Quadratic roots, full range

Average Error: 33.5 → 12.2

Time: 17.3s

Precision: binary64

Cost: 33484

Math

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

↓

\[\begin{array}{l} t_0 := a \cdot \left(c \cdot -4\right)\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -6.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4.65 \cdot 10^{-290}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\mathsf{hypot}\left(b, {\left(e^{0.25 \cdot \left(\log \left(c \cdot 4\right) - \log \left(\frac{-1}{a}\right)\right)}\right)}^{2}\right) - b}}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(b, b, t_0\right) - b \cdot b\right)}{b + \mathsf{hypot}\left(b, \sqrt{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c -4.0))))
   (if (<= b -1.2e+151)
     (- (/ c b) (/ b a))
     (if (<= b -6.8e-158)
       (/ (- (sqrt (+ (* b b) (* c (* a -4.0)))) b) (* a 2.0))
       (if (<= b 4.65e-290)
         (/
          0.5
          (/
           a
           (-
            (hypot
             b
             (pow (exp (* 0.25 (- (log (* c 4.0)) (log (/ -1.0 a))))) 2.0))
            b)))
         (if (<= b 1.26e-43)
           (/
            (* (/ 0.5 a) (- (fma b b t_0) (* b b)))
            (+ b (hypot b (sqrt t_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 t_0 = a * (c * -4.0);
	double tmp;
	if (b <= -1.2e+151) {
		tmp = (c / b) - (b / a);
	} else if (b <= -6.8e-158) {
		tmp = (sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0);
	} else if (b <= 4.65e-290) {
		tmp = 0.5 / (a / (hypot(b, pow(exp((0.25 * (log((c * 4.0)) - log((-1.0 / a))))), 2.0)) - b));
	} else if (b <= 1.26e-43) {
		tmp = ((0.5 / a) * (fma(b, b, t_0) - (b * b))) / (b + hypot(b, sqrt(t_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)
	t_0 = Float64(a * Float64(c * -4.0))
	tmp = 0.0
	if (b <= -1.2e+151)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -6.8e-158)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	elseif (b <= 4.65e-290)
		tmp = Float64(0.5 / Float64(a / Float64(hypot(b, (exp(Float64(0.25 * Float64(log(Float64(c * 4.0)) - log(Float64(-1.0 / a))))) ^ 2.0)) - b)));
	elseif (b <= 1.26e-43)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(fma(b, b, t_0) - Float64(b * b))) / Float64(b + hypot(b, sqrt(t_0))));
	else
		tmp = Float64(-Float64(c / b));
	end
	return 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_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.2e+151], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -6.8e-158], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.65e-290], N[(0.5 / N[(a / N[(N[Sqrt[b ^ 2 + N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(c * 4.0), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] ^ 2], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.26e-43], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(N[(b * b + t$95$0), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.20000000000000005e151

   1. Initial program 63.3

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

   2. Simplified 63.3

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

      [Start] 63.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 63.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 2.2

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

   4. Simplified 2.2

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

      [Start] 2.2	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 2.2	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 2.2	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -1.20000000000000005e151 < b < -6.7999999999999999e-158

   1. Initial program 5.8

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

   if -6.7999999999999999e-158 < b < 4.65000000000000014e-290

   1. Initial program 14.6

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

   2. Simplified 14.7

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

      [Start] 14.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 14.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 14.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 14.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 14.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 14.6	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 14.7	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      +-commutative [=>] 14.7	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{\frac{--1}{2}}{a} \]
      unsub-neg [=>] 14.7	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{\frac{--1}{2}}{a} \]
      fma-neg [=>] 14.7	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      associate-*l* [=>] 14.7	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      distribute-lft-neg-in [=>] 14.7	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      *-commutative [=>] 14.7	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      associate-*l* [=>] 14.7	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot \left(-4\right)\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [=>] 14.7	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-4}\right)\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [=>] 14.7	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]
      metadata-eval [=>] 14.7	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Applied egg-rr 11.5

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

   4. Applied egg-rr 11.7

      \[\leadsto \frac{0.5}{\frac{a}{\mathsf{hypot}\left(b, \color{blue}{{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}}\right) - b}} \]

   5. Taylor expanded in a around -inf 35.4

      \[\leadsto \frac{0.5}{\frac{a}{\mathsf{hypot}\left(b, {\color{blue}{\left(e^{0.25 \cdot \left(\log \left(4 \cdot c\right) + -1 \cdot \log \left(\frac{-1}{a}\right)\right)}\right)}}^{2}\right) - b}} \]

   if 4.65000000000000014e-290 < b < 1.26e-43

   1. Initial program 22.9

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

   2. Simplified 23.0

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

      [Start] 22.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 22.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 22.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 22.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 22.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 22.9	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 22.9	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      +-commutative [=>] 22.9	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{\frac{--1}{2}}{a} \]
      unsub-neg [=>] 22.9	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{\frac{--1}{2}}{a} \]
      fma-neg [=>] 22.9	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      associate-*l* [=>] 22.9	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      distribute-lft-neg-in [=>] 22.9	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      *-commutative [=>] 22.9	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      associate-*l* [=>] 23.0	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot \left(-4\right)\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [=>] 23.0	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-4}\right)\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [=>] 23.0	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]
      metadata-eval [=>] 23.0	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Applied egg-rr 23.4

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

   if 1.26e-43 < b

   1. Initial program 53.6

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

   2. Simplified 53.6

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

      [Start] 53.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 53.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 53.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 53.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 53.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 53.6	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 53.6	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      +-commutative [=>] 53.6	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{\frac{--1}{2}}{a} \]
      unsub-neg [=>] 53.6	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{\frac{--1}{2}}{a} \]
      fma-neg [=>] 53.6	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      associate-*l* [=>] 53.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      distribute-lft-neg-in [=>] 53.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      *-commutative [=>] 53.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      associate-*l* [=>] 53.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot \left(-4\right)\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [=>] 53.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-4}\right)\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [=>] 53.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]
      metadata-eval [=>] 53.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Taylor expanded in b around inf 7.8

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

   4. Simplified 7.8

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

      [Start] 7.8	\[ -1 \cdot \frac{c}{b} \]
      mul-1-neg [=>] 7.8	\[ \color{blue}{-\frac{c}{b}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 12.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -6.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4.65 \cdot 10^{-290}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\mathsf{hypot}\left(b, {\left(e^{0.25 \cdot \left(\log \left(c \cdot 4\right) - \log \left(\frac{-1}{a}\right)\right)}\right)}^{2}\right) - b}}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.2
Cost	33356

\[\begin{array}{l} t_0 := a \cdot \left(c \cdot -4\right)\\ \mathbf{if}\;b \leq -3.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-290}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\mathsf{hypot}\left(b, {\left(e^{0.25 \cdot \left(\log \left(a \cdot -4\right) + \log c\right)}\right)}^{2}\right) - b}}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(b, b, t_0\right) - b \cdot b\right)}{b + \mathsf{hypot}\left(b, \sqrt{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 2
Error	10.0
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-43}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 3
Error	9.9
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 4
Error	13.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-43}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 5
Error	23.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-231}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	40.0
Cost	256

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

Alternative 7
Error	56.2
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))