quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.6% → 87.5%

Time: 11.4s

Precision: binary64

Cost: 7820

Math

\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

↓

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - a \cdot c}\\ \mathbf{if}\;b_2 \leq -1.7 \cdot 10^{+128}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.25 \cdot 10^{-146}:\\ \;\;\;\;\frac{t_0 - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.16 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{a \cdot \left(-c\right)}{b_2 + t_0}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

↓

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* a c)))))
   (if (<= b_2 -1.7e+128)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
     (if (<= b_2 1.25e-146)
       (/ (- t_0 b_2) a)
       (if (<= b_2 1.16e+94)
         (/ (/ (* a (- c)) (+ b_2 t_0)) a)
         (* (/ c b_2) -0.5))))))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

double code(double a, double b_2, double c) {
	double t_0 = sqrt(((b_2 * b_2) - (a * c)));
	double tmp;
	if (b_2 <= -1.7e+128) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.25e-146) {
		tmp = (t_0 - b_2) / a;
	} else if (b_2 <= 1.16e+94) {
		tmp = ((a * -c) / (b_2 + t_0)) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
end function

↓

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (a * c)))
    if (b_2 <= (-1.7d+128)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 1.25d-146) then
        tmp = (t_0 - b_2) / a
    else if (b_2 <= 1.16d+94) then
        tmp = ((a * -c) / (b_2 + t_0)) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(((b_2 * b_2) - (a * c)));
	double tmp;
	if (b_2 <= -1.7e+128) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.25e-146) {
		tmp = (t_0 - b_2) / a;
	} else if (b_2 <= 1.16e+94) {
		tmp = ((a * -c) / (b_2 + t_0)) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

↓

def code(a, b_2, c):
	t_0 = math.sqrt(((b_2 * b_2) - (a * c)))
	tmp = 0
	if b_2 <= -1.7e+128:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 1.25e-146:
		tmp = (t_0 - b_2) / a
	elif b_2 <= 1.16e+94:
		tmp = ((a * -c) / (b_2 + t_0)) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

↓

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))
	tmp = 0.0
	if (b_2 <= -1.7e+128)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.25e-146)
		tmp = Float64(Float64(t_0 - b_2) / a);
	elseif (b_2 <= 1.16e+94)
		tmp = Float64(Float64(Float64(a * Float64(-c)) / Float64(b_2 + t_0)) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
end

↓

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt(((b_2 * b_2) - (a * c)));
	tmp = 0.0;
	if (b_2 <= -1.7e+128)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 1.25e-146)
		tmp = (t_0 - b_2) / a;
	elseif (b_2 <= 1.16e+94)
		tmp = ((a * -c) / (b_2 + t_0)) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -1.7e+128], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.25e-146], N[(N[(t$95$0 - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.16e+94], N[(N[(N[(a * (-c)), $MachinePrecision] / N[(b$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.6999999999999999e128

   1. Initial program 54.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Step-by-step derivation

      [Start] 54.1%	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 54.1%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 54.1%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 98.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -1.6999999999999999e128 < b_2 < 1.24999999999999989e-146

   1. Initial program 82.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Step-by-step derivation

      [Start] 82.4%	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 82.4%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 82.4%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   if 1.24999999999999989e-146 < b_2 < 1.1599999999999999e94

   1. Initial program 45.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 45.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Step-by-step derivation

      [Start] 45.6%	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 45.6%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 45.6%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 45.6%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2}} - b_2}{a} \]
      Step-by-step derivation

      [Start] 45.6%	\[ \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a} \]
      add-sqr-sqrt [=>] 45.5%	\[ \frac{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]
      pow2 [=>] 45.5%	\[ \frac{\color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}} - b_2}{a} \]
      pow1/2 [=>] 45.5%	\[ \frac{{\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2} - b_2}{a} \]
      sqrt-pow1 [=>] 45.5%	\[ \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b_2}{a} \]
      fma-neg [=>] 45.6%	\[ \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]
      *-commutative [=>] 45.6%	\[ \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]
      distribute-rgt-neg-in [=>] 45.6%	\[ \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]
      metadata-eval [=>] 45.6%	\[ \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b_2}{a} \]

   4. Applied egg-rr 45.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(b_2 \cdot b_2 - c \cdot a\right) - b_2 \cdot b_2}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.5} + b_2}}}{a} \]
      Step-by-step derivation

      [Start] 45.6%	\[ \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} - b_2}{a} \]
      flip-- [=>] 45.2%	\[ \frac{\color{blue}{\frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} \cdot {\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} - b_2 \cdot b_2}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} + b_2}}}{a} \]
      pow-pow [=>] 45.5%	\[ \frac{\frac{\color{blue}{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\left(0.25 \cdot 2\right)}} \cdot {\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} - b_2 \cdot b_2}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} + b_2}}{a} \]
      metadata-eval [=>] 45.5%	\[ \frac{\frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.5}} \cdot {\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} - b_2 \cdot b_2}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} + b_2}}{a} \]
      pow-pow [=>] 45.0%	\[ \frac{\frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.5} \cdot \color{blue}{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\left(0.25 \cdot 2\right)}} - b_2 \cdot b_2}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} + b_2}}{a} \]
      metadata-eval [=>] 45.0%	\[ \frac{\frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.5}} - b_2 \cdot b_2}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} + b_2}}{a} \]
      pow-prod-up [=>] 45.3%	\[ \frac{\frac{\color{blue}{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\left(0.5 + 0.5\right)}} - b_2 \cdot b_2}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} + b_2}}{a} \]
      metadata-eval [=>] 45.3%	\[ \frac{\frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1}} - b_2 \cdot b_2}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} + b_2}}{a} \]
      pow1 [<=] 45.3%	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)} - b_2 \cdot b_2}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} + b_2}}{a} \]
      distribute-rgt-neg-out [=>] 45.3%	\[ \frac{\frac{\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right) - b_2 \cdot b_2}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} + b_2}}{a} \]
      fma-neg [<=] 45.3%	\[ \frac{\frac{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right)} - b_2 \cdot b_2}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} + b_2}}{a} \]

   5. Simplified 45.4%

      \[\leadsto \frac{\color{blue}{\frac{b_2 \cdot b_2 - \left(c \cdot a + b_2 \cdot b_2\right)}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]
      Step-by-step derivation

      [Start] 45.4%	\[ \frac{\frac{\left(b_2 \cdot b_2 - c \cdot a\right) - b_2 \cdot b_2}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.5} + b_2}}{a} \]
      unpow2 [<=] 45.4%	\[ \frac{\frac{\left(\color{blue}{{b_2}^{2}} - c \cdot a\right) - b_2 \cdot b_2}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.5} + b_2}}{a} \]
      unpow2 [<=] 45.4%	\[ \frac{\frac{\left({b_2}^{2} - c \cdot a\right) - \color{blue}{{b_2}^{2}}}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.5} + b_2}}{a} \]
      associate--l- [=>] 45.4%	\[ \frac{\frac{\color{blue}{{b_2}^{2} - \left(c \cdot a + {b_2}^{2}\right)}}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.5} + b_2}}{a} \]
      unpow2 [=>] 45.4%	\[ \frac{\frac{\color{blue}{b_2 \cdot b_2} - \left(c \cdot a + {b_2}^{2}\right)}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.5} + b_2}}{a} \]
      unpow2 [=>] 45.4%	\[ \frac{\frac{b_2 \cdot b_2 - \left(c \cdot a + \color{blue}{b_2 \cdot b_2}\right)}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.5} + b_2}}{a} \]
      +-commutative [=>] 45.4%	\[ \frac{\frac{b_2 \cdot b_2 - \left(c \cdot a + b_2 \cdot b_2\right)}{\color{blue}{b_2 + {\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.5}}}}{a} \]
      unpow1/2 [=>] 45.4%	\[ \frac{\frac{b_2 \cdot b_2 - \left(c \cdot a + b_2 \cdot b_2\right)}{b_2 + \color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]

   6. Applied egg-rr 3.1%

      \[\leadsto \frac{\frac{\color{blue}{\log \left(\frac{e^{b_2 \cdot b_2 - c \cdot a}}{{\left(e^{b_2}\right)}^{b_2}}\right)}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      Step-by-step derivation

      [Start] 45.4%	\[ \frac{\frac{b_2 \cdot b_2 - \left(c \cdot a + b_2 \cdot b_2\right)}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      associate--r+ [=>] 45.4%	\[ \frac{\frac{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) - b_2 \cdot b_2}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      add-log-exp [=>] 4.0%	\[ \frac{\frac{\color{blue}{\log \left(e^{b_2 \cdot b_2 - c \cdot a}\right)} - b_2 \cdot b_2}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      add-log-exp [=>] 3.1%	\[ \frac{\frac{\log \left(e^{b_2 \cdot b_2 - c \cdot a}\right) - \color{blue}{\log \left(e^{b_2 \cdot b_2}\right)}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      diff-log [=>] 3.1%	\[ \frac{\frac{\color{blue}{\log \left(\frac{e^{b_2 \cdot b_2 - c \cdot a}}{e^{b_2 \cdot b_2}}\right)}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      add-log-exp [=>] 3.1%	\[ \frac{\frac{\log \left(\frac{e^{b_2 \cdot b_2 - c \cdot a}}{e^{\color{blue}{\log \left(e^{b_2}\right)} \cdot b_2}}\right)}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      exp-to-pow [=>] 3.1%	\[ \frac{\frac{\log \left(\frac{e^{b_2 \cdot b_2 - c \cdot a}}{\color{blue}{{\left(e^{b_2}\right)}^{b_2}}}\right)}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]

   7. Simplified 80.1%

      \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(-a\right) + 0}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      Step-by-step derivation

      [Start] 3.1%	\[ \frac{\frac{\log \left(\frac{e^{b_2 \cdot b_2 - c \cdot a}}{{\left(e^{b_2}\right)}^{b_2}}\right)}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      log-div [=>] 3.1%	\[ \frac{\frac{\color{blue}{\log \left(e^{b_2 \cdot b_2 - c \cdot a}\right) - \log \left({\left(e^{b_2}\right)}^{b_2}\right)}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      rem-log-exp [=>] 29.0%	\[ \frac{\frac{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right)} - \log \left({\left(e^{b_2}\right)}^{b_2}\right)}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      log-pow [=>] 29.2%	\[ \frac{\frac{\left(b_2 \cdot b_2 - c \cdot a\right) - \color{blue}{b_2 \cdot \log \left(e^{b_2}\right)}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      rem-log-exp [=>] 45.4%	\[ \frac{\frac{\left(b_2 \cdot b_2 - c \cdot a\right) - b_2 \cdot \color{blue}{b_2}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      unsub-neg [<=] 45.4%	\[ \frac{\frac{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + \left(-b_2 \cdot b_2\right)}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      sub-neg [=>] 45.4%	\[ \frac{\frac{\color{blue}{\left(b_2 \cdot b_2 + \left(-c \cdot a\right)\right)} + \left(-b_2 \cdot b_2\right)}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      +-commutative [<=] 45.4%	\[ \frac{\frac{\color{blue}{\left(\left(-c \cdot a\right) + b_2 \cdot b_2\right)} + \left(-b_2 \cdot b_2\right)}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      associate-+l+ [=>] 80.1%	\[ \frac{\frac{\color{blue}{\left(-c \cdot a\right) + \left(b_2 \cdot b_2 + \left(-b_2 \cdot b_2\right)\right)}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      distribute-rgt-neg-in [=>] 80.1%	\[ \frac{\frac{\color{blue}{c \cdot \left(-a\right)} + \left(b_2 \cdot b_2 + \left(-b_2 \cdot b_2\right)\right)}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      sub-neg [<=] 80.1%	\[ \frac{\frac{c \cdot \left(-a\right) + \color{blue}{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      +-inverses [=>] 80.1%	\[ \frac{\frac{c \cdot \left(-a\right) + \color{blue}{0}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]

   if 1.1599999999999999e94 < b_2

   1. Initial program 8.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 8.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Step-by-step derivation

      [Start] 8.4%	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 8.4%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 8.4%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 98.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.7 \cdot 10^{+128}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.25 \cdot 10^{-146}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.16 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{a \cdot \left(-c\right)}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.5%
Cost	7820

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - a \cdot c}\\ \mathbf{if}\;b_2 \leq -1.7 \cdot 10^{+128}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.25 \cdot 10^{-146}:\\ \;\;\;\;\frac{t_0 - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.16 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{a \cdot \left(-c\right)}{b_2 + t_0}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 2
Accuracy	84.9%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.65 \cdot 10^{+129}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;{\left(0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}\right)}^{-1}\\ \end{array} \]

Alternative 3
Accuracy	85.2%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{+126}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 4
Accuracy	79.7%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.9 \cdot 10^{-25}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 5
Accuracy	68.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 6
Accuracy	47.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 5.5 \cdot 10^{-270}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 7
Accuracy	68.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 3.7 \cdot 10^{-270}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 8
Accuracy	68.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 4.5 \cdot 10^{-270}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 9
Accuracy	23.4%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-288}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 10
Accuracy	10.7%
Cost	192

\[\frac{0}{a} \]

Reproduce

herbie shell --seed 2023167 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))