quadm (p42, negative)

Average Accuracy: 51.9% → 88.0%

Time: 18.7s

Precision: binary64

Cost: 14028

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot -4}{\mathsf{hypot}\left(b, \sqrt{-4 \cdot \left(c \cdot a\right)}\right) - b}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+104}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 -6.6e-52)
   (/ (- c) b)
   (if (<= b 8e-309)
     (* -0.5 (/ (* c -4.0) (- (hypot b (sqrt (* -4.0 (* c a)))) b)))
     (if (<= b 7e+104)
       (* -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 <= -6.6e-52) {
		tmp = -c / b;
	} else if (b <= 8e-309) {
		tmp = -0.5 * ((c * -4.0) / (hypot(b, sqrt((-4.0 * (c * a)))) - b));
	} else if (b <= 7e+104) {
		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 <= -6.6e-52)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 8e-309)
		tmp = Float64(-0.5 * Float64(Float64(c * -4.0) / Float64(hypot(b, sqrt(Float64(-4.0 * Float64(c * a)))) - b)));
	elseif (b <= 7e+104)
		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, -6.6e-52], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 8e-309], N[(-0.5 * N[(N[(c * -4.0), $MachinePrecision] / N[(N[Sqrt[b ^ 2 + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e+104], 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	51.9%
Target	70.1%
Herbie	88.0%

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

Derivation

1. Split input into 4 regimes

2. if b < -6.5999999999999999e-52

   1. Initial program 14.5%

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

   2. Taylor expanded in b around -inf 88.2%

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

   3. Simplified 88.2%

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

      [Start] 88.2	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 88.2	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 88.2	\[ \frac{\color{blue}{-c}}{b} \]

   if -6.5999999999999999e-52 < b < 8.0000000000000003e-309

   1. Initial program 65.9%

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

   2. Simplified 65.9%

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

      [Start] 65.9	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 65.9	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 65.9	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 65.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 65.8	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 65.8	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 65.8	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 65.8	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 65.8	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 65.8	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 65.9	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 65.9	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 65.9	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 65.9	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 65.9	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 65.9	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 65.9	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   3. Applied egg-rr 65.4%

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

      [Start] 65.9	\[ -0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \]
      flip-+ [=>] 65.4	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
      add-sqr-sqrt [<=] 65.4	\[ -0.5 \cdot \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      div-sub [=>] 65.4	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
      fma-udef [=>] 65.4	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      +-commutative [=>] 65.4	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      add-sqr-sqrt [=>] 65.4	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      hypot-def [=>] 65.4	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      associate-*r* [=>] 65.4	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right)} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]

   4. Simplified 65.4%

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

      [Start] 65.4	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)} - \frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{a} \]
      div-sub [<=] 65.4	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      *-lft-identity [<=] 65.4	\[ -0.5 \cdot \frac{\color{blue}{1 \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      metadata-eval [<=] 65.4	\[ -0.5 \cdot \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{a} \]
      times-frac [<=] 65.4	\[ -0.5 \cdot \frac{\color{blue}{\frac{-1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}{-1 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}}{a} \]
      neg-mul-1 [<=] 65.4	\[ -0.5 \cdot \frac{\frac{-1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}{\color{blue}{-\left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}}{a} \]
      neg-mul-1 [<=] 65.4	\[ -0.5 \cdot \frac{\frac{\color{blue}{-\left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}}{-\left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}{a} \]

   5. Applied egg-rr 83.1%

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

      [Start] 65.4	\[ -0.5 \cdot \frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a} \]
      add-log-exp [=>] 4.3	\[ -0.5 \cdot \color{blue}{\log \left(e^{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right)} \]
      *-un-lft-identity [=>] 4.3	\[ -0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right)} \]
      log-prod [=>] 4.3	\[ -0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right)\right)} \]
      metadata-eval [=>] 4.3	\[ -0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right)\right) \]
      add-log-exp [<=] 65.4	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right) \]
      associate-/l/ [=>] 58.0	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{c \cdot \left(a \cdot -4\right)}{a \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b\right)}}\right) \]
      *-commutative [=>] 58.0	\[ -0.5 \cdot \left(0 + \frac{\color{blue}{\left(a \cdot -4\right) \cdot c}}{a \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b\right)}\right) \]
      times-frac [=>] 83.1	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{a \cdot -4}{a} \cdot \frac{c}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}\right) \]

   6. Simplified 83.1%

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

      [Start] 83.1	\[ -0.5 \cdot \left(0 + \frac{a \cdot -4}{a} \cdot \frac{c}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}\right) \]
      +-lft-identity [=>] 83.1	\[ -0.5 \cdot \color{blue}{\left(\frac{a \cdot -4}{a} \cdot \frac{c}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}\right)} \]
      associate-*r/ [=>] 83.1	\[ -0.5 \cdot \color{blue}{\frac{\frac{a \cdot -4}{a} \cdot c}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}} \]
      associate-*l/ [=>] 65.5	\[ -0.5 \cdot \frac{\color{blue}{\frac{\left(a \cdot -4\right) \cdot c}{a}}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b} \]
      *-commutative [<=] 65.5	\[ -0.5 \cdot \frac{\frac{\color{blue}{c \cdot \left(a \cdot -4\right)}}{a}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b} \]
      associate-*r* [=>] 65.5	\[ -0.5 \cdot \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot -4}}{a}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b} \]
      associate-*l/ [<=] 65.5	\[ -0.5 \cdot \frac{\color{blue}{\frac{c \cdot a}{a} \cdot -4}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b} \]
      associate-/l* [=>] 83.1	\[ -0.5 \cdot \frac{\color{blue}{\frac{c}{\frac{a}{a}}} \cdot -4}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b} \]
      *-inverses [=>] 83.1	\[ -0.5 \cdot \frac{\frac{c}{\color{blue}{1}} \cdot -4}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b} \]
      /-rgt-identity [=>] 83.1	\[ -0.5 \cdot \frac{\color{blue}{c} \cdot -4}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b} \]
      *-commutative [<=] 83.1	\[ -0.5 \cdot \frac{\color{blue}{-4 \cdot c}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b} \]
      *-commutative [=>] 83.1	\[ -0.5 \cdot \frac{-4 \cdot c}{\mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c}}\right) - b} \]
      *-commutative [=>] 83.1	\[ -0.5 \cdot \frac{-4 \cdot c}{\mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(-4 \cdot a\right)} \cdot c}\right) - b} \]
      associate-*l* [=>] 83.1	\[ -0.5 \cdot \frac{-4 \cdot c}{\mathsf{hypot}\left(b, \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right) - b} \]
      *-commutative [<=] 83.1	\[ -0.5 \cdot \frac{-4 \cdot c}{\mathsf{hypot}\left(b, \sqrt{-4 \cdot \color{blue}{\left(c \cdot a\right)}}\right) - b} \]

   if 8.0000000000000003e-309 < b < 7.0000000000000003e104

   1. Initial program 77.1%

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

   2. Simplified 77.1%

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

      [Start] 77.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 77.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 77.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 77.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 77.0	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 77.0	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 77.0	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 77.0	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 77.0	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 77.0	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 77.1	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 77.1	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 77.1	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 77.1	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 77.1	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 77.1	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 77.1	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   if 7.0000000000000003e104 < b

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf 98.2%

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

   3. Simplified 98.2%

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

      [Start] 98.2	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 98.2	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 98.2	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot -4}{\mathsf{hypot}\left(b, \sqrt{-4 \cdot \left(c \cdot a\right)}\right) - b}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+104}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.5%
Cost	13896

\[\begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+101}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Accuracy	85.5%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-94}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+98}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Accuracy	80.6%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-20}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(-4 \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Accuracy	68.3%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Accuracy	43.9%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -15.5:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Accuracy	68.2%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-295}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Accuracy	12.5%
Cost	324

\[\begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a}\\ \end{array} \]

Alternative 8
Accuracy	5.3%
Cost	192

\[\frac{-0.5}{a} \]

Reproduce

herbie shell --seed 2023159 
(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)))