Quadratic roots, medium range

Percentage Accurate: 31.3% → 99.4%

Time: 19.1s

Alternatives: 8

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: 31.3% 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

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-2.0 * (a * c)) / (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 = ((-2.0d0) * (a * c)) / (a * (b + sqrt(((b * b) + (a * (c * (-4.0d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(a * c), $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 30.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.4%

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

   1. flip-- N/A

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

   2. div-inv N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

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

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

6. Applied egg-rr 31.3%

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

7. Taylor expanded in b around 0

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

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

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

   2. *-lowering-*.f64 99.1%

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

9. Simplified 99.1%

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

    1. *-commutative N/A

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

    2. *-commutative N/A

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

    3. *-commutative N/A

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

    4. associate-/l/ N/A

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

    5. un-div-inv N/A

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

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

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

    7. *-commutative N/A

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

    8. *-commutative N/A

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

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

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

    10. *-commutative N/A

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

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

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

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

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

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

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

    14. rem-square-sqrt N/A

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

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

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

    16. rem-square-sqrt N/A

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

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

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

11. Applied egg-rr 99.3%

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

Alternative 2: 95.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.4%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 96.8%

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

9. Applied egg-rr 96.8%

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

10. Final simplification 96.8%

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

Alternative 3: 93.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (((((a * a) * (-2.0 * (c * (c * c)))) / (b * (b * (b * b)))) - 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 = (((((a * a) * ((-2.0d0) * (c * (c * c)))) / (b * (b * (b * b)))) - c) - ((a * (c * c)) / (b * b))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.4%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 96.8%

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

9. Applied egg-rr 96.8%

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

10. Taylor expanded in a around 0

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

    1. associate-*r/ N/A

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

    2. mul-1-neg N/A

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

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

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

    4. mul-1-neg N/A

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

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

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

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

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

    7. unpow2 N/A

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

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

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

    9. unpow2 N/A

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

    10. *-lowering-*.f64 95.4%

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

12. Simplified 95.4%

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

13. Final simplification 95.4%

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

Alternative 4: 93.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(-2 \cdot \left(a \cdot c\right)\right) \cdot \frac{\frac{1}{c \cdot \left(-2 \cdot \left(\frac{a}{b} + \frac{c \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)\right) + b \cdot 2}}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (* -2.0 (* a c))
  (/
   (/
    1.0
    (+ (* c (* -2.0 (+ (/ a b) (/ (* c (* a a)) (* b (* b b)))))) (* b 2.0)))
   a)))

C

double code(double a, double b, double c) {
	return (-2.0 * (a * c)) * ((1.0 / ((c * (-2.0 * ((a / b) + ((c * (a * a)) / (b * (b * b)))))) + (b * 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 = ((-2.0d0) * (a * c)) * ((1.0d0 / ((c * ((-2.0d0) * ((a / b) + ((c * (a * a)) / (b * (b * b)))))) + (b * 2.0d0))) / a)
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * (a * c)) * ((1.0 / ((c * (-2.0 * ((a / b) + ((c * (a * a)) / (b * (b * b)))))) + (b * 2.0))) / a);
}

Python

def code(a, b, c):
	return (-2.0 * (a * c)) * ((1.0 / ((c * (-2.0 * ((a / b) + ((c * (a * a)) / (b * (b * b)))))) + (b * 2.0))) / a)

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(a * c)) * Float64(Float64(1.0 / Float64(Float64(c * Float64(-2.0 * Float64(Float64(a / b) + Float64(Float64(c * Float64(a * a)) / Float64(b * Float64(b * b)))))) + Float64(b * 2.0))) / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (a * c)) * ((1.0 / ((c * (-2.0 * ((a / b) + ((c * (a * a)) / (b * (b * b)))))) + (b * 2.0))) / a);
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(N[(c * N[(-2.0 * N[(N[(a / b), $MachinePrecision] + N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.4%

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

   1. flip-- N/A

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

   2. div-inv N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

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

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

6. Applied egg-rr 31.3%

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

7. Taylor expanded in b around 0

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

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

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

   2. *-lowering-*.f64 99.1%

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

9. Simplified 99.1%

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

10. Taylor expanded in c around 0

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

    1. +-commutative N/A

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

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

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

12. Simplified 95.1%

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

Alternative 5: 93.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(-2 \cdot \left(a \cdot c\right)\right) \cdot \frac{\frac{1}{b + \left(b + c \cdot \left(-2 \cdot \left(\frac{a}{b} + \frac{c \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)\right)\right)}}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (* -2.0 (* a c))
  (/
   (/
    1.0
    (+ b (+ b (* c (* -2.0 (+ (/ a b) (/ (* c (* a a)) (* b (* b b)))))))))
   a)))

C

double code(double a, double b, double c) {
	return (-2.0 * (a * c)) * ((1.0 / (b + (b + (c * (-2.0 * ((a / b) + ((c * (a * a)) / (b * (b * b))))))))) / a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-2.0d0) * (a * c)) * ((1.0d0 / (b + (b + (c * ((-2.0d0) * ((a / b) + ((c * (a * a)) / (b * (b * b))))))))) / a)
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * (a * c)) * ((1.0 / (b + (b + (c * (-2.0 * ((a / b) + ((c * (a * a)) / (b * (b * b))))))))) / a);
}

Python

def code(a, b, c):
	return (-2.0 * (a * c)) * ((1.0 / (b + (b + (c * (-2.0 * ((a / b) + ((c * (a * a)) / (b * (b * b))))))))) / a)

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(a * c)) * Float64(Float64(1.0 / Float64(b + Float64(b + Float64(c * Float64(-2.0 * Float64(Float64(a / b) + Float64(Float64(c * Float64(a * a)) / Float64(b * Float64(b * b))))))))) / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (a * c)) * ((1.0 / (b + (b + (c * (-2.0 * ((a / b) + ((c * (a * a)) / (b * (b * b))))))))) / a);
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(b + N[(b + N[(c * N[(-2.0 * N[(N[(a / b), $MachinePrecision] + N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.4%

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

   1. flip-- N/A

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

   2. div-inv N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

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

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

6. Applied egg-rr 31.3%

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

7. Taylor expanded in b around 0

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

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

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

   2. *-lowering-*.f64 99.1%

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

9. Simplified 99.1%

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

10. Taylor expanded in c around 0

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

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

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

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

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

    3. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

    8. *-commutative N/A

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

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

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

    10. unpow2 N/A

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

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

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

    12. cube-mult N/A

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

    13. unpow2 N/A

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

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

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

    15. unpow2 N/A

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

    16. *-lowering-*.f64 95.1%

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

12. Simplified 95.1%

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

Alternative 6: 90.7% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.4%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 96.8%

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

8. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
9. 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

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

   3. +-commutative N/A

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

   4. sub-neg N/A

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

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

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

   6. associate-*r/ N/A

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

   7. mul-1-neg N/A

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

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

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

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

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

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

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

   11. unpow2 N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 92.1%

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

10. Simplified 92.1%

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

11. Final simplification 92.1%

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

Alternative 7: 81.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.4%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 96.8%

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

8. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

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

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

   4. neg-lowering-neg.f64 82.3%

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

10. Simplified 82.3%

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

11. Final simplification 82.3%

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

Alternative 8: 10.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.4%

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

5. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. neg-lowering-neg.f64 10.0%

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

7. Simplified 10.0%

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

8. Final simplification 10.0%

   \[\leadsto 0 - \frac{b}{a} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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)))