Cubic critical, wide range

Percentage Accurate: 17.5% → 99.4%

Time: 14.6s

Alternatives: 6

Speedup: 2.9×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.5% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

4. Applied rewrites 18.4%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sub-div N/A

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

   6. associate-/l/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 18.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower-fma.f64 N/A

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

   12. *-commutative N/A

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

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

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 99.4

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

8. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

   2. lift--.f64 N/A

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

   3. +-inverses N/A

      \[\leadsto \frac{\left(a \cdot -3\right) \cdot c + \color{blue}{0}}{\left(a \cdot 3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}\right)} \]

   4. +-rgt-identity N/A

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

   5. lower-*.f64 99.4

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

10. Applied rewrites 99.4%

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

Alternative 2: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -3 \cdot \left(a \cdot c\right)\\ \frac{t\_0}{\left(a \cdot 3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, t\_0\right)}\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -3.0 (* a c))))
   (/ t_0 (* (* a 3.0) (+ b (sqrt (fma b b t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = -3.0 * (a * c);
	return t_0 / ((a * 3.0) * (b + sqrt(fma(b, b, t_0))));
}

Julia

function code(a, b, c)
	t_0 = Float64(-3.0 * Float64(a * c))
	return Float64(t_0 / Float64(Float64(a * 3.0) * Float64(b + sqrt(fma(b, b, t_0)))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 / N[(N[(a * 3.0), $MachinePrecision] * N[(b + N[Sqrt[N[(b * b + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 17.5%

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

4. Applied rewrites 17.7%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sub-div N/A

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

   6. associate-/l/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 17.9%

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

7. Taylor expanded in b around 0

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

   1. rem-square-sqrt N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. rem-square-sqrt N/A

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

   6. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))