Quadratic roots, full range

Average Accuracy: 47.0% → 86.9%

Time: 17.0s

Precision: binary64

Cost: 14028

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{c \cdot \left(-2\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.1e+83)
   (/ (- b) a)
   (if (<= b -6e-308)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (if (<= b 6.8e-24)
       (/ (* c (- 2.0)) (+ 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 tmp;
	if (b <= -3.1e+83) {
		tmp = -b / a;
	} else if (b <= -6e-308) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else if (b <= 6.8e-24) {
		tmp = (c * -2.0) / (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 tmp;
	if (b <= -3.1e+83) {
		tmp = -b / a;
	} else if (b <= -6e-308) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else if (b <= 6.8e-24) {
		tmp = (c * -2.0) / (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):
	tmp = 0
	if b <= -3.1e+83:
		tmp = -b / a
	elif b <= -6e-308:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	elif b <= 6.8e-24:
		tmp = (c * -2.0) / (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(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.1e+83)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= -6e-308)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	elseif (b <= 6.8e-24)
		tmp = Float64(Float64(c * Float64(-2.0)) / 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)
	tmp = 0.0;
	if (b <= -3.1e+83)
		tmp = -b / a;
	elseif (b <= -6e-308)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	elseif (b <= 6.8e-24)
		tmp = (c * -2.0) / (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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := If[LessEqual[b, -3.1e+83], N[((-b) / a), $MachinePrecision], If[LessEqual[b, -6e-308], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-24], N[(N[(c * (-2.0)), $MachinePrecision] / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.09999999999999992e83

   1. Initial program 33.4%

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

   2. Simplified 33.3%

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

      [Start] 33.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 33.4	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 33.4	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 33.4	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 33.4	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 33.3	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 33.3	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 33.3	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 33.3	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 33.3	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Taylor expanded in b around -inf 93.5%

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

   4. Simplified 93.5%

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

      [Start] 93.5	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 93.5	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 93.5	\[ \frac{\color{blue}{-b}}{a} \]

   if -3.09999999999999992e83 < b < -6.00000000000000044e-308

   1. Initial program 85.9%

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

   if -6.00000000000000044e-308 < b < 6.79999999999999985e-24

   1. Initial program 61.6%

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

   2. Simplified 61.5%

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

      [Start] 61.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 61.6	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 61.6	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 61.6	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 61.6	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 61.5	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 61.5	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 61.5	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 61.5	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 61.5	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Applied egg-rr 60.3%

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

      [Start] 61.5	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      fma-udef [=>] 61.5	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}\right) \cdot \frac{-0.5}{a} \]
      add-sqr-sqrt [=>] 60.3	\[ \left(b - \sqrt{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      hypot-def [=>] 60.3	\[ \left(b - \color{blue}{\mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}\right) \cdot \frac{-0.5}{a} \]

   4. Simplified 60.3%

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

      [Start] 60.3	\[ \left(b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot \frac{-0.5}{a} \]
      associate-*r* [=>] 60.3	\[ \left(b - \mathsf{hypot}\left(\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}, b\right)\right) \cdot \frac{-0.5}{a} \]
      *-commutative [<=] 60.3	\[ \left(b - \mathsf{hypot}\left(\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}, b\right)\right) \cdot \frac{-0.5}{a} \]
      associate-*l* [=>] 60.3	\[ \left(b - \mathsf{hypot}\left(\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}, b\right)\right) \cdot \frac{-0.5}{a} \]

   5. Applied egg-rr 50.9%

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

      [Start] 60.3	\[ \left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 60.3	\[ \color{blue}{\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)} \]
      clear-num [=>] 60.3	\[ \color{blue}{\frac{1}{\frac{a}{-0.5}}} \cdot \left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right) \]
      flip-- [=>] 60.3	\[ \frac{1}{\frac{a}{-0.5}} \cdot \color{blue}{\frac{b \cdot b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) \cdot \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}{b + \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}} \]
      frac-times [=>] 50.9	\[ \color{blue}{\frac{1 \cdot \left(b \cdot b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) \cdot \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)}{\frac{a}{-0.5} \cdot \left(b + \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)}} \]

   6. Simplified 54.8%

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

      [Start] 50.9	\[ \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}{\left(a \cdot -2\right) \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot -4\right) \cdot a}\right)\right)} \]
      *-lft-identity [<=] 50.9	\[ \frac{\color{blue}{1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right)}}{\left(a \cdot -2\right) \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot -4\right) \cdot a}\right)\right)} \]
      times-frac [=>] 60.2	\[ \color{blue}{\frac{1}{a \cdot -2} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot -4\right) \cdot a}\right)}} \]
      *-commutative [=>] 60.2	\[ \frac{1}{\color{blue}{-2 \cdot a}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot -4\right) \cdot a}\right)} \]
      associate-/r* [=>] 60.2	\[ \color{blue}{\frac{\frac{1}{-2}}{a}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot -4\right) \cdot a}\right)} \]
      metadata-eval [=>] 60.2	\[ \frac{\color{blue}{-0.5}}{a} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot -4\right) \cdot a}\right)} \]
      associate-*r/ [=>] 60.3	\[ \color{blue}{\frac{\frac{-0.5}{a} \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right)}{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot -4\right) \cdot a}\right)}} \]
      associate-/l* [=>] 60.2	\[ \color{blue}{\frac{\frac{-0.5}{a}}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot -4\right) \cdot a}\right)}{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}}} \]
      associate-/r/ [=>] 50.9	\[ \color{blue}{\frac{\frac{-0.5}{a}}{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot -4\right) \cdot a}\right)} \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right)} \]

   7. Applied egg-rr 54.2%

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

      [Start] 54.8	\[ \frac{\frac{-0.5}{a}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \left(-c \cdot \left(a \cdot -4\right)\right) \]
      distribute-rgt-neg-out [=>] 54.8	\[ \color{blue}{-\frac{\frac{-0.5}{a}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \left(c \cdot \left(a \cdot -4\right)\right)} \]
      neg-sub0 [=>] 54.8	\[ \color{blue}{0 - \frac{\frac{-0.5}{a}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \left(c \cdot \left(a \cdot -4\right)\right)} \]
      associate-/l/ [=>] 54.2	\[ 0 - \color{blue}{\frac{-0.5}{\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot a}} \cdot \left(c \cdot \left(a \cdot -4\right)\right) \]
      *-commutative [=>] 54.2	\[ 0 - \frac{-0.5}{\color{blue}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)}} \cdot \left(c \cdot \left(a \cdot -4\right)\right) \]
      *-commutative [=>] 54.2	\[ 0 - \frac{-0.5}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c}}\right)\right)} \cdot \left(c \cdot \left(a \cdot -4\right)\right) \]
      associate-*l* [=>] 54.2	\[ 0 - \frac{-0.5}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{a \cdot \left(-4 \cdot c\right)}}\right)\right)} \cdot \left(c \cdot \left(a \cdot -4\right)\right) \]
      *-commutative [=>] 54.2	\[ 0 - \frac{-0.5}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)\right)} \cdot \color{blue}{\left(\left(a \cdot -4\right) \cdot c\right)} \]
      associate-*l* [=>] 54.2	\[ 0 - \frac{-0.5}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)\right)} \cdot \color{blue}{\left(a \cdot \left(-4 \cdot c\right)\right)} \]

   8. Simplified 78.2%

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

      [Start] 54.2	\[ 0 - \frac{-0.5}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)\right)} \cdot \left(a \cdot \left(-4 \cdot c\right)\right) \]
      neg-sub0 [<=] 54.2	\[ \color{blue}{-\frac{-0.5}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)\right)} \cdot \left(a \cdot \left(-4 \cdot c\right)\right)} \]
      associate-*l/ [=>] 55.0	\[ -\color{blue}{\frac{-0.5 \cdot \left(a \cdot \left(-4 \cdot c\right)\right)}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)\right)}} \]
      *-commutative [=>] 55.0	\[ -\frac{\color{blue}{\left(a \cdot \left(-4 \cdot c\right)\right) \cdot -0.5}}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)\right)} \]
      times-frac [=>] 65.3	\[ -\color{blue}{\frac{a \cdot \left(-4 \cdot c\right)}{a} \cdot \frac{-0.5}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)}} \]
      associate-*r/ [=>] 65.4	\[ -\color{blue}{\frac{\frac{a \cdot \left(-4 \cdot c\right)}{a} \cdot -0.5}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)}} \]
      *-commutative [=>] 65.4	\[ -\frac{\frac{\color{blue}{\left(-4 \cdot c\right) \cdot a}}{a} \cdot -0.5}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)} \]
      associate-/l* [=>] 78.3	\[ -\frac{\color{blue}{\frac{-4 \cdot c}{\frac{a}{a}}} \cdot -0.5}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)} \]
      *-inverses [=>] 78.3	\[ -\frac{\frac{-4 \cdot c}{\color{blue}{1}} \cdot -0.5}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)} \]
      /-rgt-identity [=>] 78.3	\[ -\frac{\color{blue}{\left(-4 \cdot c\right)} \cdot -0.5}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)} \]
      *-commutative [=>] 78.3	\[ -\frac{\color{blue}{\left(c \cdot -4\right)} \cdot -0.5}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)} \]
      rem-square-sqrt [<=] 0.0	\[ -\frac{\left(c \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}\right) \cdot -0.5}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)} \]
      unpow2 [<=] 0.0	\[ -\frac{\left(c \cdot \color{blue}{{\left(\sqrt{-4}\right)}^{2}}\right) \cdot -0.5}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)} \]
      associate-*l* [=>] 0.0	\[ -\frac{\color{blue}{c \cdot \left({\left(\sqrt{-4}\right)}^{2} \cdot -0.5\right)}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)} \]
      unpow2 [=>] 0.0	\[ -\frac{c \cdot \left(\color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)} \cdot -0.5\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)} \]
      rem-square-sqrt [=>] 78.3	\[ -\frac{c \cdot \left(\color{blue}{-4} \cdot -0.5\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)} \]
      metadata-eval [=>] 78.3	\[ -\frac{c \cdot \color{blue}{2}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)} \]
      associate-*r* [=>] 78.2	\[ -\frac{c \cdot 2}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c}}\right)} \]
      *-commutative [=>] 78.2	\[ -\frac{c \cdot 2}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}\right)} \]

   if 6.79999999999999985e-24 < b

   1. Initial program 14.4%

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

   2. Simplified 14.4%

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

      [Start] 14.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 14.4	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 14.4	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 14.4	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 14.4	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 14.4	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 14.4	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 14.4	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 14.4	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 14.4	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Taylor expanded in b around inf 89.3%

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

   4. Simplified 89.3%

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

      [Start] 89.3	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 89.3	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      mul-1-neg [=>] 89.3	\[ \frac{\color{blue}{-c}}{b} \]

3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{c \cdot \left(-2\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.1%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{+83}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Accuracy	78.7%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-67}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.16 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Accuracy	64.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-281}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 4
Accuracy	38.5%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{+19}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 5
Accuracy	64.9%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-252}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6
Accuracy	2.6%
Cost	192

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

Alternative 7
Accuracy	11.7%
Cost	192

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

Reproduce

herbie shell --seed 2023147 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))