quadp (p42, positive)

Percentage Accurate: 52.2% → 86.9%

Time: 18.7s

Precision: binary64

Cost: 14476

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+129}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 10^{-300}:\\ \;\;\;\;\frac{\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot -0.5}{a}\\ \mathbf{elif}\;b \leq 200000:\\ \;\;\;\;\frac{0.5}{a \cdot \frac{-1}{\frac{c}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+129)
   (/ (- b) a)
   (if (<= b 1e-300)
     (/ (* (- b (sqrt (- (* b b) (* a (* c 4.0))))) -0.5) a)
     (if (<= b 200000.0)
       (/
        0.5
        (*
         a
         (/ -1.0 (* (/ c (+ b (hypot b (sqrt (* c (* a -4.0)))))) (* 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 tmp;
	if (b <= -5e+129) {
		tmp = -b / a;
	} else if (b <= 1e-300) {
		tmp = ((b - sqrt(((b * b) - (a * (c * 4.0))))) * -0.5) / a;
	} else if (b <= 200000.0) {
		tmp = 0.5 / (a * (-1.0 / ((c / (b + hypot(b, sqrt((c * (a * -4.0)))))) * (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 tmp;
	if (b <= -5e+129) {
		tmp = -b / a;
	} else if (b <= 1e-300) {
		tmp = ((b - Math.sqrt(((b * b) - (a * (c * 4.0))))) * -0.5) / a;
	} else if (b <= 200000.0) {
		tmp = 0.5 / (a * (-1.0 / ((c / (b + Math.hypot(b, Math.sqrt((c * (a * -4.0)))))) * (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):
	tmp = 0
	if b <= -5e+129:
		tmp = -b / a
	elif b <= 1e-300:
		tmp = ((b - math.sqrt(((b * b) - (a * (c * 4.0))))) * -0.5) / a
	elif b <= 200000.0:
		tmp = 0.5 / (a * (-1.0 / ((c / (b + math.hypot(b, math.sqrt((c * (a * -4.0)))))) * (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)
	tmp = 0.0
	if (b <= -5e+129)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1e-300)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 4.0))))) * -0.5) / a);
	elseif (b <= 200000.0)
		tmp = Float64(0.5 / Float64(a * Float64(-1.0 / Float64(Float64(c / Float64(b + hypot(b, sqrt(Float64(c * Float64(a * -4.0)))))) * 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)
	tmp = 0.0;
	if (b <= -5e+129)
		tmp = -b / a;
	elseif (b <= 1e-300)
		tmp = ((b - sqrt(((b * b) - (a * (c * 4.0))))) * -0.5) / a;
	elseif (b <= 200000.0)
		tmp = 0.5 / (a * (-1.0 / ((c / (b + hypot(b, sqrt((c * (a * -4.0)))))) * (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_] := If[LessEqual[b, -5e+129], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1e-300], N[(N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 200000.0], N[(0.5 / N[(a * N[(-1.0 / N[(N[(c / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Target

Original	52.2%
Target	70.5%
Herbie	86.9%

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

Derivation

1. Split input into 4 regimes

2. if b < -5.0000000000000003e129

   1. Initial program 55.9%

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

   2. Simplified 56.1%

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

      [Start] 55.9%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      neg-sub0 [=>] 55.9%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-+l- [=>] 55.9%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 55.9%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 55.9%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      *-commutative [=>] 55.9%	\[ \frac{\color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}}{2 \cdot a} \]
      associate-*r/ [<=] 55.9%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around -inf 98.1%

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

   4. Simplified 98.1%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
      Step-by-step derivation

      [Start] 98.1%	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 98.1%	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 98.1%	\[ \frac{\color{blue}{-b}}{a} \]

   if -5.0000000000000003e129 < b < 1.00000000000000003e-300

   1. Initial program 84.8%

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

   2. Simplified 84.6%

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

      [Start] 84.8%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      neg-sub0 [=>] 84.8%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-+l- [=>] 84.8%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 84.8%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 84.8%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      *-commutative [=>] 84.8%	\[ \frac{\color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}}{2 \cdot a} \]
      associate-*r/ [<=] 84.5%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Applied egg-rr 84.6%

      \[\leadsto \left(b - \sqrt{\color{blue}{b \cdot b - a \cdot \left(c \cdot 4\right)}}\right) \cdot \frac{-0.5}{a} \]
      Step-by-step derivation

      [Start] 84.6%	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      fma-udef [=>] 84.6%	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}\right) \cdot \frac{-0.5}{a} \]
      associate-*r* [=>] 84.5%	\[ \left(b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      metadata-eval [<=] 84.5%	\[ \left(b - \sqrt{\left(a \cdot c\right) \cdot \color{blue}{\left(-4\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      distribute-rgt-neg-in [<=] 84.5%	\[ \left(b - \sqrt{\color{blue}{\left(-\left(a \cdot c\right) \cdot 4\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [<=] 84.5%	\[ \left(b - \sqrt{\left(-\color{blue}{4 \cdot \left(a \cdot c\right)}\right) + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      +-commutative [=>] 84.5%	\[ \left(b - \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \cdot \frac{-0.5}{a} \]
      sub-neg [<=] 84.5%	\[ \left(b - \sqrt{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 84.5%	\[ \left(b - \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 4}}\right) \cdot \frac{-0.5}{a} \]
      associate-*l* [=>] 84.6%	\[ \left(b - \sqrt{b \cdot b - \color{blue}{a \cdot \left(c \cdot 4\right)}}\right) \cdot \frac{-0.5}{a} \]

   4. Applied egg-rr 84.8%

      \[\leadsto \color{blue}{\frac{\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot -0.5}{a}} \]
      Step-by-step derivation

      [Start] 84.6%	\[ \left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot \frac{-0.5}{a} \]
      associate-*r/ [=>] 84.8%	\[ \color{blue}{\frac{\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot -0.5}{a}} \]

   if 1.00000000000000003e-300 < b < 2e5

   1. Initial program 63.2%

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

   2. Simplified 63.1%

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

      [Start] 63.2%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      neg-sub0 [=>] 63.2%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-+l- [=>] 63.2%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 63.2%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 63.2%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      *-commutative [=>] 63.2%	\[ \frac{\color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}}{2 \cdot a} \]
      associate-*r/ [<=] 63.1%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Applied egg-rr 63.2%

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

      [Start] 63.1%	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      associate-*r/ [=>] 63.2%	\[ \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot -0.5}{a}} \]
      frac-2neg [=>] 63.2%	\[ \color{blue}{\frac{-\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot -0.5}{-a}} \]

   4. Simplified 63.2%

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

      [Start] 63.2%	\[ \frac{-\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot -0.5}{-a} \]
      distribute-rgt-neg-in [=>] 63.2%	\[ \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \left(--0.5\right)}}{-a} \]
      metadata-eval [=>] 63.2%	\[ \frac{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \color{blue}{0.5}}{-a} \]
      *-commutative [=>] 63.2%	\[ \frac{\color{blue}{0.5 \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}}{-a} \]
      associate-/l* [=>] 63.2%	\[ \color{blue}{\frac{0.5}{\frac{-a}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}} \]
      fma-def [<=] 63.2%	\[ \frac{0.5}{\frac{-a}{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}}} \]
      +-commutative [<=] 63.2%	\[ \frac{0.5}{\frac{-a}{b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}}} \]
      fma-def [=>] 63.2%	\[ \frac{0.5}{\frac{-a}{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}} \]

   5. Applied egg-rr 62.2%

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

      [Start] 63.2%	\[ \frac{0.5}{\frac{-a}{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}} \]
      div-inv [=>] 63.1%	\[ \frac{0.5}{\color{blue}{\left(-a\right) \cdot \frac{1}{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}} \]
      fma-udef [=>] 63.1%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}}} \]
      add-sqr-sqrt [=>] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}}} \]
      hypot-def [=>] 62.2%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]

   6. Applied egg-rr 62.0%

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

      [Start] 62.2%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}} \]
      flip-- [=>] 61.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\color{blue}{\frac{b \cdot b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) \cdot \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}}} \]
      hypot-udef [=>] 61.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \color{blue}{\sqrt{b \cdot b + \sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}} \cdot \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      add-sqr-sqrt [<=] 61.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -4\right)}} \cdot \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      fma-udef [<=] 61.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} \cdot \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      hypot-udef [=>] 61.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \color{blue}{\sqrt{b \cdot b + \sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      add-sqr-sqrt [<=] 61.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -4\right)}}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      fma-udef [<=] 61.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      add-sqr-sqrt [<=] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]

   7. Simplified 72.9%

      \[\leadsto \frac{0.5}{\left(-a\right) \cdot \frac{1}{\color{blue}{\frac{0 + c \cdot \left(4 \cdot a\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}}} \]
      Step-by-step derivation

      [Start] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      fma-udef [=>] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \color{blue}{\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      associate-*r* [=>] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \left(b \cdot b + \color{blue}{\left(a \cdot c\right) \cdot -4}\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      *-commutative [<=] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \left(b \cdot b + \color{blue}{-4 \cdot \left(a \cdot c\right)}\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      *-commutative [<=] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \left(b \cdot b + -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      +-commutative [=>] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \color{blue}{\left(-4 \cdot \left(c \cdot a\right) + b \cdot b\right)}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      associate-*r* [=>] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \left(\color{blue}{\left(-4 \cdot c\right) \cdot a} + b \cdot b\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      *-commutative [<=] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \left(\color{blue}{\left(c \cdot -4\right)} \cdot a + b \cdot b\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      metadata-eval [<=] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \left(\left(c \cdot \color{blue}{\left(-4\right)}\right) \cdot a + b \cdot b\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      distribute-rgt-neg-in [<=] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \left(\color{blue}{\left(-c \cdot 4\right)} \cdot a + b \cdot b\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      *-commutative [<=] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \left(\color{blue}{a \cdot \left(-c \cdot 4\right)} + b \cdot b\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      distribute-rgt-neg-in [<=] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \left(\color{blue}{\left(-a \cdot \left(c \cdot 4\right)\right)} + b \cdot b\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      +-commutative [<=] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \color{blue}{\left(b \cdot b + \left(-a \cdot \left(c \cdot 4\right)\right)\right)}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      sub-neg [<=] 62.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{b \cdot b - \color{blue}{\left(b \cdot b - a \cdot \left(c \cdot 4\right)\right)}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      associate-+l- [<=] 72.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{\color{blue}{\left(b \cdot b - b \cdot b\right) + a \cdot \left(c \cdot 4\right)}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      +-inverses [=>] 72.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{\color{blue}{0} + a \cdot \left(c \cdot 4\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      *-commutative [=>] 72.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{0 + \color{blue}{\left(c \cdot 4\right) \cdot a}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      associate-*l* [=>] 72.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{0 + \color{blue}{c \cdot \left(4 \cdot a\right)}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]
      associate-*r* [=>] 72.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{0 + c \cdot \left(4 \cdot a\right)}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right)}}} \]
      rem-square-sqrt [<=] 0.0%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{0 + c \cdot \left(4 \cdot a\right)}{b + \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}\right)}}} \]

   8. Applied egg-rr 72.9%

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

      [Start] 72.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\frac{0 + c \cdot \left(4 \cdot a\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}} \]
      *-un-lft-identity [=>] 72.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\color{blue}{1 \cdot \frac{0 + c \cdot \left(4 \cdot a\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}}} \]
      +-lft-identity [=>] 72.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{1 \cdot \frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}} \]
      *-commutative [=>] 72.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{1 \cdot \frac{c \cdot \color{blue}{\left(a \cdot 4\right)}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}} \]

   9. Simplified 79.1%

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

      [Start] 72.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{1 \cdot \frac{c \cdot \left(a \cdot 4\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}} \]
      *-lft-identity [=>] 72.9%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\color{blue}{\frac{c \cdot \left(a \cdot 4\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}}} \]
      associate-/l* [=>] 79.1%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\color{blue}{\frac{c}{\frac{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}{a \cdot 4}}}}} \]
      associate-/r/ [=>] 79.1%	\[ \frac{0.5}{\left(-a\right) \cdot \frac{1}{\color{blue}{\frac{c}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \left(a \cdot 4\right)}}} \]

   if 2e5 < b

   1. Initial program 15.6%

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

   2. Simplified 15.6%

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

      [Start] 15.6%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      neg-sub0 [=>] 15.6%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-+l- [=>] 15.6%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 15.6%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 15.6%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      *-commutative [=>] 15.6%	\[ \frac{\color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}}{2 \cdot a} \]
      associate-*r/ [<=] 15.6%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around inf 95.6%

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

   4. Simplified 95.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
      Step-by-step derivation

      [Start] 95.6%	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 95.6%	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 95.6%	\[ \frac{\color{blue}{-c}}{b} \]

3. Recombined 4 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+129}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 10^{-300}:\\ \;\;\;\;\frac{\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot -0.5}{a}\\ \mathbf{elif}\;b \leq 200000:\\ \;\;\;\;\frac{0.5}{a \cdot \frac{-1}{\frac{c}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.9%
Cost	14476

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+129}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 10^{-300}:\\ \;\;\;\;\frac{\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot -0.5}{a}\\ \mathbf{elif}\;b \leq 200000:\\ \;\;\;\;\frac{0.5}{a \cdot \frac{-1}{\frac{c}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Accuracy	85.2%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+116}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-92}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Accuracy	85.4%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+124}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Accuracy	79.8%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Accuracy	79.8%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{-0.5 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6
Accuracy	67.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7
Accuracy	67.1%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-307}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8
Accuracy	34.6%
Cost	256

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

Alternative 9
Accuracy	2.6%
Cost	192

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

Reproduce

herbie shell --seed 2023271 
(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)))