quadp (p42, positive)

Average Error: 33.8 → 8.3

Time: 18.3s

Precision: binary64

Cost: 13964

Math

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

↓

\[\begin{array}{l} t_0 := a \cdot \left(c \cdot -4\right)\\ \mathbf{if}\;b \leq -4.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-292}:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b + t_0}}{a} \cdot -0.5\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;c \cdot \frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 -4.6e+103)
     (/ (fma 2.0 (/ c (/ b a)) (* b -2.0)) (* 2.0 a))
     (if (<= b -1.4e-292)
       (* (/ (- b (sqrt (+ (* b b) t_0))) a) -0.5)
       (if (<= b 2.1e-10)
         (* c (/ -2.0 (+ 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 <= -4.6e+103) {
		tmp = fma(2.0, (c / (b / a)), (b * -2.0)) / (2.0 * a);
	} else if (b <= -1.4e-292) {
		tmp = ((b - sqrt(((b * b) + t_0))) / a) * -0.5;
	} else if (b <= 2.1e-10) {
		tmp = c * (-2.0 / (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(4.0 * Float64(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 <= -4.6e+103)
		tmp = Float64(fma(2.0, Float64(c / Float64(b / a)), Float64(b * -2.0)) / Float64(2.0 * a));
	elseif (b <= -1.4e-292)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(Float64(b * b) + t_0))) / a) * -0.5);
	elseif (b <= 2.1e-10)
		tmp = Float64(c * Float64(-2.0 / 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[(4.0 * N[(a * c), $MachinePrecision]), $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, -4.6e+103], N[(N[(2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.4e-292], N[(N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[b, 2.1e-10], N[(c * N[(-2.0 / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]

Target

Original	33.8
Target	21.5
Herbie	8.3

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

Derivation

1. Split input into 4 regimes

2. if b < -4.60000000000000017e103

   1. Initial program 49.2

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

   2. Taylor expanded in b around -inf 11.0

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}}{2 \cdot a} \]

   3. Simplified 3.4

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

      [Start] 11.0	\[ \frac{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}{2 \cdot a} \]
      fma-def [=>] 11.0	\[ \frac{\color{blue}{\mathsf{fma}\left(2, \frac{c \cdot a}{b}, -2 \cdot b\right)}}{2 \cdot a} \]
      associate-/l* [=>] 3.4	\[ \frac{\mathsf{fma}\left(2, \color{blue}{\frac{c}{\frac{b}{a}}}, -2 \cdot b\right)}{2 \cdot a} \]
      *-commutative [=>] 3.4	\[ \frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot -2}\right)}{2 \cdot a} \]

   if -4.60000000000000017e103 < b < -1.4000000000000001e-292

   1. Initial program 9.4

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

   2. Simplified 9.4

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

      [Start] 9.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-/r* [=>] 9.4	\[ \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}} \]
      /-rgt-identity [<=] 9.4	\[ \frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{\color{blue}{\frac{a}{1}}} \]
      metadata-eval [<=] 9.4	\[ \frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{\frac{a}{\color{blue}{-1 \cdot -1}}} \]
      associate-/l/ [<=] 9.4	\[ \frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{\color{blue}{\frac{\frac{a}{-1}}{-1}}} \]
      associate-/l* [<=] 9.4	\[ \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot -1}{\frac{a}{-1}}} \]
      associate-*r/ [<=] 9.5	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{-1}{\frac{a}{-1}}} \]
      times-frac [<=] 9.4	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}{2 \cdot \frac{a}{-1}}} \]
      *-commutative [=>] 9.4	\[ \frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}{\color{blue}{\frac{a}{-1} \cdot 2}} \]
      times-frac [=>] 9.4	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \cdot \frac{-1}{2}} \]

   3. Applied egg-rr 9.4

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

   if -1.4000000000000001e-292 < b < 2.1e-10

   1. Initial program 23.7

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

   2. Simplified 23.8

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

      [Start] 23.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 23.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 23.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 23.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 23.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 23.7	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 23.8	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 23.8	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 23.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Applied egg-rr 24.2

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

   4. Simplified 29.4

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

      [Start] 24.2	\[ \frac{\left(-\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b\right)\right) \cdot \frac{0.5}{a}}{-\left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)} \]
      associate-/l* [=>] 29.4	\[ \color{blue}{\frac{-\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b\right)}{\frac{-\left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}{\frac{0.5}{a}}}} \]
      fma-def [<=] 29.4	\[ \frac{-\left(\color{blue}{\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)} - b \cdot b\right)}{\frac{-\left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}{\frac{0.5}{a}}} \]
      +-commutative [=>] 29.4	\[ \frac{-\left(\color{blue}{\left(a \cdot \left(c \cdot -4\right) + b \cdot b\right)} - b \cdot b\right)}{\frac{-\left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}{\frac{0.5}{a}}} \]
      fma-def [=>] 29.4	\[ \frac{-\left(\color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b \cdot b\right)}{\frac{-\left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}{\frac{0.5}{a}}} \]

   5. Applied egg-rr 24.1

      \[\leadsto \color{blue}{\frac{1}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{a \cdot 2}} \]

   6. Simplified 20.7

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

      [Start] 24.1	\[ \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{a \cdot 2} \]
      associate-/r* [=>] 24.1	\[ \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{a}}{2}} \]
      associate-*r/ [=>] 24.1	\[ \color{blue}{\frac{\frac{1}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{a}}{2}} \]
      associate-*l/ [<=] 24.1	\[ \color{blue}{\frac{\frac{1}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}{2} \cdot \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{a}} \]
      associate-/r* [<=] 24.1	\[ \color{blue}{\frac{1}{\left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \cdot 2}} \cdot \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{a} \]
      *-commutative [=>] 24.1	\[ \frac{1}{\color{blue}{2 \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}} \cdot \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{a} \]
      associate-/r* [=>] 24.1	\[ \color{blue}{\frac{\frac{1}{2}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}} \cdot \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{a} \]
      metadata-eval [=>] 24.1	\[ \frac{\color{blue}{0.5}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{a} \]
      associate-*r* [=>] 24.2	\[ \frac{0.5}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right)} \cdot \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{a} \]
      *-commutative [<=] 24.2	\[ \frac{0.5}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}\right)} \cdot \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{a} \]
      *-commutative [<=] 24.2	\[ \frac{0.5}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right)}}\right)} \cdot \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{a} \]

   7. Applied egg-rr 53.2

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)} \cdot \left(a \cdot c\right)\right)} - 1} \]

   8. Simplified 13.0

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

      [Start] 53.2	\[ e^{\mathsf{log1p}\left(\frac{-2}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)} \cdot \left(a \cdot c\right)\right)} - 1 \]
      expm1-def [=>] 36.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-2}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)} \cdot \left(a \cdot c\right)\right)\right)} \]
      expm1-log1p [=>] 26.6	\[ \color{blue}{\frac{-2}{a \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)} \cdot \left(a \cdot c\right)} \]
      *-commutative [=>] 26.6	\[ \frac{-2}{\color{blue}{\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot a}} \cdot \left(a \cdot c\right) \]
      associate-/r* [=>] 26.4	\[ \color{blue}{\frac{\frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}{a}} \cdot \left(a \cdot c\right) \]
      associate-/r/ [<=] 20.7	\[ \color{blue}{\frac{\frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}{\frac{a}{a \cdot c}}} \]
      associate-/r* [=>] 13.1	\[ \frac{\frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}{\color{blue}{\frac{\frac{a}{a}}{c}}} \]
      associate-/r/ [=>] 13.0	\[ \color{blue}{\frac{\frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}{\frac{a}{a}} \cdot c} \]
      *-commutative [=>] 13.0	\[ \color{blue}{c \cdot \frac{\frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}{\frac{a}{a}}} \]
      *-inverses [=>] 13.0	\[ c \cdot \frac{\frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}{\color{blue}{1}} \]
      associate-/l/ [=>] 13.0	\[ c \cdot \color{blue}{\frac{-2}{1 \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)}} \]
      associate-/r* [=>] 13.0	\[ c \cdot \color{blue}{\frac{\frac{-2}{1}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}} \]
      metadata-eval [=>] 13.0	\[ c \cdot \frac{\color{blue}{-2}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]
      associate-*r* [=>] 13.0	\[ c \cdot \frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4}}\right)} \]
      *-commutative [=>] 13.0	\[ c \cdot \frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4}\right)} \]
      associate-*l* [=>] 13.0	\[ c \cdot \frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right)} \]

   if 2.1e-10 < b

   1. Initial program 55.3

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

   2. Taylor expanded in b around inf 6.6

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

   3. Simplified 6.6

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

      [Start] 6.6	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 6.6	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 6.6	\[ \frac{\color{blue}{-c}}{b} \]

3. Recombined 4 regimes into one program.

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-292}:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a} \cdot -0.5\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;c \cdot \frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.5
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Error	14.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-14}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.7 \cdot 10^{-25}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Error	14.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.7 \cdot 10^{-25}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Error	23.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-187}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Error	45.5
Cost	256

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

Reproduce

herbie shell --seed 2023053 
(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)))