \]

13. Final simplification 90.2%

    \[\leadsto \frac{c \cdot -4}{b \cdot 4 + -4 \cdot \frac{c \cdot a}{b}} \]
14. Add Preprocessing

Alternative 8: 90.8% accurate, 12.9× speedup

Math

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

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (- (/ a b) (/ b c))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 32.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot 2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{1}{a \cdot \color{blue}{\frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{a}\right), \color{blue}{\left(\frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \left(\frac{\color{blue}{2}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

   12. *-lowering-*.f64 32.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right) \]

6. Applied egg-rr 32.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

7. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-1 \cdot \frac{b}{a} + c \cdot \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{3}} + \frac{a}{{b}^{3}}\right)\right) + \frac{1}{b}\right)}{c}\right)}\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\left(-1 \cdot \frac{b}{a} + c \cdot \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{3}} + \frac{a}{{b}^{3}}\right)\right) + \frac{1}{b}\right)\right), \color{blue}{c}\right)\right) \]

9. Simplified 93.2%

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

10. Taylor expanded in a around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-1 \cdot \frac{b}{c} + \frac{a}{b}}{a}\right)}\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\left(-1 \cdot \frac{b}{c} + \frac{a}{b}\right), \color{blue}{a}\right)\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\left(\frac{a}{b} + -1 \cdot \frac{b}{c}\right), a\right)\right) \]

    3. mul-1-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\left(\frac{a}{b} + \left(\mathsf{neg}\left(\frac{b}{c}\right)\right)\right), a\right)\right) \]

    4. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\left(\frac{a}{b} - \frac{b}{c}\right), a\right)\right) \]

    5. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{a}{b}\right), \left(\frac{b}{c}\right)\right), a\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, b\right), \left(\frac{b}{c}\right)\right), a\right)\right) \]

    7. /-lowering-/.f64 89.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(b, c\right)\right), a\right)\right) \]

12. Simplified 89.8%

    \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{a}{b} - \frac{b}{c}}{a}}} \]
13. Step-by-step derivation

    1. associate-/r/ N/A

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

    2. associate-*l/ N/A

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

    3. inv-pow N/A

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

    4. pow-plus N/A

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

    5. metadata-eval N/A

       \[\leadsto \frac{{a}^{0}}{\frac{a}{\color{blue}{b}} - \frac{b}{c}} \]

    6. metadata-eval N/A

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

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{a}{b} - \frac{b}{c}\right)}\right) \]

    8. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\frac{a}{b}\right), \color{blue}{\left(\frac{b}{c}\right)}\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, b\right), \left(\frac{\color{blue}{b}}{c}\right)\right)\right) \]

    10. /-lowering-/.f64 90.0%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(b, \color{blue}{c}\right)\right)\right) \]

14. Applied egg-rr 90.0%

    \[\leadsto \color{blue}{\frac{1}{\frac{a}{b} - \frac{b}{c}}} \]
15. Add Preprocessing

Alternative 9: 81.2% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (- 0.0 (/ c b)))

C

double code(double a, double b, double c) {
	return 0.0 - (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 - (c / b);
}

Python

def code(a, b, c):
	return 0.0 - (c / b)

Julia

function code(a, b, c)
	return Float64(0.0 - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 32.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

   4. /-lowering-/.f64 80.1%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

7. Simplified 80.1%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

8. Taylor expanded in c around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{-1 \cdot c}{\color{blue}{b}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot c\right), \color{blue}{b}\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(c\right)\right), b\right) \]

   4. neg-lowering-neg.f64 80.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(c\right), b\right) \]

10. Simplified 80.1%

    \[\leadsto \color{blue}{\frac{-c}{b}} \]

11. Final simplification 80.1%

    \[\leadsto 0 - \frac{c}{b} \]
12. Add Preprocessing

Alternative 10: 1.6% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c b))

C

double code(double a, double b, double c) {
	return c / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / b
end function

Java

public static double code(double a, double b, double c) {
	return c / b;
}

Python

def code(a, b, c):
	return c / b

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / b;
end

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 32.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 32.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right), \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a} + -1 \cdot \frac{c}{{b}^{2}}\right), \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a} + \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a} - \frac{c}{{b}^{2}}\right), \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{a}\right), \left(\frac{c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \left(\frac{c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

   13. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \left(0 - \color{blue}{b}\right)\right) \]

   14. --lowering--.f64 10.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{b}\right)\right) \]

7. Simplified 10.2%

   \[\leadsto \color{blue}{\left(\frac{1}{a} - \frac{c}{b \cdot b}\right) \cdot \left(0 - b\right)} \]

8. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{c}{b}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 1.6%

      \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{b}\right) \]

10. Simplified 1.6%

    \[\leadsto \color{blue}{\frac{c}{b}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))