Cubic critical

Average Error: 33.5 → 12.1

Time: 21.5s

Precision: binary64

Cost: 13900

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+151}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq -6.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-290}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot -3} \cdot \sqrt{c}}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

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 -1.7e+151)
   (/ b (* a -1.5))
   (if (<= b -6.8e-158)
     (/ (/ (- b (sqrt (+ (* b b) (* a (* c -3.0))))) a) -3.0)
     (if (<= b 8.6e-290)
       (/ (+ b (* (sqrt (* a -3.0)) (sqrt c))) (* a 3.0))
       (if (<= b 1.4e-43)
         (/ (- (sqrt (+ (* b b) (* c (* a -3.0)))) 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 <= -1.7e+151) {
		tmp = b / (a * -1.5);
	} else if (b <= -6.8e-158) {
		tmp = ((b - sqrt(((b * b) + (a * (c * -3.0))))) / a) / -3.0;
	} else if (b <= 8.6e-290) {
		tmp = (b + (sqrt((a * -3.0)) * sqrt(c))) / (a * 3.0);
	} else if (b <= 1.4e-43) {
		tmp = (sqrt(((b * b) + (c * (a * -3.0)))) - 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 <= (-1.7d+151)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= (-6.8d-158)) then
        tmp = ((b - sqrt(((b * b) + (a * (c * (-3.0d0)))))) / a) / (-3.0d0)
    else if (b <= 8.6d-290) then
        tmp = (b + (sqrt((a * (-3.0d0))) * sqrt(c))) / (a * 3.0d0)
    else if (b <= 1.4d-43) then
        tmp = (sqrt(((b * b) + (c * (a * (-3.0d0))))) - 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 <= -1.7e+151) {
		tmp = b / (a * -1.5);
	} else if (b <= -6.8e-158) {
		tmp = ((b - Math.sqrt(((b * b) + (a * (c * -3.0))))) / a) / -3.0;
	} else if (b <= 8.6e-290) {
		tmp = (b + (Math.sqrt((a * -3.0)) * Math.sqrt(c))) / (a * 3.0);
	} else if (b <= 1.4e-43) {
		tmp = (Math.sqrt(((b * b) + (c * (a * -3.0)))) - 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 <= -1.7e+151:
		tmp = b / (a * -1.5)
	elif b <= -6.8e-158:
		tmp = ((b - math.sqrt(((b * b) + (a * (c * -3.0))))) / a) / -3.0
	elif b <= 8.6e-290:
		tmp = (b + (math.sqrt((a * -3.0)) * math.sqrt(c))) / (a * 3.0)
	elif b <= 1.4e-43:
		tmp = (math.sqrt(((b * b) + (c * (a * -3.0)))) - 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 <= -1.7e+151)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= -6.8e-158)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -3.0))))) / a) / -3.0);
	elseif (b <= 8.6e-290)
		tmp = Float64(Float64(b + Float64(sqrt(Float64(a * -3.0)) * sqrt(c))) / Float64(a * 3.0));
	elseif (b <= 1.4e-43)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) - 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 <= -1.7e+151)
		tmp = b / (a * -1.5);
	elseif (b <= -6.8e-158)
		tmp = ((b - sqrt(((b * b) + (a * (c * -3.0))))) / a) / -3.0;
	elseif (b <= 8.6e-290)
		tmp = (b + (sqrt((a * -3.0)) * sqrt(c))) / (a * 3.0);
	elseif (b <= 1.4e-43)
		tmp = (sqrt(((b * b) + (c * (a * -3.0)))) - 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, -1.7e+151], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -6.8e-158], N[(N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 8.6e-290], N[(N[(b + N[(N[Sqrt[N[(a * -3.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.4e-43], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.7e151

   1. Initial program 63.4

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

   2. Simplified 63.4

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

      [Start] 63.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 63.4	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 63.4	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 63.4	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 63.4	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 63.4	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 63.4	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 63.4	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 63.4	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 63.4	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 63.4	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 63.4	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 63.4	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 63.4	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 63.4	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   3. Applied egg-rr 63.4

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

   4. Taylor expanded in a around 0 63.3

      \[\leadsto \frac{\frac{b - \sqrt{\color{blue}{{b}^{2}}}}{a}}{-3} \]

   5. Simplified 63.3

      \[\leadsto \frac{\frac{b - \sqrt{\color{blue}{b \cdot b}}}{a}}{-3} \]
      Proof

      [Start] 63.3	\[ \frac{\frac{b - \sqrt{{b}^{2}}}{a}}{-3} \]
      unpow2 [=>] 63.3	\[ \frac{\frac{b - \sqrt{\color{blue}{b \cdot b}}}{a}}{-3} \]

   6. Taylor expanded in b around -inf 2.6

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

   7. Simplified 2.6

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

      [Start] 2.6	\[ -0.6666666666666666 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 2.6	\[ \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]
      associate-*l/ [<=] 2.6	\[ \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]
      *-commutative [=>] 2.6	\[ \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   8. Applied egg-rr 2.5

      \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if -1.7e151 < b < -6.7999999999999999e-158

   1. Initial program 6.0

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

   2. Simplified 6.2

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

      [Start] 6.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 6.0	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 6.0	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 6.0	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 6.0	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 6.0	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 6.0	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 6.0	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 6.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 6.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 6.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 6.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 6.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 6.0	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 6.0	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   3. Applied egg-rr 6.1

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

   4. Applied egg-rr 6.1

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

   if -6.7999999999999999e-158 < b < 8.6000000000000004e-290

   1. Initial program 14.6

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

   2. Taylor expanded in b around 0 14.8

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

   3. Simplified 14.7

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

      [Start] 14.8	\[ \frac{\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a} \]
      *-commutative [<=] 14.8	\[ \frac{\left(-b\right) + \sqrt{-3 \cdot \color{blue}{\left(a \cdot c\right)}}}{3 \cdot a} \]
      associate-*r* [=>] 14.7	\[ \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-3 \cdot a\right) \cdot c}}}{3 \cdot a} \]
      *-commutative [<=] 14.7	\[ \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot -3\right)} \cdot c}}{3 \cdot a} \]
      *-commutative [=>] 14.7	\[ \frac{\left(-b\right) + \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{3 \cdot a} \]

   4. Applied egg-rr 34.3

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

   5. Simplified 34.3

      \[\leadsto \frac{\color{blue}{b + \sqrt{a \cdot -3} \cdot \sqrt{c}}}{3 \cdot a} \]
      Proof

      [Start] 34.3	\[ \frac{b - \left(-\sqrt{c}\right) \cdot \sqrt{a \cdot -3}}{3 \cdot a} \]
      cancel-sign-sub [=>] 34.3	\[ \frac{\color{blue}{b + \sqrt{c} \cdot \sqrt{a \cdot -3}}}{3 \cdot a} \]
      *-commutative [<=] 34.3	\[ \frac{b + \color{blue}{\sqrt{a \cdot -3} \cdot \sqrt{c}}}{3 \cdot a} \]

   if 8.6000000000000004e-290 < b < 1.3999999999999999e-43

   1. Initial program 22.9

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

   if 1.3999999999999999e-43 < b

   1. Initial program 53.6

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

   2. Simplified 53.6

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

      [Start] 53.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 53.6	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 53.6	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 54.5	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 54.5	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 55.4	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 55.4	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 57.9	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 57.9	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 57.9	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 57.9	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 57.9	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 57.9	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 55.4	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 55.4	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   3. Taylor expanded in b around inf 7.8

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+151}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq -6.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-290}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot -3} \cdot \sqrt{c}}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.1
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-43}:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2
Error	10.1
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3
Error	13.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 1.5\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-43}:\\ \;\;\;\;\left(b - \sqrt{-3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4
Error	13.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 1.5\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-43}:\\ \;\;\;\;\left(b - \sqrt{a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5
Error	13.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 1.5\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6
Error	13.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 1.5\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-43}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7
Error	13.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 1.5\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8
Error	23.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{b}{a} \cdot 2}{-3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9
Error	23.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-231}:\\ \;\;\;\;\frac{\frac{b + b}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10
Error	23.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-227}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11
Error	23.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-228}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12
Error	40.0
Cost	320

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

Alternative 13
Error	56.2
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))