Quadratic roots, wide range

Average Accuracy: 18.3% → 99.4%

Time: 17.2s

Precision: binary64

Cost: 15040

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

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

↓

\[\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{\frac{c \cdot \left(c \cdot \left(\left(a \cdot a\right) \cdot 16\right)\right) - {b}^{4}}{\left(c \cdot a\right) \cdot -4 - b \cdot b}}}}{a \cdot 2} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (* 4.0 (* c a))
   (-
    (- b)
    (sqrt
     (/
      (- (* c (* c (* (* a a) 16.0))) (pow b 4.0))
      (- (* (* c a) -4.0) (* b b))))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	return ((4.0 * (c * a)) / (-b - sqrt((((c * (c * ((a * a) * 16.0))) - pow(b, 4.0)) / (((c * a) * -4.0) - (b * b)))))) / (a * 2.0);
}

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) - ((4.0d0 * a) * c)))) / (2.0d0 * 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
    code = ((4.0d0 * (c * a)) / (-b - sqrt((((c * (c * ((a * a) * 16.0d0))) - (b ** 4.0d0)) / (((c * a) * (-4.0d0)) - (b * b)))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

↓

public static double code(double a, double b, double c) {
	return ((4.0 * (c * a)) / (-b - Math.sqrt((((c * (c * ((a * a) * 16.0))) - Math.pow(b, 4.0)) / (((c * a) * -4.0) - (b * b)))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

↓

def code(a, b, c):
	return ((4.0 * (c * a)) / (-b - math.sqrt((((c * (c * ((a * a) * 16.0))) - math.pow(b, 4.0)) / (((c * a) * -4.0) - (b * b)))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	return Float64(Float64(Float64(4.0 * Float64(c * a)) / Float64(Float64(-b) - sqrt(Float64(Float64(Float64(c * Float64(c * Float64(Float64(a * a) * 16.0))) - (b ^ 4.0)) / Float64(Float64(Float64(c * a) * -4.0) - Float64(b * b)))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

↓

function tmp = code(a, b, c)
	tmp = ((4.0 * (c * a)) / (-b - sqrt((((c * (c * ((a * a) * 16.0))) - (b ^ 4.0)) / (((c * a) * -4.0) - (b * b)))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := N[(N[(N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(N[(c * N[(c * N[(N[(a * a), $MachinePrecision] * 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

2. Simplified 18.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   Proof

   [Start] 18.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 18.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 18.7%

   \[\leadsto \frac{\color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
   Proof

   [Start] 18.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   flip-+ [=>] 18.3	\[ \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]
   sub-neg [=>] 18.3	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\left(-b\right) + \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{a \cdot 2} \]
   add-sqr-sqrt [=>] 18.3	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-\color{blue}{\sqrt{b} \cdot \sqrt{b}}\right) + \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]
   sqrt-prod [<=] 18.3	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-\color{blue}{\sqrt{b \cdot b}}\right) + \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]
   sqr-neg [<=] 18.3	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}\right) + \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]
   sqrt-unprod [<=] 0.0	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}\right) + \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]
   add-sqr-sqrt [<=] 0.4	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-\color{blue}{\left(-b\right)}\right) + \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]
   distribute-neg-in [<=] 0.4	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{-\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{a \cdot 2} \]
   add-sqr-sqrt [=>] 0.0	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{-\color{blue}{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{a \cdot 2} \]
   distribute-rgt-neg-in [=>] 0.0	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(-\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}}}{a \cdot 2} \]

4. Simplified 18.8%

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}}{a \cdot 2} \]
   Proof

   [Start] 18.7	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   associate-/l/ [=>] 18.7	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
   /-rgt-identity [<=] 18.7	\[ \frac{\frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{1}}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   /-rgt-identity [=>] 18.7	\[ \frac{\frac{\color{blue}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [<=] 18.8	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 18.8	\[ \frac{\frac{b \cdot b - \color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [=>] 18.8	\[ \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   distribute-lft-neg-in [<=] 18.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{\color{blue}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
   rem-square-sqrt [=>] 18.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}}{a \cdot 2} \]
   *-lft-identity [<=] 18.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\color{blue}{1 \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}}{a \cdot 2} \]
   *-lft-identity [=>] 18.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}}{a \cdot 2} \]

5. Taylor expanded in b around 0 99.4%

   \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{-\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}{a \cdot 2} \]

6. Applied egg-rr 99.4%

   \[\leadsto \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\color{blue}{\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \left(c \cdot \left(a \cdot -4\right)\right) - {b}^{4}}{c \cdot \left(a \cdot -4\right) - b \cdot b}}}\right)}}{a \cdot 2} \]
   Proof

   [Start] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}{a \cdot 2} \]
   fma-udef [=>] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}\right)}}{a \cdot 2} \]
   flip-+ [=>] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\color{blue}{\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \left(c \cdot \left(a \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{c \cdot \left(a \cdot -4\right) - b \cdot b}}}\right)}}{a \cdot 2} \]
   pow2 [=>] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \left(c \cdot \left(a \cdot -4\right)\right) - \color{blue}{{b}^{2}} \cdot \left(b \cdot b\right)}{c \cdot \left(a \cdot -4\right) - b \cdot b}}\right)}}{a \cdot 2} \]
   pow2 [=>] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \left(c \cdot \left(a \cdot -4\right)\right) - {b}^{2} \cdot \color{blue}{{b}^{2}}}{c \cdot \left(a \cdot -4\right) - b \cdot b}}\right)}}{a \cdot 2} \]
   pow-sqr [=>] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \left(c \cdot \left(a \cdot -4\right)\right) - \color{blue}{{b}^{\left(2 \cdot 2\right)}}}{c \cdot \left(a \cdot -4\right) - b \cdot b}}\right)}}{a \cdot 2} \]
   metadata-eval [=>] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \left(c \cdot \left(a \cdot -4\right)\right) - {b}^{\color{blue}{4}}}{c \cdot \left(a \cdot -4\right) - b \cdot b}}\right)}}{a \cdot 2} \]

7. Simplified 99.4%

   \[\leadsto \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\color{blue}{\frac{c \cdot \left(c \cdot \left(\left(a \cdot a\right) \cdot 16\right)\right) - {b}^{4}}{\left(c \cdot a\right) \cdot -4 - b \cdot b}}}\right)}}{a \cdot 2} \]
   Proof

   [Start] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \left(c \cdot \left(a \cdot -4\right)\right) - {b}^{4}}{c \cdot \left(a \cdot -4\right) - b \cdot b}}\right)}}{a \cdot 2} \]
   swap-sqr [=>] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\color{blue}{\left(c \cdot c\right) \cdot \left(\left(a \cdot -4\right) \cdot \left(a \cdot -4\right)\right)} - {b}^{4}}{c \cdot \left(a \cdot -4\right) - b \cdot b}}\right)}}{a \cdot 2} \]
   associate-*l* [=>] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\color{blue}{c \cdot \left(c \cdot \left(\left(a \cdot -4\right) \cdot \left(a \cdot -4\right)\right)\right)} - {b}^{4}}{c \cdot \left(a \cdot -4\right) - b \cdot b}}\right)}}{a \cdot 2} \]
   swap-sqr [=>] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{c \cdot \left(c \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(-4 \cdot -4\right)\right)}\right) - {b}^{4}}{c \cdot \left(a \cdot -4\right) - b \cdot b}}\right)}}{a \cdot 2} \]
   metadata-eval [=>] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{c \cdot \left(c \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{16}\right)\right) - {b}^{4}}{c \cdot \left(a \cdot -4\right) - b \cdot b}}\right)}}{a \cdot 2} \]
   associate-*r* [=>] 99.4	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{c \cdot \left(c \cdot \left(\left(a \cdot a\right) \cdot 16\right)\right) - {b}^{4}}{\color{blue}{\left(c \cdot a\right) \cdot -4} - b \cdot b}}\right)}}{a \cdot 2} \]

8. Final simplification 99.4%

   \[\leadsto \frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{\frac{c \cdot \left(c \cdot \left(\left(a \cdot a\right) \cdot 16\right)\right) - {b}^{4}}{\left(c \cdot a\right) \cdot -4 - b \cdot b}}}}{a \cdot 2} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	14080

\[\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2} \]

Alternative 2
Accuracy	99.4%
Cost	7744

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \frac{t_0}{a \cdot \left(b + \sqrt{b \cdot b + t_0}\right)} \cdot 0.5 \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	7744

\[\frac{-4 \cdot \frac{c \cdot a}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}}{a \cdot 2} \]

Alternative 4
Accuracy	95.1%
Cost	7232

\[\frac{-c}{b} - a \cdot \frac{c}{\frac{{b}^{3}}{c}} \]

Alternative 5
Accuracy	95.0%
Cost	1600

\[\frac{-2 \cdot \left(a \cdot \frac{c}{b} + \left(a \cdot a\right) \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\right)}{a \cdot 2} \]

Alternative 6
Accuracy	94.9%
Cost	1344

\[\frac{\frac{4 \cdot \left(c \cdot a\right)}{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}}{a \cdot 2} \]

Alternative 7
Accuracy	90.1%
Cost	256

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

Alternative 8
Accuracy	1.7%
Cost	192

\[\frac{c}{b} \]

Reproduce

herbie shell --seed 2023138 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))