Cubic critical

Average Error: 34.1 → 10.7

Time: 19.3s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{b - \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)}}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b}{-0.5}}\\ \end{array} \]

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 -3.1e+95)
   (/ b (* a -1.5))
   (if (<= b 2.9e-131)
     (/ (/ (- b (sqrt (+ (* b b) (* -3.0 (* a c))))) -3.0) a)
     (/ c (/ b -0.5)))))

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 <= -3.1e+95) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.9e-131) {
		tmp = ((b - sqrt(((b * b) + (-3.0 * (a * c))))) / -3.0) / a;
	} else {
		tmp = c / (b / -0.5);
	}
	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 <= (-3.1d+95)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 2.9d-131) then
        tmp = ((b - sqrt(((b * b) + ((-3.0d0) * (a * c))))) / (-3.0d0)) / a
    else
        tmp = c / (b / (-0.5d0))
    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 <= -3.1e+95) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.9e-131) {
		tmp = ((b - Math.sqrt(((b * b) + (-3.0 * (a * c))))) / -3.0) / a;
	} else {
		tmp = c / (b / -0.5);
	}
	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 <= -3.1e+95:
		tmp = b / (a * -1.5)
	elif b <= 2.9e-131:
		tmp = ((b - math.sqrt(((b * b) + (-3.0 * (a * c))))) / -3.0) / a
	else:
		tmp = c / (b / -0.5)
	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 <= -3.1e+95)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 2.9e-131)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(Float64(b * b) + Float64(-3.0 * Float64(a * c))))) / -3.0) / a);
	else
		tmp = Float64(c / Float64(b / -0.5));
	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 <= -3.1e+95)
		tmp = b / (a * -1.5);
	elseif (b <= 2.9e-131)
		tmp = ((b - sqrt(((b * b) + (-3.0 * (a * c))))) / -3.0) / a;
	else
		tmp = c / (b / -0.5);
	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, -3.1e+95], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e-131], N[(N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], N[(c / N[(b / -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.1000000000000003e95

   1. Initial program 47.5

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

   2. Simplified 47.5

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

      [Start] 47.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 47.5	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 47.5	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 47.5	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 47.5	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 47.5	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 47.5	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 47.5	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 47.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 47.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 47.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 47.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 47.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 47.5	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 47.5	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      associate-*r/ [=>] 47.5	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{\frac{-1 \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      associate-*l/ [<=] 47.6	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{\frac{-1}{3 \cdot a} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \]
      distribute-lft-out-- [=>] 47.6	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \left(\frac{-b}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 47.6	\[ \color{blue}{\left(\frac{-b}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a}} \]
      neg-mul-1 [=>] 47.6	\[ \left(\frac{\color{blue}{-1 \cdot b}}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      *-commutative [=>] 47.6	\[ \left(\frac{\color{blue}{b \cdot -1}}{-1} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      associate-/l* [=>] 47.6	\[ \left(\color{blue}{\frac{b}{\frac{-1}{-1}}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      metadata-eval [=>] 47.6	\[ \left(\frac{b}{\color{blue}{1}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      /-rgt-identity [=>] 47.6	\[ \left(\color{blue}{b} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{3 \cdot a} \]
      cancel-sign-sub-inv [=>] 47.6	\[ \left(b - \sqrt{\color{blue}{b \cdot b + \left(-3 \cdot a\right) \cdot c}}\right) \cdot \frac{-1}{3 \cdot a} \]
      +-commutative [=>] 47.6	\[ \left(b - \sqrt{\color{blue}{\left(-3 \cdot a\right) \cdot c + b \cdot b}}\right) \cdot \frac{-1}{3 \cdot a} \]
      *-commutative [=>] 47.6	\[ \left(b - \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
      distribute-lft-neg-in [=>] 47.6	\[ \left(b - \sqrt{c \cdot \color{blue}{\left(\left(-3\right) \cdot a\right)} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
      associate-*r* [=>] 47.6	\[ \left(b - \sqrt{\color{blue}{\left(c \cdot \left(-3\right)\right) \cdot a} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
      *-commutative [=>] 47.6	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot \left(-3\right)\right)} + b \cdot b}\right) \cdot \frac{-1}{3 \cdot a} \]
      fma-def [=>] 47.6	\[ \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot \left(-3\right), b \cdot b\right)}}\right) \cdot \frac{-1}{3 \cdot a} \]
      metadata-eval [=>] 47.6	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot \color{blue}{-3}, b \cdot b\right)}\right) \cdot \frac{-1}{3 \cdot a} \]
      associate-/r* [=>] 47.5	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{\frac{-1}{3}}{a}} \]
      metadata-eval [=>] 47.5	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{\color{blue}{-0.3333333333333333}}{a} \]

   3. Taylor expanded in b around -inf 4.1

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

   4. Simplified 4.1

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

      [Start] 4.1	\[ -0.6666666666666666 \cdot \frac{b}{a} \]
      *-commutative [=>] 4.1	\[ \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   5. Taylor expanded in b around 0 4.1

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

   6. Simplified 3.9

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

      [Start] 4.1	\[ -0.6666666666666666 \cdot \frac{b}{a} \]
      *-commutative [<=] 4.1	\[ \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
      /-rgt-identity [<=] 4.1	\[ \color{blue}{\frac{\frac{b}{a} \cdot -0.6666666666666666}{1}} \]
      associate-/l* [=>] 4.0	\[ \color{blue}{\frac{\frac{b}{a}}{\frac{1}{-0.6666666666666666}}} \]
      associate-/r* [<=] 3.9	\[ \color{blue}{\frac{b}{a \cdot \frac{1}{-0.6666666666666666}}} \]
      metadata-eval [=>] 3.9	\[ \frac{b}{a \cdot \color{blue}{-1.5}} \]

   if -3.1000000000000003e95 < b < 2.9000000000000002e-131

   1. Initial program 11.4

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

   2. Applied egg-rr 11.5

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}{-3} \cdot \frac{1}{a}} \]

   3. Simplified 11.4

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

      [Start] 11.5	\[ \frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}{-3} \cdot \frac{1}{a} \]
      *-commutative [<=] 11.5	\[ \color{blue}{\frac{1}{a} \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}{-3}} \]
      associate-*l/ [=>] 11.4	\[ \color{blue}{\frac{1 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}{-3}}{a}} \]
      *-lft-identity [=>] 11.4	\[ \frac{\color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}{-3}}}{a} \]
      fma-def [<=] 11.4	\[ \frac{\frac{b - \sqrt{\color{blue}{b \cdot b + -3 \cdot \left(a \cdot c\right)}}}{-3}}{a} \]
      +-commutative [=>] 11.4	\[ \frac{\frac{b - \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right) + b \cdot b}}}{-3}}{a} \]
      fma-def [=>] 11.4	\[ \frac{\frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}{-3}}{a} \]
      *-commutative [=>] 11.4	\[ \frac{\frac{b - \sqrt{\mathsf{fma}\left(-3, \color{blue}{c \cdot a}, b \cdot b\right)}}{-3}}{a} \]

   4. Applied egg-rr 11.4

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

   if 2.9000000000000002e-131 < b

   1. Initial program 50.6

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

   2. Simplified 50.7

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

      [Start] 50.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      *-lft-identity [<=] 50.6	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [<=] 50.6	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 50.6	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      neg-mul-1 [<=] 50.6	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
      distribute-rgt-neg-in [=>] 50.6	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
      times-frac [=>] 50.7	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
      *-commutative [=>] 50.7	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]
      neg-mul-1 [=>] 50.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \cdot \frac{-1}{3} \]
      neg-sub0 [=>] 50.7	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1 \cdot a} \cdot \frac{-1}{3} \]
      associate-+l- [=>] 50.7	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{-1 \cdot a} \cdot \frac{-1}{3} \]
      sub0-neg [=>] 50.7	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{-1 \cdot a} \cdot \frac{-1}{3} \]
      neg-mul-1 [=>] 50.7	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{-1 \cdot a} \cdot \frac{-1}{3} \]
      times-frac [=>] 50.7	\[ \color{blue}{\left(\frac{-1}{-1} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}\right)} \cdot \frac{-1}{3} \]
      metadata-eval [=>] 50.7	\[ \left(\color{blue}{1} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}\right) \cdot \frac{-1}{3} \]
      *-lft-identity [=>] 50.7	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}} \cdot \frac{-1}{3} \]
      cancel-sign-sub-inv [=>] 50.7	\[ \frac{b - \sqrt{\color{blue}{b \cdot b + \left(-3 \cdot a\right) \cdot c}}}{a} \cdot \frac{-1}{3} \]
      +-commutative [=>] 50.7	\[ \frac{b - \sqrt{\color{blue}{\left(-3 \cdot a\right) \cdot c + b \cdot b}}}{a} \cdot \frac{-1}{3} \]
      *-commutative [=>] 50.7	\[ \frac{b - \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}}{a} \cdot \frac{-1}{3} \]
      distribute-lft-neg-in [=>] 50.7	\[ \frac{b - \sqrt{c \cdot \color{blue}{\left(\left(-3\right) \cdot a\right)} + b \cdot b}}{a} \cdot \frac{-1}{3} \]
      associate-*r* [=>] 50.7	\[ \frac{b - \sqrt{\color{blue}{\left(c \cdot \left(-3\right)\right) \cdot a} + b \cdot b}}{a} \cdot \frac{-1}{3} \]
      *-commutative [=>] 50.7	\[ \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot \left(-3\right)\right)} + b \cdot b}}{a} \cdot \frac{-1}{3} \]
      fma-def [=>] 50.7	\[ \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot \left(-3\right), b \cdot b\right)}}}{a} \cdot \frac{-1}{3} \]
      metadata-eval [=>] 50.7	\[ \frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot \color{blue}{-3}, b \cdot b\right)}}{a} \cdot \frac{-1}{3} \]
      metadata-eval [=>] 50.7	\[ \frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \color{blue}{-0.3333333333333333} \]

   3. Taylor expanded in b around inf 12.6

      \[\leadsto \color{blue}{\left(1.5 \cdot \frac{c}{b}\right)} \cdot -0.3333333333333333 \]

   4. Simplified 12.7

      \[\leadsto \color{blue}{\frac{1.5 \cdot c}{b}} \cdot -0.3333333333333333 \]
      Proof

      [Start] 12.6	\[ \left(1.5 \cdot \frac{c}{b}\right) \cdot -0.3333333333333333 \]
      associate-*r/ [=>] 12.7	\[ \color{blue}{\frac{1.5 \cdot c}{b}} \cdot -0.3333333333333333 \]

   5. Taylor expanded in c around 0 12.4

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]

   6. Simplified 12.5

      \[\leadsto \color{blue}{\frac{c}{\frac{b}{-0.5}}} \]
      Proof

      [Start] 12.4	\[ -0.5 \cdot \frac{c}{b} \]
      *-commutative [=>] 12.4	\[ \color{blue}{\frac{c}{b} \cdot -0.5} \]
      associate-/r/ [<=] 12.5	\[ \color{blue}{\frac{c}{\frac{b}{-0.5}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{b - \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)}}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b}{-0.5}}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.7
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b}{-0.5}}\\ \end{array} \]

Alternative 2
Error	14.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-131}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b}{-0.5}}\\ \end{array} \]

Alternative 3
Error	22.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 1.56 \cdot 10^{-257}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b}{-0.5}}\\ \end{array} \]

Alternative 4
Error	36.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{-275}:\\ \;\;\;\;\frac{b}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-268}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-252}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-257}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-255}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-273}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b}{-0.5}}\\ \end{array} \]

Alternative 10
Error	39.9
Cost	320

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

Reproduce

herbie shell --seed 2023011 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))