Quadratic roots, narrow range

Percentage Accurate: 55.6% → 99.5%

Time: 17.5s

Alternatives: 9

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / (a * ((b + sqrt(((b * b) + (c * ((-4.0d0) * a))))) / (a / (-0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 52.8%

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

   1. div-sub N/A

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

   2. flip-- N/A

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

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

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

6. Applied egg-rr 53.6%

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

7. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

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

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

   4. neg-lowering-neg.f64 99.2%

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

9. Simplified 99.2%

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

    1. associate-/l/ N/A

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

    2. frac-2neg N/A

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

    3. remove-double-neg N/A

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

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

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

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

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

    6. *-commutative N/A

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

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

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

    8. *-commutative N/A

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

    9. metadata-eval N/A

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

    10. associate-/r* N/A

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

    11. *-commutative N/A

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

    12. un-div-inv N/A

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

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

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

11. Applied egg-rr 99.5%

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

    1. *-commutative N/A

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

    2. distribute-lft-neg-in N/A

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

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

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

13. Applied egg-rr 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 91.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(c \cdot c\right)\\ t_1 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -90:\\ \;\;\;\;\frac{0.5}{a \cdot \frac{1}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot t\_0\right)}{b \cdot t\_1} + \left(\frac{-0.25}{\frac{a \cdot \left(t\_1 \cdot t\_1\right)}{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot t\_0\right) \cdot 20\right)}} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* c c))) (t_1 (* b (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -90.0)
     (/ 0.5 (* a (/ 1.0 (- (sqrt (+ (* b b) (* a (* c -4.0)))) b))))
     (/
      (-
       (+
        (/ (* (* a a) (* -2.0 t_0)) (* b t_1))
        (-
         (/
          -0.25
          (/ (* a (* t_1 t_1)) (* (* a (* a (* a a))) (* (* c t_0) 20.0))))
         (/ (* a (* c c)) (* b b))))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = b * (b * b);
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -90.0) {
		tmp = 0.5 / (a * (1.0 / (sqrt(((b * b) + (a * (c * -4.0)))) - b)));
	} else {
		tmp = (((((a * a) * (-2.0 * t_0)) / (b * t_1)) + ((-0.25 / ((a * (t_1 * t_1)) / ((a * (a * (a * a))) * ((c * t_0) * 20.0)))) - ((a * (c * c)) / (b * b)))) - c) / b;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (c * c)
    t_1 = b * (b * b)
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-90.0d0)) then
        tmp = 0.5d0 / (a * (1.0d0 / (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b)))
    else
        tmp = (((((a * a) * ((-2.0d0) * t_0)) / (b * t_1)) + (((-0.25d0) / ((a * (t_1 * t_1)) / ((a * (a * (a * a))) * ((c * t_0) * 20.0d0)))) - ((a * (c * c)) / (b * b)))) - c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = b * (b * b);
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -90.0) {
		tmp = 0.5 / (a * (1.0 / (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b)));
	} else {
		tmp = (((((a * a) * (-2.0 * t_0)) / (b * t_1)) + ((-0.25 / ((a * (t_1 * t_1)) / ((a * (a * (a * a))) * ((c * t_0) * 20.0)))) - ((a * (c * c)) / (b * b)))) - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (c * c)
	t_1 = b * (b * b)
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -90.0:
		tmp = 0.5 / (a * (1.0 / (math.sqrt(((b * b) + (a * (c * -4.0)))) - b)))
	else:
		tmp = (((((a * a) * (-2.0 * t_0)) / (b * t_1)) + ((-0.25 / ((a * (t_1 * t_1)) / ((a * (a * (a * a))) * ((c * t_0) * 20.0)))) - ((a * (c * c)) / (b * b)))) - c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(c * c))
	t_1 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -90.0)
		tmp = Float64(0.5 / Float64(a * Float64(1.0 / Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(a * a) * Float64(-2.0 * t_0)) / Float64(b * t_1)) + Float64(Float64(-0.25 / Float64(Float64(a * Float64(t_1 * t_1)) / Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(Float64(c * t_0) * 20.0)))) - Float64(Float64(a * Float64(c * c)) / Float64(b * b)))) - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (c * c);
	t_1 = b * (b * b);
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -90.0)
		tmp = 0.5 / (a * (1.0 / (sqrt(((b * b) + (a * (c * -4.0)))) - b)));
	else
		tmp = (((((a * a) * (-2.0 * t_0)) / (b * t_1)) + ((-0.25 / ((a * (t_1 * t_1)) / ((a * (a * (a * a))) * ((c * t_0) * 20.0)))) - ((a * (c * c)) / (b * b)))) - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -90.0], N[(0.5 / N[(a * N[(1.0 / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 / N[(N[(a * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c * t$95$0), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -90

   1. Initial program 90.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 90.0%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

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

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

   6. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b} \cdot a}} \]

   if -90 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 50.3%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

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

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

   3. Simplified 50.3%

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

   5. Taylor expanded in b around inf

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

   7. Simplified 93.3%

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

   9. Applied egg-rr 93.3%

      \[\leadsto \frac{\color{blue}{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(\frac{-0.25}{\frac{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 20\right)}} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right) - c}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -90:\\ \;\;\;\;\frac{0.5}{a \cdot \frac{1}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(\frac{-0.25}{\frac{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 20\right)}} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(c \cdot c\right)\\ t_1 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -90:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot t\_0\right)}{b \cdot t\_1} + \left(\frac{-0.25}{\frac{a \cdot \left(t\_1 \cdot t\_1\right)}{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot t\_0\right) \cdot 20\right)}} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* c c))) (t_1 (* b (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -90.0)
     (/ 0.5 (/ a (- (sqrt (+ (* b b) (* a (* c -4.0)))) b)))
     (/
      (-
       (+
        (/ (* (* a a) (* -2.0 t_0)) (* b t_1))
        (-
         (/
          -0.25
          (/ (* a (* t_1 t_1)) (* (* a (* a (* a a))) (* (* c t_0) 20.0))))
         (/ (* a (* c c)) (* b b))))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = b * (b * b);
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -90.0) {
		tmp = 0.5 / (a / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
	} else {
		tmp = (((((a * a) * (-2.0 * t_0)) / (b * t_1)) + ((-0.25 / ((a * (t_1 * t_1)) / ((a * (a * (a * a))) * ((c * t_0) * 20.0)))) - ((a * (c * c)) / (b * b)))) - c) / b;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (c * c)
    t_1 = b * (b * b)
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-90.0d0)) then
        tmp = 0.5d0 / (a / (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b))
    else
        tmp = (((((a * a) * ((-2.0d0) * t_0)) / (b * t_1)) + (((-0.25d0) / ((a * (t_1 * t_1)) / ((a * (a * (a * a))) * ((c * t_0) * 20.0d0)))) - ((a * (c * c)) / (b * b)))) - c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = b * (b * b);
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -90.0) {
		tmp = 0.5 / (a / (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b));
	} else {
		tmp = (((((a * a) * (-2.0 * t_0)) / (b * t_1)) + ((-0.25 / ((a * (t_1 * t_1)) / ((a * (a * (a * a))) * ((c * t_0) * 20.0)))) - ((a * (c * c)) / (b * b)))) - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (c * c)
	t_1 = b * (b * b)
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -90.0:
		tmp = 0.5 / (a / (math.sqrt(((b * b) + (a * (c * -4.0)))) - b))
	else:
		tmp = (((((a * a) * (-2.0 * t_0)) / (b * t_1)) + ((-0.25 / ((a * (t_1 * t_1)) / ((a * (a * (a * a))) * ((c * t_0) * 20.0)))) - ((a * (c * c)) / (b * b)))) - c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(c * c))
	t_1 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -90.0)
		tmp = Float64(0.5 / Float64(a / Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(a * a) * Float64(-2.0 * t_0)) / Float64(b * t_1)) + Float64(Float64(-0.25 / Float64(Float64(a * Float64(t_1 * t_1)) / Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(Float64(c * t_0) * 20.0)))) - Float64(Float64(a * Float64(c * c)) / Float64(b * b)))) - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (c * c);
	t_1 = b * (b * b);
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -90.0)
		tmp = 0.5 / (a / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
	else
		tmp = (((((a * a) * (-2.0 * t_0)) / (b * t_1)) + ((-0.25 / ((a * (t_1 * t_1)) / ((a * (a * (a * a))) * ((c * t_0) * 20.0)))) - ((a * (c * c)) / (b * b)))) - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -90.0], N[(0.5 / N[(a / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 / N[(N[(a * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c * t$95$0), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -90

   1. Initial program 90.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 90.0%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 90.1%

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

   6. Applied egg-rr 90.1%

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

   if -90 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 50.3%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

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

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

   3. Simplified 50.3%

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

   5. Taylor expanded in b around inf

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

   7. Simplified 93.3%

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

   9. Applied egg-rr 93.3%

      \[\leadsto \frac{\color{blue}{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(\frac{-0.25}{\frac{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 20\right)}} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right) - c}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -90:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(\frac{-0.25}{\frac{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 20\right)}} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := c \cdot \left(c \cdot c\right)\\ \frac{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot t\_1\right)}{b \cdot t\_0} + \left(\frac{-0.25}{\frac{a \cdot \left(t\_0 \cdot t\_0\right)}{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot t\_1\right) \cdot 20\right)}} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right) - c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* c (* c c))))
   (/
    (-
     (+
      (/ (* (* a a) (* -2.0 t_1)) (* b t_0))
      (-
       (/
        -0.25
        (/ (* a (* t_0 t_0)) (* (* a (* a (* a a))) (* (* c t_1) 20.0))))
       (/ (* a (* c c)) (* b b))))
     c)
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = c * (c * c);
	return (((((a * a) * (-2.0 * t_1)) / (b * t_0)) + ((-0.25 / ((a * (t_0 * t_0)) / ((a * (a * (a * a))) * ((c * t_1) * 20.0)))) - ((a * (c * c)) / (b * b)))) - c) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    t_0 = b * (b * b)
    t_1 = c * (c * c)
    code = (((((a * a) * ((-2.0d0) * t_1)) / (b * t_0)) + (((-0.25d0) / ((a * (t_0 * t_0)) / ((a * (a * (a * a))) * ((c * t_1) * 20.0d0)))) - ((a * (c * c)) / (b * b)))) - c) / b
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = c * (c * c);
	return (((((a * a) * (-2.0 * t_1)) / (b * t_0)) + ((-0.25 / ((a * (t_0 * t_0)) / ((a * (a * (a * a))) * ((c * t_1) * 20.0)))) - ((a * (c * c)) / (b * b)))) - c) / b;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = c * (c * c)
	return (((((a * a) * (-2.0 * t_1)) / (b * t_0)) + ((-0.25 / ((a * (t_0 * t_0)) / ((a * (a * (a * a))) * ((c * t_1) * 20.0)))) - ((a * (c * c)) / (b * b)))) - c) / b

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(c * Float64(c * c))
	return Float64(Float64(Float64(Float64(Float64(Float64(a * a) * Float64(-2.0 * t_1)) / Float64(b * t_0)) + Float64(Float64(-0.25 / Float64(Float64(a * Float64(t_0 * t_0)) / Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(Float64(c * t_1) * 20.0)))) - Float64(Float64(a * Float64(c * c)) / Float64(b * b)))) - c) / b)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = c * (c * c);
	tmp = (((((a * a) * (-2.0 * t_1)) / (b * t_0)) + ((-0.25 / ((a * (t_0 * t_0)) / ((a * (a * (a * a))) * ((c * t_1) * 20.0)))) - ((a * (c * c)) / (b * b)))) - c) / b;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 / N[(N[(a * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c * t$95$1), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Initial program 52.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 52.8%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 91.3%

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

9. Applied egg-rr 91.3%

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

Alternative 5: 90.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = (b * b) * (b * b)
    code = ((((a * a) * (c * ((-2.0d0) * (c * c)))) / t_0) + ((((-0.25d0) / a) / (((b * b) * t_0) / ((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0d0)))))) - (c + ((c * (c * a)) / (b * b))))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 52.8%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 91.3%

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

9. Applied egg-rr 91.1%

   \[\leadsto \color{blue}{\frac{1}{b} \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(\frac{-0.25}{\frac{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 20\right)}} - \left(c + \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)\right)} \]

11. Applied egg-rr 91.3%

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

12. Final simplification 91.3%

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

Alternative 6: 88.4% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 52.8%

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

   1. div-sub N/A

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

   2. flip-- N/A

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

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

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

6. Applied egg-rr 53.6%

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

7. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

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

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

   4. neg-lowering-neg.f64 99.2%

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

9. Simplified 99.2%

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

    1. associate-/l/ N/A

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

    2. frac-2neg N/A

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

    3. remove-double-neg N/A

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

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

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

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

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

    6. *-commutative N/A

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

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

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

    8. *-commutative N/A

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

    9. metadata-eval N/A

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

    10. associate-/r* N/A

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

    11. *-commutative N/A

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

    12. un-div-inv N/A

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

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

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

11. Applied egg-rr 99.5%

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

12. Taylor expanded in a around 0

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

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

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

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

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

    3. cancel-sign-sub-inv N/A

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

    4. metadata-eval N/A

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

    5. *-lft-identity N/A

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

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

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

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

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

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

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

    9. unpow2 N/A

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

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

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

    11. cube-mult N/A

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

    12. unpow2 N/A

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

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

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

    14. unpow2 N/A

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

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

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

    16. /-lowering-/.f64 88.5%

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

14. Simplified 88.5%

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

Alternative 7: 82.3% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 52.8%

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

   1. div-sub N/A

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

   2. flip-- N/A

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

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

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

6. Applied egg-rr 53.6%

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

7. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

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

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

   4. neg-lowering-neg.f64 99.2%

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

9. Simplified 99.2%

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

    1. associate-/l/ N/A

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

    2. frac-2neg N/A

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

    3. remove-double-neg N/A

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

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

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

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

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

    6. *-commutative N/A

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

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

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

    8. *-commutative N/A

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

    9. metadata-eval N/A

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

    10. associate-/r* N/A

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

    11. *-commutative N/A

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

    12. un-div-inv N/A

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

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

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

11. Applied egg-rr 99.5%

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

12. Taylor expanded in a around 0

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

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

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

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

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

    3. *-commutative N/A

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

    4. *-lowering-*.f64 82.1%

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

14. Simplified 82.1%

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

Alternative 8: 64.3% 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 52.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 52.8%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 91.3%

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

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 66.1%

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

10. Simplified 66.1%

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

11. Final simplification 66.1%

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

Alternative 9: 3.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 52.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 52.8%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. associate-/r/ N/A

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

   6. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 52.2%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2} + \frac{0.5}{a} \cdot \left(0 - b\right)} \]

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 3.2%

      \[\leadsto 0 \]

9. Simplified 3.2%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))