Cubic critical, narrow range

Percentage Accurate: 55.1% → 91.4%

Time: 16.4s

Alternatives: 12

Speedup: 2.9×

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

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

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: 55.1% 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: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)\\ t_2 := b + \sqrt{t\_1}\\ t_3 := c \cdot \left(c \cdot c\right)\\ \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\frac{t\_1}{t\_2} - \frac{b \cdot b}{t\_2}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(\frac{\left(t\_3 \cdot \left(c \cdot 6.328125\right)\right) \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(t\_0 \cdot t\_0\right)}, -0.16666666666666666, \frac{\mathsf{fma}\left(-0.375, a \cdot \left(c \cdot c\right), \frac{t\_3 \cdot \left(\left(a \cdot a\right) \cdot -0.5625\right)}{b \cdot b}\right)}{b \cdot b}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (fma a (* -3.0 c) (* b b)))
        (t_2 (+ b (sqrt t_1)))
        (t_3 (* c (* c c))))
   (if (<= b 4.8)
     (/ (- (/ t_1 t_2) (/ (* b b) t_2)) (* a 3.0))
     (/
      (fma
       c
       -0.5
       (fma
        (/ (* (* t_3 (* c 6.328125)) (* a (* a (* a a)))) (* a (* t_0 t_0)))
        -0.16666666666666666
        (/
         (fma -0.375 (* a (* c c)) (/ (* t_3 (* (* a a) -0.5625)) (* b b)))
         (* b b))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = fma(a, (-3.0 * c), (b * b));
	double t_2 = b + sqrt(t_1);
	double t_3 = c * (c * c);
	double tmp;
	if (b <= 4.8) {
		tmp = ((t_1 / t_2) - ((b * b) / t_2)) / (a * 3.0);
	} else {
		tmp = fma(c, -0.5, fma((((t_3 * (c * 6.328125)) * (a * (a * (a * a)))) / (a * (t_0 * t_0))), -0.16666666666666666, (fma(-0.375, (a * (c * c)), ((t_3 * ((a * a) * -0.5625)) / (b * b))) / (b * b)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = fma(a, Float64(-3.0 * c), Float64(b * b))
	t_2 = Float64(b + sqrt(t_1))
	t_3 = Float64(c * Float64(c * c))
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(Float64(t_1 / t_2) - Float64(Float64(b * b) / t_2)) / Float64(a * 3.0));
	else
		tmp = Float64(fma(c, -0.5, fma(Float64(Float64(Float64(t_3 * Float64(c * 6.328125)) * Float64(a * Float64(a * Float64(a * a)))) / Float64(a * Float64(t_0 * t_0))), -0.16666666666666666, Float64(fma(-0.375, Float64(a * Float64(c * c)), Float64(Float64(t_3 * Float64(Float64(a * a) * -0.5625)) / Float64(b * b))) / Float64(b * b)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.8], N[(N[(N[(t$95$1 / t$95$2), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5 + N[(N[(N[(N[(t$95$3 * N[(c * 6.328125), $MachinePrecision]), $MachinePrecision] * N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 * N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 84.2%

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

   4. Applied rewrites 84.6%

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

   if 4.79999999999999982 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b}} \]

   7. Applied rewrites 93.7%

      \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, \frac{\left(-0.16666666666666666 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right) + \frac{c \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b} \]

   9. Applied rewrites 93.7%

      \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot 6.328125\right)\right) \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, -0.16666666666666666, \mathsf{fma}\left(a, \frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot b}, \frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(\left(a \cdot a\right) \cdot -0.5625\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)\right)}{b} \]

   10. Taylor expanded in b around inf

       \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \frac{405}{64}\right)\right) \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{-1}{6}, \frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right)\right)}{b} \]
   11. Step-by-step derivation

   12. Applied rewrites 93.7%

       \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot 6.328125\right)\right) \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, -0.16666666666666666, \frac{\mathsf{fma}\left(-0.375, a \cdot \left(c \cdot c\right), \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(\left(a \cdot a\right) \cdot -0.5625\right)}{b \cdot b}\right)}{b \cdot b}\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}} - \frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot 6.328125\right)\right) \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, -0.16666666666666666, \frac{\mathsf{fma}\left(-0.375, a \cdot \left(c \cdot c\right), \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(\left(a \cdot a\right) \cdot -0.5625\right)}{b \cdot b}\right)}{b \cdot b}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)\\ t_1 := b + \sqrt{t\_0}\\ \mathbf{if}\;b \leq 5.6:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1} - \frac{b \cdot b}{t\_1}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -0.5, \frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot \left(a \cdot c\right)\right), \frac{-0.5625}{b \cdot b}, a \cdot \left(\left(c \cdot c\right) \cdot -0.375\right)\right)}{b \cdot b}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* -3.0 c) (* b b))) (t_1 (+ b (sqrt t_0))))
   (if (<= b 5.6)
     (/ (- (/ t_0 t_1) (/ (* b b) t_1)) (* a 3.0))
     (/
      (fma
       c
       -0.5
       (/
        (fma
         (* (* c c) (* a (* a c)))
         (/ -0.5625 (* b b))
         (* a (* (* c c) -0.375)))
        (* b b)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (-3.0 * c), (b * b));
	double t_1 = b + sqrt(t_0);
	double tmp;
	if (b <= 5.6) {
		tmp = ((t_0 / t_1) - ((b * b) / t_1)) / (a * 3.0);
	} else {
		tmp = fma(c, -0.5, (fma(((c * c) * (a * (a * c))), (-0.5625 / (b * b)), (a * ((c * c) * -0.375))) / (b * b))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(-3.0 * c), Float64(b * b))
	t_1 = Float64(b + sqrt(t_0))
	tmp = 0.0
	if (b <= 5.6)
		tmp = Float64(Float64(Float64(t_0 / t_1) - Float64(Float64(b * b) / t_1)) / Float64(a * 3.0));
	else
		tmp = Float64(fma(c, -0.5, Float64(fma(Float64(Float64(c * c) * Float64(a * Float64(a * c))), Float64(-0.5625 / Float64(b * b)), Float64(a * Float64(Float64(c * c) * -0.375))) / Float64(b * b))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 5.6], N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5 + N[(N[(N[(N[(c * c), $MachinePrecision] * N[(a * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5625 / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5999999999999996

   1. Initial program 80.0%

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

   4. Applied rewrites 80.7%

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

   if 5.5999999999999996 < b

   1. Initial program 48.6%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b}} \]

   7. Applied rewrites 94.1%

      \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, \frac{\left(-0.16666666666666666 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right) + \frac{c \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b} \]

   8. Taylor expanded in b around inf

      \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right)}{b} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.9%

       \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot \left(a \cdot c\right)\right), \frac{-0.5625}{b \cdot b}, a \cdot \left(-0.375 \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.6:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}} - \frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -0.5, \frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot \left(a \cdot c\right)\right), \frac{-0.5625}{b \cdot b}, a \cdot \left(\left(c \cdot c\right) \cdot -0.375\right)\right)}{b \cdot b}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))