Cubic critical, medium range

Percentage Accurate: 31.5% → 99.4%

Time: 14.0s

Alternatives: 9

Speedup: 2.9×

Specification

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

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 31.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(3 \cdot a\right)}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot \mathsf{fma}\left(b \cdot -3, a \cdot \frac{c}{b \cdot b}, b\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* c (* 3.0 a))
  (* (* 3.0 a) (- (- b) (sqrt (* b (fma (* b -3.0) (* a (/ c (* b b))) b)))))))

C

double code(double a, double b, double c) {
	return (c * (3.0 * a)) / ((3.0 * a) * (-b - sqrt((b * fma((b * -3.0), (a * (c / (b * b))), b)))));
}

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(3.0 * a)) / Float64(Float64(3.0 * a) * Float64(Float64(-b) - sqrt(Float64(b * fma(Float64(b * -3.0), Float64(a * Float64(c / Float64(b * b))), b))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[((-b) - N[Sqrt[N[(b * N[(N[(b * -3.0), $MachinePrecision] * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \left(b \cdot \left(-3 \cdot \frac{a \cdot c}{{b}^{2}}\right) + \color{blue}{b}\right)}}{3 \cdot a} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \color{blue}{\mathsf{fma}\left(b, -3 \cdot \frac{a \cdot c}{{b}^{2}}, b\right)}}}{3 \cdot a} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \mathsf{fma}\left(b, \color{blue}{-3 \cdot \frac{a \cdot c}{{b}^{2}}}, b\right)}}{3 \cdot a} \]

   9. associate-/l* N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{2}}\right)}, b\right)}}{3 \cdot a} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{2}}\right)}, b\right)}}{3 \cdot a} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \color{blue}{\frac{c}{{b}^{2}}}\right), b\right)}}{3 \cdot a} \]

   12. unpow2 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{\color{blue}{b \cdot b}}\right), b\right)}}{3 \cdot a} \]

   13. lower-*.f64 32.7

       \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{\color{blue}{b \cdot b}}\right), b\right)}}{3 \cdot a} \]

5. Applied rewrites 32.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)}}}{3 \cdot a} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)}}}{3 \cdot a} \]

   2. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)} \cdot \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)}}}}{3 \cdot a} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)} \cdot \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)}}}}{3 \cdot a} \]

7. Applied rewrites 33.3%

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b - b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}{\left(-b\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}}{3 \cdot a} \]

8. Taylor expanded in b around 0

   \[\leadsto \frac{\frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}{3 \cdot a} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(3 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}{3 \cdot a} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(3 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}{3 \cdot a} \]

   3. lower-*.f64 99.3

      \[\leadsto \frac{\frac{\color{blue}{\left(3 \cdot a\right)} \cdot c}{\left(-b\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}{3 \cdot a} \]

10. Applied rewrites 99.3%

    \[\leadsto \frac{\frac{\color{blue}{\left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}{3 \cdot a} \]
11. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{\left(3 \cdot a\right) \cdot c}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}{3 \cdot a}} \]

    2. lift-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\left(3 \cdot a\right) \cdot c}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}}{3 \cdot a} \]

    3. associate-/l/ N/A

       \[\leadsto \color{blue}{\frac{\left(3 \cdot a\right) \cdot c}{\left(3 \cdot a\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}\right)}} \]

    4. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\left(3 \cdot a\right) \cdot c}{\left(3 \cdot a\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}\right)}} \]

12. Applied rewrites 99.4%

    \[\leadsto \color{blue}{\frac{c \cdot \left(a \cdot 3\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot \mathsf{fma}\left(b \cdot -3, a \cdot \frac{c}{b \cdot b}, b\right)}\right)}} \]

13. Final simplification 99.4%

    \[\leadsto \frac{c \cdot \left(3 \cdot a\right)}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot \mathsf{fma}\left(b \cdot -3, a \cdot \frac{c}{b \cdot b}, b\right)}\right)} \]
14. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ (* c (* 3.0 a)) (- (- b) (sqrt (fma -3.0 (* a c) (* b b))))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return ((c * (3.0 * a)) / (-b - sqrt(fma(-3.0, (a * c), (b * b))))) / (3.0 * a);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * Float64(3.0 * a)) / Float64(Float64(-b) - sqrt(fma(-3.0, Float64(a * c), Float64(b * b))))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \left(b \cdot \left(-3 \cdot \frac{a \cdot c}{{b}^{2}}\right) + \color{blue}{b}\right)}}{3 \cdot a} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \color{blue}{\mathsf{fma}\left(b, -3 \cdot \frac{a \cdot c}{{b}^{2}}, b\right)}}}{3 \cdot a} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \mathsf{fma}\left(b, \color{blue}{-3 \cdot \frac{a \cdot c}{{b}^{2}}}, b\right)}}{3 \cdot a} \]

   9. associate-/l* N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{2}}\right)}, b\right)}}{3 \cdot a} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{2}}\right)}, b\right)}}{3 \cdot a} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \color{blue}{\frac{c}{{b}^{2}}}\right), b\right)}}{3 \cdot a} \]

   12. unpow2 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{\color{blue}{b \cdot b}}\right), b\right)}}{3 \cdot a} \]

   13. lower-*.f64 31.5

       \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{\color{blue}{b \cdot b}}\right), b\right)}}{3 \cdot a} \]

5. Applied rewrites 31.5%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)}}}{3 \cdot a} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)}}}{3 \cdot a} \]

   2. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)} \cdot \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)}}}}{3 \cdot a} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)} \cdot \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(b, -3 \cdot \left(a \cdot \frac{c}{b \cdot b}\right), b\right)}}}}{3 \cdot a} \]

7. Applied rewrites 31.9%

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b - b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}{\left(-b\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}}{3 \cdot a} \]

8. Taylor expanded in b around 0

   \[\leadsto \frac{\frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}{3 \cdot a} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(3 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}{3 \cdot a} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(3 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}{3 \cdot a} \]

   3. lower-*.f64 99.4

      \[\leadsto \frac{\frac{\color{blue}{\left(3 \cdot a\right)} \cdot c}{\left(-b\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}{3 \cdot a} \]

10. Applied rewrites 99.4%

    \[\leadsto \frac{\frac{\color{blue}{\left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot \mathsf{fma}\left(\left(b \cdot -3\right) \cdot a, \frac{c}{b \cdot b}, b\right)}}}{3 \cdot a} \]

11. Taylor expanded in b around 0

    \[\leadsto \frac{\frac{\left(3 \cdot a\right) \cdot c}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{-3 \cdot \left(a \cdot c\right) + \color{blue}{{b}^{2}}}}}{3 \cdot a} \]
12. Step-by-step derivation

13. Applied rewrites 99.4%

    \[\leadsto \frac{\frac{\left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, \color{blue}{a \cdot c}, b \cdot b\right)}}}{3 \cdot a} \]

14. Final simplification 99.4%

    \[\leadsto \frac{\frac{c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}{3 \cdot a} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))