quadm (p42, negative)

Average Accuracy: 42.9% → 79.1%

Time: 20.1s

Precision: binary64

Cost: 14344

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-46}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{-251}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot \frac{1}{-0.25 \cdot \frac{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}{a}}}{a}\\ \mathbf{elif}\;b \leq 10^{+109}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 -3.3e-46)
   (/ (- c) b)
   (if (<= b -1.8e-251)
     (*
      -0.5
      (/
       (* c (/ 1.0 (* -0.25 (/ (- (hypot b (sqrt (* c (* a -4.0)))) b) a))))
       a))
     (if (<= b 1e+109)
       (* -0.5 (/ (+ b (sqrt (fma a (* c -4.0) (* b b)))) a))
       (- (/ c b) (/ b a))))))

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 <= -3.3e-46) {
		tmp = -c / b;
	} else if (b <= -1.8e-251) {
		tmp = -0.5 * ((c * (1.0 / (-0.25 * ((hypot(b, sqrt((c * (a * -4.0)))) - b) / a)))) / a);
	} else if (b <= 1e+109) {
		tmp = -0.5 * ((b + sqrt(fma(a, (c * -4.0), (b * b)))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= -3.3e-46)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= -1.8e-251)
		tmp = Float64(-0.5 * Float64(Float64(c * Float64(1.0 / Float64(-0.25 * Float64(Float64(hypot(b, sqrt(Float64(c * Float64(a * -4.0)))) - b) / a)))) / a));
	elseif (b <= 1e+109)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))) / a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return 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, -3.3e-46], N[((-c) / b), $MachinePrecision], If[LessEqual[b, -1.8e-251], N[(-0.5 * N[(N[(c * N[(1.0 / N[(-0.25 * N[(N[(N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+109], N[(-0.5 * N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	42.9%
Target	61.6%
Herbie	79.1%

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

Derivation

1. Split input into 4 regimes

2. if b < -3.30000000000000013e-46

   1. Initial program 15.1%

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

   2. Taylor expanded in b around -inf 88.4%

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

   3. Simplified 88.4%

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

      [Start] 88.4	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 88.4	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 88.4	\[ \frac{\color{blue}{-c}}{b} \]

   if -3.30000000000000013e-46 < b < -1.8000000000000001e-251

   1. Initial program 62.2%

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

   2. Simplified 62.4%

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

      [Start] 62.2	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 62.2	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 62.2	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 62.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 62.3	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 62.3	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 62.3	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 62.3	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 62.3	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 62.3	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 62.2	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 62.2	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 62.3	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 62.3	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 62.3	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 62.3	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 62.3	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   3. Applied egg-rr 61.7%

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

      [Start] 62.4	\[ -0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \]
      flip-+ [=>] 62.3	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
      add-sqr-sqrt [<=] 62.3	\[ -0.5 \cdot \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      div-sub [=>] 62.3	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
      fma-udef [=>] 62.3	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      +-commutative [=>] 62.3	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      add-sqr-sqrt [=>] 61.7	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      hypot-def [=>] 61.7	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      associate-*r* [=>] 61.7	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right)} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]

   4. Simplified 65.7%

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

      [Start] 61.7	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)} - \frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{a} \]
      div-sub [<=] 61.7	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      *-lft-identity [<=] 61.7	\[ -0.5 \cdot \frac{\color{blue}{1 \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      metadata-eval [<=] 61.7	\[ -0.5 \cdot \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{a} \]
      times-frac [<=] 61.7	\[ -0.5 \cdot \frac{\color{blue}{\frac{-1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}{-1 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}}{a} \]
      neg-mul-1 [<=] 61.7	\[ -0.5 \cdot \frac{\frac{-1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}{\color{blue}{-\left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}}{a} \]
      neg-mul-1 [<=] 61.7	\[ -0.5 \cdot \frac{\frac{\color{blue}{-\left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}}{-\left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}{a} \]

   5. Applied egg-rr 75.4%

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

      [Start] 65.7	\[ -0.5 \cdot \frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a} \]
      associate-/l* [=>] 75.5	\[ -0.5 \cdot \frac{\color{blue}{\frac{c}{\frac{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}{a \cdot -4}}}}{a} \]
      div-inv [=>] 75.4	\[ -0.5 \cdot \frac{\color{blue}{c \cdot \frac{1}{\frac{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}{a \cdot -4}}}}{a} \]
      *-un-lft-identity [=>] 75.4	\[ -0.5 \cdot \frac{c \cdot \frac{1}{\frac{\color{blue}{1 \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b\right)}}{a \cdot -4}}}{a} \]
      *-commutative [=>] 75.4	\[ -0.5 \cdot \frac{c \cdot \frac{1}{\frac{1 \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b\right)}{\color{blue}{-4 \cdot a}}}}{a} \]
      times-frac [=>] 75.4	\[ -0.5 \cdot \frac{c \cdot \frac{1}{\color{blue}{\frac{1}{-4} \cdot \frac{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}{a}}}}{a} \]
      metadata-eval [=>] 75.4	\[ -0.5 \cdot \frac{c \cdot \frac{1}{\color{blue}{-0.25} \cdot \frac{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}{a}}}{a} \]

   if -1.8000000000000001e-251 < b < 9.99999999999999982e108

   1. Initial program 81.7%

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

   2. Simplified 81.7%

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

      [Start] 81.7	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 81.7	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 81.7	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 81.5	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 81.4	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 81.4	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 81.4	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 81.4	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 81.4	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 81.4	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 81.7	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 81.7	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 81.7	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 81.7	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 81.7	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 81.7	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 81.7	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   if 9.99999999999999982e108 < b

   1. Initial program 15.8%

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

   2. Taylor expanded in b around inf 62.5%

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

   3. Simplified 62.5%

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

      [Start] 62.5	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 62.5	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 62.5	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-46}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{-251}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot \frac{1}{-0.25 \cdot \frac{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}{a}}}{a}\\ \mathbf{elif}\;b \leq 10^{+109}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	77.6%
Cost	13896

\[\begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+109}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Accuracy	77.6%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+107}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Accuracy	72.8%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-61}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Accuracy	59.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Accuracy	35.0%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Accuracy	58.9%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-168}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Accuracy	10.6%
Cost	192

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

Reproduce

herbie shell --seed 2023153 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

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

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