Cubic critical

Average Error: 34.1 → 11.6

Time: 14.3s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e+154)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 9.5e-167)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

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

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e+154) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 9.5e-167) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (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
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.1d+154)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 9.5d-167) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
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);
}

↓

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e+154) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 9.5e-167) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

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

↓

def code(a, b, c):
	tmp = 0
	if b <= -1.1e+154:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 9.5e-167:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

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

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e+154)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 9.5e-167)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.1e+154)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 9.5e-167)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
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]

↓

code[a_, b_, c_] := If[LessEqual[b, -1.1e+154], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 9.5e-167], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1000000000000001e154

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}} \]
      Proof

      [Start] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 64.0	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 64.0	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 64.0	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 64.0	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 64.0	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 64.0	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 64.0	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 64.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 64.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 64.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 64.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 64.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 64.0	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 64.0	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      associate-*r/ [=>] 64.0	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{\frac{-1 \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      associate-*l/ [<=] 64.0	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{\frac{-1}{3 \cdot a} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \]
      distribute-lft-out-- [=>] 64.0	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \left(\frac{-b}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 64.0	\[ \color{blue}{\left(\frac{-b}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a}} \]
      neg-mul-1 [=>] 64.0	\[ \left(\frac{\color{blue}{-1 \cdot b}}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      *-commutative [=>] 64.0	\[ \left(\frac{\color{blue}{b \cdot -1}}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      associate-/l* [=>] 64.0	\[ \left(\color{blue}{\frac{b}{\frac{-1}{-1}}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      metadata-eval [=>] 64.0	\[ \left(\frac{b}{\color{blue}{1}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      /-rgt-identity [=>] 64.0	\[ \left(\color{blue}{b} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      cancel-sign-sub-inv [=>] 64.0	\[ \left(b - \sqrt{\color{blue}{b \cdot b + \left(-3 \cdot a\right) \cdot c}}\right) \cdot \frac{-1}{3 \cdot a} \]
      +-commutative [=>] 64.0	\[ \left(b - \sqrt{\color{blue}{\left(-3 \cdot a\right) \cdot c + b \cdot b}}\right) \cdot \frac{-1}{3 \cdot a} \]
      *-commutative [=>] 64.0	\[ \left(b - \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
      distribute-lft-neg-in [=>] 64.0	\[ \left(b - \sqrt{c \cdot \color{blue}{\left(\left(-3\right) \cdot a\right)} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
      associate-*r* [=>] 64.0	\[ \left(b - \sqrt{\color{blue}{\left(c \cdot \left(-3\right)\right) \cdot a} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
      *-commutative [=>] 64.0	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot \left(-3\right)\right)} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
      fma-def [=>] 64.0	\[ \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot \left(-3\right), b \cdot b\right)}}\right) \cdot \frac{-1}{3 \cdot a} \]
      metadata-eval [=>] 64.0	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot \color{blue}{-3}, b \cdot b\right)}\right) \cdot \frac{-1}{3 \cdot a} \]
      associate-/r* [=>] 64.0	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{\frac{-1}{3}}{a}} \]
      metadata-eval [=>] 64.0	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{\color{blue}{-0.3333333333333333}}{a} \]

   3. Taylor expanded in b around -inf 3.0

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

   4. Simplified 3.0

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
      Proof

      [Start] 3.0	\[ -0.6666666666666666 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 3.0	\[ \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]
      *-commutative [=>] 3.0	\[ \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   if -1.1000000000000001e154 < b < 9.49999999999999955e-167

   1. Initial program 10.7

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

   if 9.49999999999999955e-167 < b

   1. Initial program 48.4

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

   2. Simplified 48.4

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}} \]
      Proof

      [Start] 48.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 48.4	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 48.4	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 49.0	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 49.0	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 49.7	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 49.7	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 51.5	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 51.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 51.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 51.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 51.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 51.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 49.7	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 49.7	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      associate-*r/ [=>] 49.7	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{\frac{-1 \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      associate-*l/ [<=] 49.0	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{\frac{-1}{3 \cdot a} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \]
      distribute-lft-out-- [=>] 48.4	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \left(\frac{-b}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 48.4	\[ \color{blue}{\left(\frac{-b}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a}} \]
      neg-mul-1 [=>] 48.4	\[ \left(\frac{\color{blue}{-1 \cdot b}}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      *-commutative [=>] 48.4	\[ \left(\frac{\color{blue}{b \cdot -1}}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      associate-/l* [=>] 48.4	\[ \left(\color{blue}{\frac{b}{\frac{-1}{-1}}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      metadata-eval [=>] 48.4	\[ \left(\frac{b}{\color{blue}{1}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      /-rgt-identity [=>] 48.4	\[ \left(\color{blue}{b} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      cancel-sign-sub-inv [=>] 48.4	\[ \left(b - \sqrt{\color{blue}{b \cdot b + \left(-3 \cdot a\right) \cdot c}}\right) \cdot \frac{-1}{3 \cdot a} \]
      +-commutative [=>] 48.4	\[ \left(b - \sqrt{\color{blue}{\left(-3 \cdot a\right) \cdot c + b \cdot b}}\right) \cdot \frac{-1}{3 \cdot a} \]
      *-commutative [=>] 48.4	\[ \left(b - \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
      distribute-lft-neg-in [=>] 48.4	\[ \left(b - \sqrt{c \cdot \color{blue}{\left(\left(-3\right) \cdot a\right)} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
      associate-*r* [=>] 48.4	\[ \left(b - \sqrt{\color{blue}{\left(c \cdot \left(-3\right)\right) \cdot a} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
      *-commutative [=>] 48.4	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot \left(-3\right)\right)} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
      fma-def [=>] 48.4	\[ \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot \left(-3\right), b \cdot b\right)}}\right) \cdot \frac{-1}{3 \cdot a} \]
      metadata-eval [=>] 48.4	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot \color{blue}{-3}, b \cdot b\right)}\right) \cdot \frac{-1}{3 \cdot a} \]
      associate-/r* [=>] 48.4	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{\frac{-1}{3}}{a}} \]
      metadata-eval [=>] 48.4	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{\color{blue}{-0.3333333333333333}}{a} \]

   3. Taylor expanded in b around inf 14.6

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 11.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.42 \cdot 10^{-128}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2
Error	37.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-214}:\\ \;\;\;\;\frac{b}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3
Error	23.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-218}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4
Error	23.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-214}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5
Error	23.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-220}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6
Error	40.4
Cost	320

\[-0.5 \cdot \frac{c}{b} \]

Reproduce

herbie shell --seed 2022356 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))