quadp (p42, positive)

Average Accuracy: 47.7% → 82.9%

Time: 21.0s

Precision: binary64

Cost: 27084

Math

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

↓

\[\begin{array}{l} t_0 := 4 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;b \leq -2.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - t_0} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -1.55 \cdot 10^{-235}:\\ \;\;\;\;-0.5 \cdot \frac{b - \mathsf{hypot}\left({\left({\left(4 \cdot c\right)}^{0.25} \cdot {\left(\frac{-1}{a}\right)}^{-0.25}\right)}^{2}, b\right)}{a}\\ \mathbf{elif}\;b \leq 440000000:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{-2}{t_0} \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 4.0 (* a c))))
   (if (<= b -2.7e+104)
     (/ (- b) a)
     (if (<= b -4.2e-162)
       (/ (- (sqrt (- (* b b) t_0)) b) (* a 2.0))
       (if (<= b -1.55e-235)
         (*
          -0.5
          (/
           (-
            b
            (hypot
             (pow (* (pow (* 4.0 c) 0.25) (pow (/ -1.0 a) -0.25)) 2.0)
             b))
           a))
         (if (<= b 440000000.0)
           (/
            1.0
            (* a (* (/ -2.0 t_0) (+ b (hypot b (sqrt (* c (* a -4.0))))))))
           (/ (- c) b)))))))

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) {
	double t_0 = 4.0 * (a * c);
	double tmp;
	if (b <= -2.7e+104) {
		tmp = -b / a;
	} else if (b <= -4.2e-162) {
		tmp = (sqrt(((b * b) - t_0)) - b) / (a * 2.0);
	} else if (b <= -1.55e-235) {
		tmp = -0.5 * ((b - hypot(pow((pow((4.0 * c), 0.25) * pow((-1.0 / a), -0.25)), 2.0), b)) / a);
	} else if (b <= 440000000.0) {
		tmp = 1.0 / (a * ((-2.0 / t_0) * (b + hypot(b, sqrt((c * (a * -4.0)))))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

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) {
	double t_0 = 4.0 * (a * c);
	double tmp;
	if (b <= -2.7e+104) {
		tmp = -b / a;
	} else if (b <= -4.2e-162) {
		tmp = (Math.sqrt(((b * b) - t_0)) - b) / (a * 2.0);
	} else if (b <= -1.55e-235) {
		tmp = -0.5 * ((b - Math.hypot(Math.pow((Math.pow((4.0 * c), 0.25) * Math.pow((-1.0 / a), -0.25)), 2.0), b)) / a);
	} else if (b <= 440000000.0) {
		tmp = 1.0 / (a * ((-2.0 / t_0) * (b + Math.hypot(b, Math.sqrt((c * (a * -4.0)))))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

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

↓

def code(a, b, c):
	t_0 = 4.0 * (a * c)
	tmp = 0
	if b <= -2.7e+104:
		tmp = -b / a
	elif b <= -4.2e-162:
		tmp = (math.sqrt(((b * b) - t_0)) - b) / (a * 2.0)
	elif b <= -1.55e-235:
		tmp = -0.5 * ((b - math.hypot(math.pow((math.pow((4.0 * c), 0.25) * math.pow((-1.0 / a), -0.25)), 2.0), b)) / a)
	elif b <= 440000000.0:
		tmp = 1.0 / (a * ((-2.0 / t_0) * (b + math.hypot(b, math.sqrt((c * (a * -4.0)))))))
	else:
		tmp = -c / b
	return tmp

Julia

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

↓

function code(a, b, c)
	t_0 = Float64(4.0 * Float64(a * c))
	tmp = 0.0
	if (b <= -2.7e+104)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= -4.2e-162)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(a * 2.0));
	elseif (b <= -1.55e-235)
		tmp = Float64(-0.5 * Float64(Float64(b - hypot((Float64((Float64(4.0 * c) ^ 0.25) * (Float64(-1.0 / a) ^ -0.25)) ^ 2.0), b)) / a));
	elseif (b <= 440000000.0)
		tmp = Float64(1.0 / Float64(a * Float64(Float64(-2.0 / t_0) * Float64(b + hypot(b, sqrt(Float64(c * Float64(a * -4.0))))))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, c)
	t_0 = 4.0 * (a * c);
	tmp = 0.0;
	if (b <= -2.7e+104)
		tmp = -b / a;
	elseif (b <= -4.2e-162)
		tmp = (sqrt(((b * b) - t_0)) - b) / (a * 2.0);
	elseif (b <= -1.55e-235)
		tmp = -0.5 * ((b - hypot(((((4.0 * c) ^ 0.25) * ((-1.0 / a) ^ -0.25)) ^ 2.0), b)) / a);
	elseif (b <= 440000000.0)
		tmp = 1.0 / (a * ((-2.0 / t_0) * (b + hypot(b, sqrt((c * (a * -4.0)))))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[a_, b_, c_] := Block[{t$95$0 = N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.7e+104], N[((-b) / a), $MachinePrecision], If[LessEqual[b, -4.2e-162], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.55e-235], N[(-0.5 * N[(N[(b - N[Sqrt[N[Power[N[(N[Power[N[(4.0 * c), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[(-1.0 / a), $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 440000000.0], N[(1.0 / N[(a * N[(N[(-2.0 / t$95$0), $MachinePrecision] * N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]]

Target

Original	47.7%
Target	68.0%
Herbie	82.9%

\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

1. Split input into 5 regimes

2. if b < -2.69999999999999985e104

   1. Initial program 28.2%

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

   2. Simplified 28.1%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]
      Proof

      [Start] 28.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      neg-sub0 [=>] 28.2	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-+l- [=>] 28.2	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 28.2	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 28.2	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      *-commutative [=>] 28.2	\[ \frac{\color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}}{2 \cdot a} \]
      associate-*r/ [<=] 28.1	\[ \color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around -inf 93.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]

   4. Simplified 93.7%

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

      [Start] 93.7	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 93.7	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 93.7	\[ \frac{\color{blue}{-b}}{a} \]

   if -2.69999999999999985e104 < b < -4.2e-162

   1. Initial program 90.4%

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

   if -4.2e-162 < b < -1.55e-235

   1. Initial program 72.1%

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

   2. Simplified 72.0%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]
      Proof

      [Start] 72.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      neg-sub0 [=>] 72.1	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-+l- [=>] 72.1	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 72.1	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 72.1	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      *-commutative [=>] 72.1	\[ \frac{\color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}}{2 \cdot a} \]
      associate-*r/ [<=] 72.0	\[ \color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Applied egg-rr 81.8%

      \[\leadsto \left(b - \color{blue}{\mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}\right) \cdot \frac{-0.5}{a} \]
      Proof

      [Start] 72.0	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      fma-udef [=>] 72.0	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}\right) \cdot \frac{-0.5}{a} \]
      add-sqr-sqrt [=>] 72.0	\[ \left(b - \sqrt{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      hypot-def [=>] 81.8	\[ \left(b - \color{blue}{\mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}\right) \cdot \frac{-0.5}{a} \]

   4. Simplified 81.8%

      \[\leadsto \left(b - \color{blue}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\right) \cdot \frac{-0.5}{a} \]
      Proof

      [Start] 81.8	\[ \left(b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot \frac{-0.5}{a} \]
      associate-*r* [=>] 81.8	\[ \left(b - \mathsf{hypot}\left(\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}, b\right)\right) \cdot \frac{-0.5}{a} \]
      *-commutative [<=] 81.8	\[ \left(b - \mathsf{hypot}\left(\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}, b\right)\right) \cdot \frac{-0.5}{a} \]
      associate-*l* [=>] 81.8	\[ \left(b - \mathsf{hypot}\left(\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}, b\right)\right) \cdot \frac{-0.5}{a} \]

   5. Applied egg-rr 81.6%

      \[\leadsto \left(b - \mathsf{hypot}\left(\color{blue}{{\left({\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}\right)}^{2}}, b\right)\right) \cdot \frac{-0.5}{a} \]
      Proof

      [Start] 81.8	\[ \left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right) \cdot \frac{-0.5}{a} \]
      add-sqr-sqrt [=>] 81.6	\[ \left(b - \mathsf{hypot}\left(\color{blue}{\sqrt{\sqrt{c \cdot \left(a \cdot -4\right)}} \cdot \sqrt{\sqrt{c \cdot \left(a \cdot -4\right)}}}, b\right)\right) \cdot \frac{-0.5}{a} \]
      pow2 [=>] 81.6	\[ \left(b - \mathsf{hypot}\left(\color{blue}{{\left(\sqrt{\sqrt{c \cdot \left(a \cdot -4\right)}}\right)}^{2}}, b\right)\right) \cdot \frac{-0.5}{a} \]
      pow1/2 [=>] 81.6	\[ \left(b - \mathsf{hypot}\left({\left(\sqrt{\color{blue}{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.5}}}\right)}^{2}, b\right)\right) \cdot \frac{-0.5}{a} \]
      sqrt-pow1 [=>] 81.6	\[ \left(b - \mathsf{hypot}\left({\color{blue}{\left({\left(c \cdot \left(a \cdot -4\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}, b\right)\right) \cdot \frac{-0.5}{a} \]
      metadata-eval [=>] 81.6	\[ \left(b - \mathsf{hypot}\left({\left({\left(c \cdot \left(a \cdot -4\right)\right)}^{\color{blue}{0.25}}\right)}^{2}, b\right)\right) \cdot \frac{-0.5}{a} \]

   6. Taylor expanded in a around -inf 32.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b - \sqrt{{b}^{2} + {\left(e^{0.25 \cdot \left(\log \left(4 \cdot c\right) + -1 \cdot \log \left(\frac{-1}{a}\right)\right)}\right)}^{4}}}{a}} \]

   7. Simplified 47.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b - \mathsf{hypot}\left({\left({\left(c \cdot 4\right)}^{0.25} \cdot {\left(\frac{-1}{a}\right)}^{-0.25}\right)}^{2}, b\right)}{a}} \]
      Proof

      [Start] 32.4	\[ -0.5 \cdot \frac{b - \sqrt{{b}^{2} + {\left(e^{0.25 \cdot \left(\log \left(4 \cdot c\right) + -1 \cdot \log \left(\frac{-1}{a}\right)\right)}\right)}^{4}}}{a} \]

   if -1.55e-235 < b < 4.4e8

   1. Initial program 62.0%

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

   2. Simplified 61.9%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]
      Proof

      [Start] 62.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      neg-sub0 [=>] 62.0	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-+l- [=>] 62.0	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 62.0	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 62.0	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      *-commutative [=>] 62.0	\[ \frac{\color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}}{2 \cdot a} \]
      associate-*r/ [<=] 61.9	\[ \color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Applied egg-rr 61.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \cdot -0.5}}} \]
      Proof

      [Start] 61.9	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      associate-*r/ [=>] 62.0	\[ \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot -0.5}{a}} \]
      clear-num [=>] 61.9	\[ \color{blue}{\frac{1}{\frac{a}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot -0.5}}} \]
      fma-udef [=>] 61.9	\[ \frac{1}{\frac{a}{\left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}\right) \cdot -0.5}} \]
      +-commutative [=>] 61.9	\[ \frac{1}{\frac{a}{\left(b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right) \cdot -0.5}} \]
      add-sqr-sqrt [=>] 61.2	\[ \frac{1}{\frac{a}{\left(b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}\right) \cdot -0.5}} \]
      hypot-def [=>] 61.3	\[ \frac{1}{\frac{a}{\left(b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right) \cdot -0.5}} \]

   4. Applied egg-rr 61.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{-2}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)} \cdot a}} \]
      Proof

      [Start] 61.3	\[ \frac{1}{\frac{a}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \cdot -0.5}} \]
      clear-num [=>] 61.3	\[ \frac{1}{\color{blue}{\frac{1}{\frac{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \cdot -0.5}{a}}}} \]
      associate-/r/ [=>] 61.3	\[ \frac{1}{\color{blue}{\frac{1}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \cdot -0.5} \cdot a}} \]
      *-commutative [=>] 61.3	\[ \frac{1}{\frac{1}{\color{blue}{-0.5 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}} \cdot a} \]
      associate-/r* [=>] 61.3	\[ \frac{1}{\color{blue}{\frac{\frac{1}{-0.5}}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}} \cdot a} \]
      metadata-eval [=>] 61.3	\[ \frac{1}{\frac{\color{blue}{-2}}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot a} \]
      hypot-udef [=>] 61.1	\[ \frac{1}{\frac{-2}{b - \color{blue}{\sqrt{b \cdot b + \sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}} \cdot a} \]
      add-sqr-sqrt [<=] 61.9	\[ \frac{1}{\frac{-2}{b - \sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -4\right)}}} \cdot a} \]
      +-commutative [=>] 61.9	\[ \frac{1}{\frac{-2}{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}} \cdot a} \]
      add-sqr-sqrt [=>] 61.1	\[ \frac{1}{\frac{-2}{b - \sqrt{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}} + b \cdot b}} \cdot a} \]
      hypot-def [=>] 61.3	\[ \frac{1}{\frac{-2}{b - \color{blue}{\mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}} \cdot a} \]

   5. Applied egg-rr 60.7%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{-2}{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\right)} \cdot a} \]
      Proof

      [Start] 61.3	\[ \frac{1}{\frac{-2}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)} \cdot a} \]
      flip-- [=>] 60.9	\[ \frac{1}{\frac{-2}{\color{blue}{\frac{b \cdot b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right) \cdot \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}}} \cdot a} \]
      associate-/r/ [=>] 60.6	\[ \frac{1}{\color{blue}{\left(\frac{-2}{b \cdot b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right) \cdot \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right)} \cdot a} \]
      hypot-udef [=>] 60.6	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \color{blue}{\sqrt{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)} + b \cdot b}} \cdot \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right) \cdot a} \]
      hypot-udef [=>] 60.6	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \sqrt{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)} + b \cdot b} \cdot \color{blue}{\sqrt{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)} + b \cdot b}}} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right) \cdot a} \]
      add-sqr-sqrt [<=] 60.6	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \color{blue}{\left(\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)} + b \cdot b\right)}} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right) \cdot a} \]
      add-sqr-sqrt [<=] 60.7	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + b \cdot b\right)} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right) \cdot a} \]
      +-commutative [=>] 60.7	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \color{blue}{\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)}} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right) \cdot a} \]
      fma-def [=>] 60.7	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right) \cdot a} \]
      hypot-udef [=>] 60.7	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \left(b + \color{blue}{\sqrt{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)} + b \cdot b}}\right)\right) \cdot a} \]

   6. Simplified 66.5%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{-2}{4 \cdot \left(c \cdot a\right)} \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)\right)} \cdot a} \]
      Proof

      [Start] 60.7	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\right) \cdot a} \]

   if 4.4e8 < b

   1. Initial program 12.0%

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

   2. Simplified 12.0%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]
      Proof

      [Start] 12.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      neg-sub0 [=>] 12.0	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-+l- [=>] 12.0	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 12.0	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 12.0	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      *-commutative [=>] 12.0	\[ \frac{\color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}}{2 \cdot a} \]
      associate-*r/ [<=] 12.0	\[ \color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around inf 91.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   4. Simplified 91.5%

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

      [Start] 91.5	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 91.5	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      mul-1-neg [=>] 91.5	\[ \frac{\color{blue}{-c}}{b} \]

3. Recombined 5 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -1.55 \cdot 10^{-235}:\\ \;\;\;\;-0.5 \cdot \frac{b - \mathsf{hypot}\left({\left({\left(4 \cdot c\right)}^{0.25} \cdot {\left(\frac{-1}{a}\right)}^{-0.25}\right)}^{2}, b\right)}{a}\\ \mathbf{elif}\;b \leq 440000000:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{-2}{4 \cdot \left(a \cdot c\right)} \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	82.8%
Cost	26956

\[\begin{array}{l} t_0 := 4 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - t_0} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -1.7 \cdot 10^{-235}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left({\left({\left(a \cdot -4\right)}^{0.25} \cdot {c}^{0.25}\right)}^{2}, b\right)\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq 3300000000:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{-2}{t_0} \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Accuracy	82.9%
Cost	20364

\[\begin{array}{l} t_0 := 4 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;b \leq -1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - t_0} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -1.95 \cdot 10^{-237}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(\sqrt{a \cdot -4} \cdot \sqrt{c}, b\right)\right)\\ \mathbf{elif}\;b \leq 3500000:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{-2}{t_0} \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Accuracy	83.9%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+103}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Accuracy	78.1%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Accuracy	37.3%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1450000:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6
Accuracy	63.9%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{-257}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7
Accuracy	2.6%
Cost	192

\[\frac{b}{a} \]

Alternative 8
Accuracy	11.3%
Cost	192

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

Reproduce

herbie shell --seed 2023151 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))