quadp (p42, positive)

Average Accuracy: 46.5% → 87.0%

Time: 23.6s

Precision: binary64

Cost: 14220

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+117}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -2.7 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 0.14:\\ \;\;\;\;\frac{c}{\frac{a}{a}} \cdot \frac{-2}{b + \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - c \cdot \frac{a}{\frac{{b}^{3}}{c}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+117)
   (- (/ c b) (/ b a))
   (if (<= b -2.7e-266)
     (/ (- (sqrt (+ (* b b) (* (* c a) -4.0))) b) (* a 2.0))
     (if (<= b 0.14)
       (* (/ c (/ a a)) (/ -2.0 (+ b (hypot (sqrt (* c (* a -4.0))) b))))
       (- (/ (- c) b) (* c (/ a (/ (pow b 3.0) c))))))))

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 tmp;
	if (b <= -5e+117) {
		tmp = (c / b) - (b / a);
	} else if (b <= -2.7e-266) {
		tmp = (sqrt(((b * b) + ((c * a) * -4.0))) - b) / (a * 2.0);
	} else if (b <= 0.14) {
		tmp = (c / (a / a)) * (-2.0 / (b + hypot(sqrt((c * (a * -4.0))), b)));
	} else {
		tmp = (-c / b) - (c * (a / (pow(b, 3.0) / c)));
	}
	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 tmp;
	if (b <= -5e+117) {
		tmp = (c / b) - (b / a);
	} else if (b <= -2.7e-266) {
		tmp = (Math.sqrt(((b * b) + ((c * a) * -4.0))) - b) / (a * 2.0);
	} else if (b <= 0.14) {
		tmp = (c / (a / a)) * (-2.0 / (b + Math.hypot(Math.sqrt((c * (a * -4.0))), b)));
	} else {
		tmp = (-c / b) - (c * (a / (Math.pow(b, 3.0) / c)));
	}
	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):
	tmp = 0
	if b <= -5e+117:
		tmp = (c / b) - (b / a)
	elif b <= -2.7e-266:
		tmp = (math.sqrt(((b * b) + ((c * a) * -4.0))) - b) / (a * 2.0)
	elif b <= 0.14:
		tmp = (c / (a / a)) * (-2.0 / (b + math.hypot(math.sqrt((c * (a * -4.0))), b)))
	else:
		tmp = (-c / b) - (c * (a / (math.pow(b, 3.0) / c)))
	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)
	tmp = 0.0
	if (b <= -5e+117)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -2.7e-266)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(Float64(c * a) * -4.0))) - b) / Float64(a * 2.0));
	elseif (b <= 0.14)
		tmp = Float64(Float64(c / Float64(a / a)) * Float64(-2.0 / Float64(b + hypot(sqrt(Float64(c * Float64(a * -4.0))), b))));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(c * Float64(a / Float64((b ^ 3.0) / c))));
	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)
	tmp = 0.0;
	if (b <= -5e+117)
		tmp = (c / b) - (b / a);
	elseif (b <= -2.7e-266)
		tmp = (sqrt(((b * b) + ((c * a) * -4.0))) - b) / (a * 2.0);
	elseif (b <= 0.14)
		tmp = (c / (a / a)) * (-2.0 / (b + hypot(sqrt((c * (a * -4.0))), b)));
	else
		tmp = (-c / b) - (c * (a / ((b ^ 3.0) / c)));
	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_] := If[LessEqual[b, -5e+117], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -2.7e-266], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.14], N[(N[(c / N[(a / a), $MachinePrecision]), $MachinePrecision] * N[(-2.0 / N[(b + N[Sqrt[N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(c * N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	46.5%
Target	66.9%
Herbie	87.0%

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

Derivation

1. Split input into 4 regimes

2. if b < -4.99999999999999983e117

   1. Initial program 19.4%

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

   2. Taylor expanded in b around -inf 95.9%

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

   3. Simplified 95.9%

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

      [Start] 95.9	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 95.9	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 95.9	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -4.99999999999999983e117 < b < -2.69999999999999996e-266

   1. Initial program 87.4%

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

   if -2.69999999999999996e-266 < b < 0.14000000000000001

   1. Initial program 60.4%

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

   2. Simplified 60.3%

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

      [Start] 60.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 60.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 60.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [<=] 60.4	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 60.4	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      +-commutative [=>] 60.4	\[ \color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]
      unsub-neg [=>] 60.4	\[ \color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)} \cdot \frac{--1}{2 \cdot a} \]
      fma-neg [=>] 60.4	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      *-commutative [=>] 60.4	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      distribute-rgt-neg-in [=>] 60.4	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      associate-*l* [=>] 60.3	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot \left(-4\right)\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      metadata-eval [=>] 60.3	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-4}\right)\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      associate-/r* [=>] 60.3	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]
      metadata-eval [=>] 60.3	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]
      metadata-eval [=>] 60.3	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Applied egg-rr 59.3%

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

      [Start] 60.3	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a} \]
      flip-- [=>] 60.2	\[ \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} + b}} \cdot \frac{0.5}{a} \]
      div-inv [=>] 60.1	\[ \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b \cdot b\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} + b}\right)} \cdot \frac{0.5}{a} \]
      add-sqr-sqrt [<=] 60.2	\[ \left(\left(\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b \cdot b\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} + b}\right) \cdot \frac{0.5}{a} \]
      fma-udef [=>] 60.2	\[ \left(\left(\color{blue}{\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)} - b \cdot b\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} + b}\right) \cdot \frac{0.5}{a} \]
      +-commutative [=>] 60.2	\[ \left(\left(\color{blue}{\left(a \cdot \left(c \cdot -4\right) + b \cdot b\right)} - b \cdot b\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} + b}\right) \cdot \frac{0.5}{a} \]
      fma-def [=>] 60.2	\[ \left(\left(\color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b \cdot b\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} + b}\right) \cdot \frac{0.5}{a} \]
      +-commutative [=>] 60.2	\[ \left(\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b\right) \cdot \frac{1}{\color{blue}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}\right) \cdot \frac{0.5}{a} \]
      fma-udef [=>] 60.2	\[ \left(\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b\right) \cdot \frac{1}{b + \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}}\right) \cdot \frac{0.5}{a} \]
      add-sqr-sqrt [=>] 59.3	\[ \left(\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b\right) \cdot \frac{1}{b + \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}}\right) \cdot \frac{0.5}{a} \]
      hypot-def [=>] 59.3	\[ \left(\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b\right) \cdot \frac{1}{b + \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}\right) \cdot \frac{0.5}{a} \]

   4. Simplified 65.3%

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

      [Start] 59.3	\[ \left(\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b\right) \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right) \cdot \frac{0.5}{a} \]
      associate-*r/ [=>] 59.3	\[ \color{blue}{\frac{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b\right) \cdot 1}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}} \cdot \frac{0.5}{a} \]
      *-rgt-identity [=>] 59.3	\[ \frac{\color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{0.5}{a} \]
      fma-def [<=] 59.3	\[ \frac{\color{blue}{\left(a \cdot \left(c \cdot -4\right) + b \cdot b\right)} - b \cdot b}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{0.5}{a} \]
      associate--l+ [=>] 65.3	\[ \frac{\color{blue}{a \cdot \left(c \cdot -4\right) + \left(b \cdot b - b \cdot b\right)}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{0.5}{a} \]
      +-inverses [=>] 65.3	\[ \frac{a \cdot \left(c \cdot -4\right) + \color{blue}{0}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{0.5}{a} \]
      +-rgt-identity [=>] 65.3	\[ \frac{\color{blue}{a \cdot \left(c \cdot -4\right)}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{0.5}{a} \]
      associate-*r* [=>] 65.3	\[ \frac{\color{blue}{\left(a \cdot c\right) \cdot -4}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{0.5}{a} \]
      *-commutative [<=] 65.3	\[ \frac{\color{blue}{\left(c \cdot a\right)} \cdot -4}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{0.5}{a} \]
      associate-*l* [=>] 65.2	\[ \frac{\color{blue}{c \cdot \left(a \cdot -4\right)}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{0.5}{a} \]
      associate-*r* [=>] 65.3	\[ \frac{c \cdot \left(a \cdot -4\right)}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right)} \cdot \frac{0.5}{a} \]
      *-commutative [<=] 65.3	\[ \frac{c \cdot \left(a \cdot -4\right)}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}\right)} \cdot \frac{0.5}{a} \]
      associate-*l* [=>] 65.3	\[ \frac{c \cdot \left(a \cdot -4\right)}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}\right)} \cdot \frac{0.5}{a} \]

   5. Applied egg-rr 56.2%

      \[\leadsto \color{blue}{0 + \frac{\left(c \cdot a\right) \cdot -2}{a \cdot \left(b + \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)}} \]
      Proof

      [Start] 65.3	\[ \frac{c \cdot \left(a \cdot -4\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \frac{0.5}{a} \]
      add-log-exp [=>] 4.3	\[ \color{blue}{\log \left(e^{\frac{c \cdot \left(a \cdot -4\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \frac{0.5}{a}}\right)} \]
      *-un-lft-identity [=>] 4.3	\[ \log \color{blue}{\left(1 \cdot e^{\frac{c \cdot \left(a \cdot -4\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \frac{0.5}{a}}\right)} \]
      log-prod [=>] 4.3	\[ \color{blue}{\log 1 + \log \left(e^{\frac{c \cdot \left(a \cdot -4\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \frac{0.5}{a}}\right)} \]
      metadata-eval [=>] 4.3	\[ \color{blue}{0} + \log \left(e^{\frac{c \cdot \left(a \cdot -4\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \frac{0.5}{a}}\right) \]
      add-log-exp [<=] 65.3	\[ 0 + \color{blue}{\frac{c \cdot \left(a \cdot -4\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \frac{0.5}{a}} \]
      frac-times [=>] 56.2	\[ 0 + \color{blue}{\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot 0.5}{\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot a}} \]
      associate-*r* [=>] 56.2	\[ 0 + \frac{\color{blue}{\left(\left(c \cdot a\right) \cdot -4\right)} \cdot 0.5}{\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot a} \]
      associate-*l* [=>] 56.2	\[ 0 + \frac{\color{blue}{\left(c \cdot a\right) \cdot \left(-4 \cdot 0.5\right)}}{\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot a} \]
      metadata-eval [=>] 56.2	\[ 0 + \frac{\left(c \cdot a\right) \cdot \color{blue}{-2}}{\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot a} \]
      *-commutative [=>] 56.2	\[ 0 + \frac{\left(c \cdot a\right) \cdot -2}{\color{blue}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)}} \]
      hypot-udef [=>] 56.2	\[ 0 + \frac{\left(c \cdot a\right) \cdot -2}{a \cdot \left(b + \color{blue}{\sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{c \cdot \left(a \cdot -4\right)}}}\right)} \]
      add-sqr-sqrt [<=] 62.1	\[ 0 + \frac{\left(c \cdot a\right) \cdot -2}{a \cdot \left(b + \sqrt{b \cdot b + \color{blue}{c \cdot \left(a \cdot -4\right)}}\right)} \]
      +-commutative [=>] 62.1	\[ 0 + \frac{\left(c \cdot a\right) \cdot -2}{a \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}\right)} \]

   6. Simplified 76.9%

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

      [Start] 56.2	\[ 0 + \frac{\left(c \cdot a\right) \cdot -2}{a \cdot \left(b + \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)} \]
      +-lft-identity [=>] 56.2	\[ \color{blue}{\frac{\left(c \cdot a\right) \cdot -2}{a \cdot \left(b + \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)}} \]
      times-frac [=>] 65.4	\[ \color{blue}{\frac{c \cdot a}{a} \cdot \frac{-2}{b + \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}} \]
      associate-/l* [=>] 76.9	\[ \color{blue}{\frac{c}{\frac{a}{a}}} \cdot \frac{-2}{b + \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)} \]

   if 0.14000000000000001 < b

   1. Initial program 13.5%

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

   2. Taylor expanded in b around inf 69.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]

   3. Simplified 73.2%

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

      [Start] 69.7	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b} \]
      +-commutative [=>] 69.7	\[ \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
      mul-1-neg [=>] 69.7	\[ -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
      unsub-neg [=>] 69.7	\[ \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
      associate-*r/ [=>] 69.7	\[ \color{blue}{\frac{-1 \cdot c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]
      neg-mul-1 [<=] 69.7	\[ \frac{\color{blue}{-c}}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]
      *-commutative [=>] 69.7	\[ \frac{-c}{b} - \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}} \]
      associate-/l* [=>] 73.2	\[ \frac{-c}{b} - \color{blue}{\frac{a}{\frac{{b}^{3}}{{c}^{2}}}} \]
      unpow2 [=>] 73.2	\[ \frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{\color{blue}{c \cdot c}}} \]

   4. Applied egg-rr 90.0%

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

      [Start] 73.2	\[ \frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]
      associate-/r* [=>] 89.3	\[ \frac{-c}{b} - \frac{a}{\color{blue}{\frac{\frac{{b}^{3}}{c}}{c}}} \]
      associate-/r/ [=>] 90.0	\[ \frac{-c}{b} - \color{blue}{\frac{a}{\frac{{b}^{3}}{c}} \cdot c} \]

3. Recombined 4 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+117}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -2.7 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 0.14:\\ \;\;\;\;\frac{c}{\frac{a}{a}} \cdot \frac{-2}{b + \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - c \cdot \frac{a}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.0%
Cost	14092

\[\begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{+116}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 0.0175:\\ \;\;\;\;\frac{1}{\frac{b + \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}{c \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - c \cdot \frac{a}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 2
Accuracy	87.0%
Cost	13964

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+117}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -5.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 0.105:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - c \cdot \frac{a}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 3
Accuracy	83.7%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-63}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - c \cdot \frac{a}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 4
Accuracy	84.0%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+117}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - c \cdot \frac{a}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 5
Accuracy	78.7%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-60}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - c \cdot \frac{a}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 6
Accuracy	78.9%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7
Accuracy	79.1%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.72 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8
Accuracy	64.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 9
Accuracy	38.1%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 0.09:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 10
Accuracy	64.4%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-302}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 11
Accuracy	2.6%
Cost	192

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

Alternative 12
Accuracy	11.5%
Cost	192

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

Reproduce

herbie shell --seed 2023136 
(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)))