Quadratic roots, medium range

Percentage Accurate: 31.7% → 99.4%

Time: 16.7s

Alternatives: 9

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: 31.7% 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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (c * (-2.0d0))) / (a * (b + sqrt(((b * b) + (a * (c * (-4.0d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 29.3%

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

   1. div-inv N/A

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

   2. flip-- N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

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

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

6. Applied egg-rr 30.4%

   \[\leadsto \color{blue}{\left(b \cdot b + \left(a \cdot \left(c \cdot -4\right) - b \cdot b\right)\right) \cdot \frac{\frac{0.5}{a}}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}} \]

7. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 99.3%

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

9. Simplified 99.3%

   \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot -4\right)} \cdot \frac{\frac{0.5}{a}}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \]
10. Step-by-step derivation

    1. associate-*l* N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

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

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

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

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

    6. associate-/l/ N/A

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

    7. associate-*r/ N/A

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

    8. metadata-eval N/A

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

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

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

    10. *-commutative N/A

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

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

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

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

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

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

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

11. Applied egg-rr 99.3%

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. associate-*r/ N/A

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

    4. *-commutative N/A

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

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

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

    6. *-commutative N/A

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

    7. *-commutative N/A

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

    8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

    13. rem-square-sqrt N/A

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

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

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

    15. rem-square-sqrt N/A

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

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

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

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

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

13. Applied egg-rr 99.4%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (c * ((-2.0d0) / (a * (b + sqrt(((b * b) + (c * (a * (-4.0d0)))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 29.3%

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

   1. div-inv N/A

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

   2. flip-- N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

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

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

6. Applied egg-rr 30.4%

   \[\leadsto \color{blue}{\left(b \cdot b + \left(a \cdot \left(c \cdot -4\right) - b \cdot b\right)\right) \cdot \frac{\frac{0.5}{a}}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}} \]

7. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 99.3%

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

9. Simplified 99.3%

   \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot -4\right)} \cdot \frac{\frac{0.5}{a}}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \]
10. Step-by-step derivation

    1. associate-*l* N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

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

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

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

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

    6. associate-/l/ N/A

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

    7. associate-*r/ N/A

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

    8. metadata-eval N/A

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

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

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

    10. *-commutative N/A

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

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

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

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

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

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

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

11. Applied egg-rr 99.3%

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

Alternative 3: 95.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \frac{\left(\frac{\left(c \cdot c\right) \cdot \left(-2 \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot t\_0} + \left(\frac{-0.25 \cdot \left(\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{a \cdot \left(t\_0 \cdot t\_0\right)} - \frac{\frac{a \cdot \left(c \cdot c\right)}{b}}{b}\right)\right) - c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (/
    (-
     (+
      (/ (* (* c c) (* -2.0 (* c (* a a)))) (* b t_0))
      (-
       (/
        (* -0.25 (* (* a (* a (* a a))) (* (* c c) (* (* c c) 20.0))))
        (* a (* t_0 t_0)))
       (/ (/ (* a (* c c)) b) b)))
     c)
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (((((c * c) * (-2.0 * (c * (a * a)))) / (b * t_0)) + (((-0.25 * ((a * (a * (a * a))) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0))) - (((a * (c * c)) / 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
    real(8) :: t_0
    t_0 = b * (b * b)
    code = (((((c * c) * ((-2.0d0) * (c * (a * a)))) / (b * t_0)) + ((((-0.25d0) * ((a * (a * (a * a))) * ((c * c) * ((c * c) * 20.0d0)))) / (a * (t_0 * t_0))) - (((a * (c * c)) / b) / b))) - c) / b
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (((((c * c) * (-2.0 * (c * (a * a)))) / (b * t_0)) + (((-0.25 * ((a * (a * (a * a))) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0))) - (((a * (c * c)) / b) / b))) - c) / b;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	return (((((c * c) * (-2.0 * (c * (a * a)))) / (b * t_0)) + (((-0.25 * ((a * (a * (a * a))) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0))) - (((a * (c * c)) / b) / b))) - c) / b

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(Float64(Float64(Float64(Float64(Float64(c * c) * Float64(-2.0 * Float64(c * Float64(a * a)))) / Float64(b * t_0)) + Float64(Float64(Float64(-0.25 * Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(Float64(c * c) * Float64(Float64(c * c) * 20.0)))) / Float64(a * Float64(t_0 * t_0))) - Float64(Float64(Float64(a * Float64(c * c)) / b) / b))) - c) / b)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	tmp = (((((c * c) * (-2.0 * (c * (a * a)))) / (b * t_0)) + (((-0.25 * ((a * (a * (a * a))) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0))) - (((a * (c * c)) / b) / b))) - c) / b;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(c * c), $MachinePrecision] * N[(-2.0 * N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 29.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 29.3%

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

5. Taylor expanded in b around inf

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

7. Simplified 98.0%

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

9. Applied egg-rr 98.0%

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

10. Final simplification 98.0%

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

Alternative 4: 93.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{a \cdot \left(c \cdot -4\right)}{\frac{a}{0.5}}}{\left(c \cdot -2\right) \cdot \left(\frac{a}{b} + \frac{a \cdot \left(a \cdot c\right)}{b \cdot \left(b \cdot b\right)}\right) + b \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (* a (* c -4.0)) (/ a 0.5))
  (+ (* (* c -2.0) (+ (/ a b) (/ (* a (* a c)) (* b (* b b))))) (* b 2.0))))

C

double code(double a, double b, double c) {
	return ((a * (c * -4.0)) / (a / 0.5)) / (((c * -2.0) * ((a / b) + ((a * (a * c)) / (b * (b * b))))) + (b * 2.0));
}

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 * (c * (-4.0d0))) / (a / 0.5d0)) / (((c * (-2.0d0)) * ((a / b) + ((a * (a * c)) / (b * (b * b))))) + (b * 2.0d0))
end function

Java

public static double code(double a, double b, double c) {
	return ((a * (c * -4.0)) / (a / 0.5)) / (((c * -2.0) * ((a / b) + ((a * (a * c)) / (b * (b * b))))) + (b * 2.0));
}

Python

def code(a, b, c):
	return ((a * (c * -4.0)) / (a / 0.5)) / (((c * -2.0) * ((a / b) + ((a * (a * c)) / (b * (b * b))))) + (b * 2.0))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(a * Float64(c * -4.0)) / Float64(a / 0.5)) / Float64(Float64(Float64(c * -2.0) * Float64(Float64(a / b) + Float64(Float64(a * Float64(a * c)) / Float64(b * Float64(b * b))))) + Float64(b * 2.0)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((a * (c * -4.0)) / (a / 0.5)) / (((c * -2.0) * ((a / b) + ((a * (a * c)) / (b * (b * b))))) + (b * 2.0));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] / N[(a / 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(c * -2.0), $MachinePrecision] * N[(N[(a / b), $MachinePrecision] + N[(N[(a * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 29.3%

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

   1. div-inv N/A

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

   2. flip-- N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

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

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

6. Applied egg-rr 30.4%

   \[\leadsto \color{blue}{\left(b \cdot b + \left(a \cdot \left(c \cdot -4\right) - b \cdot b\right)\right) \cdot \frac{\frac{0.5}{a}}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}} \]

7. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 99.3%

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

9. Simplified 99.3%

   \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot -4\right)} \cdot \frac{\frac{0.5}{a}}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \]

10. Taylor expanded in c around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \color{blue}{\left(2 \cdot b + c \cdot \left(-2 \cdot \frac{a}{b} + -2 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right)}\right)\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(c \cdot \left(-2 \cdot \frac{a}{b} + -2 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right) + \color{blue}{2 \cdot b}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(\left(c \cdot \left(-2 \cdot \frac{a}{b} + -2 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right), \color{blue}{\left(2 \cdot b\right)}\right)\right)\right) \]

12. Simplified 96.4%

    \[\leadsto \left(\left(c \cdot a\right) \cdot -4\right) \cdot \frac{\frac{0.5}{a}}{\color{blue}{c \cdot \left(-2 \cdot \left(\frac{a}{b} + \frac{c \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)\right) + b \cdot 2}} \]
13. Step-by-step derivation

    1. associate-*r* N/A

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

    2. associate-*r/ N/A

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

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

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

    4. clear-num N/A

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

    5. un-div-inv N/A

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

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

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

    7. *-commutative N/A

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

    8. associate-*l* N/A

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

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

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

    10. *-commutative N/A

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

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

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

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

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

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

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

14. Applied egg-rr 96.8%

    \[\leadsto \color{blue}{\frac{\frac{a \cdot \left(c \cdot -4\right)}{\frac{a}{0.5}}}{\left(\frac{a}{b} + \frac{a \cdot \left(a \cdot c\right)}{b \cdot \left(b \cdot b\right)}\right) \cdot \left(c \cdot -2\right) + b \cdot 2}} \]

15. Final simplification 96.8%

    \[\leadsto \frac{\frac{a \cdot \left(c \cdot -4\right)}{\frac{a}{0.5}}}{\left(c \cdot -2\right) \cdot \left(\frac{a}{b} + \frac{a \cdot \left(a \cdot c\right)}{b \cdot \left(b \cdot b\right)}\right) + b \cdot 2} \]
16. Add Preprocessing

Alternative 5: 93.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(a \cdot \left(c \cdot -4\right)\right) \cdot 0.5}{a \cdot \left(\left(c \cdot -2\right) \cdot \left(\frac{a}{b} + \frac{a \cdot \left(a \cdot c\right)}{b \cdot \left(b \cdot b\right)}\right) + b \cdot 2\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (* a (* c -4.0)) 0.5)
  (*
   a
   (+ (* (* c -2.0) (+ (/ a b) (/ (* a (* a c)) (* b (* b b))))) (* b 2.0)))))

C

double code(double a, double b, double c) {
	return ((a * (c * -4.0)) * 0.5) / (a * (((c * -2.0) * ((a / b) + ((a * (a * c)) / (b * (b * b))))) + (b * 2.0)));
}

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 * (c * (-4.0d0))) * 0.5d0) / (a * (((c * (-2.0d0)) * ((a / b) + ((a * (a * c)) / (b * (b * b))))) + (b * 2.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return ((a * (c * -4.0)) * 0.5) / (a * (((c * -2.0) * ((a / b) + ((a * (a * c)) / (b * (b * b))))) + (b * 2.0)));
}

Python

def code(a, b, c):
	return ((a * (c * -4.0)) * 0.5) / (a * (((c * -2.0) * ((a / b) + ((a * (a * c)) / (b * (b * b))))) + (b * 2.0)))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(a * Float64(c * -4.0)) * 0.5) / Float64(a * Float64(Float64(Float64(c * -2.0) * Float64(Float64(a / b) + Float64(Float64(a * Float64(a * c)) / Float64(b * Float64(b * b))))) + Float64(b * 2.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((a * (c * -4.0)) * 0.5) / (a * (((c * -2.0) * ((a / b) + ((a * (a * c)) / (b * (b * b))))) + (b * 2.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / N[(a * N[(N[(N[(c * -2.0), $MachinePrecision] * N[(N[(a / b), $MachinePrecision] + N[(N[(a * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 29.3%

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

   1. div-inv N/A

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

   2. flip-- N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

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

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

6. Applied egg-rr 30.4%

   \[\leadsto \color{blue}{\left(b \cdot b + \left(a \cdot \left(c \cdot -4\right) - b \cdot b\right)\right) \cdot \frac{\frac{0.5}{a}}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}} \]

7. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 99.3%

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

9. Simplified 99.3%

   \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot -4\right)} \cdot \frac{\frac{0.5}{a}}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \]

10. Taylor expanded in c around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \color{blue}{\left(2 \cdot b + c \cdot \left(-2 \cdot \frac{a}{b} + -2 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right)}\right)\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(c \cdot \left(-2 \cdot \frac{a}{b} + -2 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right) + \color{blue}{2 \cdot b}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(\left(c \cdot \left(-2 \cdot \frac{a}{b} + -2 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right), \color{blue}{\left(2 \cdot b\right)}\right)\right)\right) \]

12. Simplified 96.4%

    \[\leadsto \left(\left(c \cdot a\right) \cdot -4\right) \cdot \frac{\frac{0.5}{a}}{\color{blue}{c \cdot \left(-2 \cdot \left(\frac{a}{b} + \frac{c \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)\right) + b \cdot 2}} \]
13. Step-by-step derivation

    1. associate-*r* N/A

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

    2. associate-/l/ N/A

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

    3. associate-*r/ N/A

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

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

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

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

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

    6. *-commutative N/A

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

    7. associate-*l* N/A

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

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

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

    9. *-commutative N/A

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

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

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

    11. *-commutative N/A

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

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

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

14. Applied egg-rr 96.5%

    \[\leadsto \color{blue}{\frac{\left(a \cdot \left(c \cdot -4\right)\right) \cdot 0.5}{a \cdot \left(\left(\frac{a}{b} + \frac{a \cdot \left(a \cdot c\right)}{b \cdot \left(b \cdot b\right)}\right) \cdot \left(c \cdot -2\right) + b \cdot 2\right)}} \]

15. Final simplification 96.5%

    \[\leadsto \frac{\left(a \cdot \left(c \cdot -4\right)\right) \cdot 0.5}{a \cdot \left(\left(c \cdot -2\right) \cdot \left(\frac{a}{b} + \frac{a \cdot \left(a \cdot c\right)}{b \cdot \left(b \cdot b\right)}\right) + b \cdot 2\right)} \]
16. Add Preprocessing

Alternative 6: 93.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

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 * (c * ((-2.0d0) / (a * (((c * (-2.0d0)) * ((a / b) + ((a * (a * c)) / (b * (b * b))))) + (b * 2.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 29.3%

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

   1. div-inv N/A

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

   2. flip-- N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

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

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

6. Applied egg-rr 30.4%

   \[\leadsto \color{blue}{\left(b \cdot b + \left(a \cdot \left(c \cdot -4\right) - b \cdot b\right)\right) \cdot \frac{\frac{0.5}{a}}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}} \]

7. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 99.3%

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

9. Simplified 99.3%

   \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot -4\right)} \cdot \frac{\frac{0.5}{a}}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \]

10. Taylor expanded in c around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \color{blue}{\left(2 \cdot b + c \cdot \left(-2 \cdot \frac{a}{b} + -2 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right)}\right)\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(c \cdot \left(-2 \cdot \frac{a}{b} + -2 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right) + \color{blue}{2 \cdot b}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(\left(c \cdot \left(-2 \cdot \frac{a}{b} + -2 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right), \color{blue}{\left(2 \cdot b\right)}\right)\right)\right) \]

12. Simplified 96.4%

    \[\leadsto \left(\left(c \cdot a\right) \cdot -4\right) \cdot \frac{\frac{0.5}{a}}{\color{blue}{c \cdot \left(-2 \cdot \left(\frac{a}{b} + \frac{c \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)\right) + b \cdot 2}} \]
13. Step-by-step derivation

    1. associate-*l* N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

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

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

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

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

    6. associate-/l/ N/A

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

    7. associate-*r/ N/A

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

    8. metadata-eval N/A

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

14. Applied egg-rr 96.4%

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

15. Final simplification 96.4%

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

Alternative 7: 90.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{\left(0 - c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return ((0.0 - c) - ((a * (c * c)) / (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 = ((0.0d0 - c) - ((a * (c * c)) / (b * b))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 29.3%

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

5. Taylor expanded in b around inf

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

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

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - c\right), \left(\frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \left(\frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\left(a \cdot {c}^{2}\right), \left({b}^{2}\right)\right)\right), b\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({c}^{2}\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot c\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

   12. unpow2 N/A

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

   13. *-lowering-*.f64 93.5%

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

7. Simplified 93.5%

   \[\leadsto \color{blue}{\frac{\left(0 - c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}} \]
8. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 93.5%

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

9. Applied egg-rr 93.5%

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

10. Final simplification 93.5%

    \[\leadsto \frac{\left(0 - c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b} \]
11. Add Preprocessing

Alternative 8: 81.1% accurate, 23.2× speedup

Math

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

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(Float64(0.0 - c) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 29.3%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 83.1%

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

7. Simplified 83.1%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
8. Step-by-step derivation

   1. sub0-neg N/A

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

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

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

   3. /-lowering-/.f64 83.1%

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

9. Applied egg-rr 83.1%

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

10. Final simplification 83.1%

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

Alternative 9: 3.2% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 29.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 29.3%

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

   1. div-sub N/A

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

   2. div-inv N/A

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

   3. div-inv N/A

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

   4. prod-diff N/A

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

   5. associate-/r/ N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}, \frac{1}{a \cdot 2}, \mathsf{neg}\left(\frac{1}{\frac{a \cdot 2}{b}}\right)\right) + \mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{a \cdot 2}\right), b, \frac{1}{a \cdot 2} \cdot b\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}, \frac{1}{a \cdot 2}, \mathsf{neg}\left(\frac{b}{a \cdot 2}\right)\right) + \mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{a \cdot 2}\right), b, \frac{1}{a \cdot 2} \cdot b\right) \]

   7. fmm-def N/A

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

   8. div-inv N/A

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

   9. div-sub N/A

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

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

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

6. Applied egg-rr 30.5%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2} + \mathsf{fma}\left(\frac{-0.5}{a}, b, \frac{b}{a \cdot 2}\right)} \]

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{b}{a} \cdot 0 \]

   3. mul0-rgt 3.2%

      \[\leadsto 0 \]

9. Simplified 3.2%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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)))