quadm (p42, negative)

Average Error: 33.8 → 8.3

Time: 20.4s

Precision: binary64

Cost: 14096

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-c}{b}\\ t_1 := 2 \cdot \frac{c}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}\\ \mathbf{if}\;b \leq -8.5 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- c) b))
        (t_1 (* 2.0 (/ c (- (hypot b (sqrt (* c (* a -4.0)))) b)))))
   (if (<= b -8.5e+55)
     t_0
     (if (<= b -7.5e+27)
       t_1
       (if (<= b -1e-20)
         t_0
         (if (<= b 3.2e-261)
           t_1
           (if (<= b 2.25e+65)
             (/ (- (- b) (sqrt (+ (* b b) (* -4.0 (* c a))))) (* 2.0 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 t_0 = -c / b;
	double t_1 = 2.0 * (c / (hypot(b, sqrt((c * (a * -4.0)))) - b));
	double tmp;
	if (b <= -8.5e+55) {
		tmp = t_0;
	} else if (b <= -7.5e+27) {
		tmp = t_1;
	} else if (b <= -1e-20) {
		tmp = t_0;
	} else if (b <= 3.2e-261) {
		tmp = t_1;
	} else if (b <= 2.25e+65) {
		tmp = (-b - sqrt(((b * b) + (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = (c / b) - (b / a);
	}
	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 t_0 = -c / b;
	double t_1 = 2.0 * (c / (Math.hypot(b, Math.sqrt((c * (a * -4.0)))) - b));
	double tmp;
	if (b <= -8.5e+55) {
		tmp = t_0;
	} else if (b <= -7.5e+27) {
		tmp = t_1;
	} else if (b <= -1e-20) {
		tmp = t_0;
	} else if (b <= 3.2e-261) {
		tmp = t_1;
	} else if (b <= 2.25e+65) {
		tmp = (-b - Math.sqrt(((b * b) + (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = (c / b) - (b / a);
	}
	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):
	t_0 = -c / b
	t_1 = 2.0 * (c / (math.hypot(b, math.sqrt((c * (a * -4.0)))) - b))
	tmp = 0
	if b <= -8.5e+55:
		tmp = t_0
	elif b <= -7.5e+27:
		tmp = t_1
	elif b <= -1e-20:
		tmp = t_0
	elif b <= 3.2e-261:
		tmp = t_1
	elif b <= 2.25e+65:
		tmp = (-b - math.sqrt(((b * b) + (-4.0 * (c * a))))) / (2.0 * 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)
	t_0 = Float64(Float64(-c) / b)
	t_1 = Float64(2.0 * Float64(c / Float64(hypot(b, sqrt(Float64(c * Float64(a * -4.0)))) - b)))
	tmp = 0.0
	if (b <= -8.5e+55)
		tmp = t_0;
	elseif (b <= -7.5e+27)
		tmp = t_1;
	elseif (b <= -1e-20)
		tmp = t_0;
	elseif (b <= 3.2e-261)
		tmp = t_1;
	elseif (b <= 2.25e+65)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	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)
	t_0 = -c / b;
	t_1 = 2.0 * (c / (hypot(b, sqrt((c * (a * -4.0)))) - b));
	tmp = 0.0;
	if (b <= -8.5e+55)
		tmp = t_0;
	elseif (b <= -7.5e+27)
		tmp = t_1;
	elseif (b <= -1e-20)
		tmp = t_0;
	elseif (b <= 3.2e-261)
		tmp = t_1;
	elseif (b <= 2.25e+65)
		tmp = (-b - sqrt(((b * b) + (-4.0 * (c * a))))) / (2.0 * a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = 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[((-c) / b), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(c / N[(N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.5e+55], t$95$0, If[LessEqual[b, -7.5e+27], t$95$1, If[LessEqual[b, -1e-20], t$95$0, If[LessEqual[b, 3.2e-261], t$95$1, If[LessEqual[b, 2.25e+65], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	33.8
Target	21.0
Herbie	8.3

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

Derivation

1. Split input into 4 regimes

2. if b < -8.50000000000000002e55 or -7.5000000000000002e27 < b < -9.99999999999999945e-21

   1. Initial program 56.0

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

   2. Simplified 56.0

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

      [Start] 56.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 56.0	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 56.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 56.0	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 56.0	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 56.0	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 56.0	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 56.0	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around -inf 5.7

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

   4. Simplified 5.7

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

      [Start] 5.7	\[ -1 \cdot \frac{c}{b} \]
      mul-1-neg [=>] 5.7	\[ \color{blue}{-\frac{c}{b}} \]
      distribute-neg-frac [=>] 5.7	\[ \color{blue}{\frac{-c}{b}} \]

   if -8.50000000000000002e55 < b < -7.5000000000000002e27 or -9.99999999999999945e-21 < b < 3.20000000000000004e-261

   1. Initial program 24.8

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

   2. Simplified 24.8

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

      [Start] 24.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 24.8	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 24.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 24.8	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 24.8	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 24.8	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 24.8	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 24.8	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Applied egg-rr 31.2

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

   4. Simplified 31.2

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

      [Start] 31.2	\[ \frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b}{\left(a \cdot -2\right) \cdot \left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b\right)} \]
      fma-def [<=] 31.2	\[ \frac{\color{blue}{\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)} - b \cdot b}{\left(a \cdot -2\right) \cdot \left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b\right)} \]
      +-commutative [=>] 31.2	\[ \frac{\color{blue}{\left(a \cdot \left(c \cdot -4\right) + b \cdot b\right)} - b \cdot b}{\left(a \cdot -2\right) \cdot \left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b\right)} \]
      fma-def [=>] 31.2	\[ \frac{\color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b \cdot b}{\left(a \cdot -2\right) \cdot \left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b\right)} \]
      *-commutative [=>] 31.2	\[ \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b\right) \cdot \left(a \cdot -2\right)}} \]
      associate-*r* [=>] 31.2	\[ \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{\left(\mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) - b\right) \cdot \left(a \cdot -2\right)} \]
      *-commutative [<=] 31.2	\[ \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{\left(\mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}\right) - b\right) \cdot \left(a \cdot -2\right)} \]
      associate-*l* [=>] 31.2	\[ \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{\left(\mathsf{hypot}\left(b, \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}\right) - b\right) \cdot \left(a \cdot -2\right)} \]

   5. Taylor expanded in a around 0 27.1

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

   6. Simplified 27.1

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

      [Start] 27.1	\[ \frac{-4 \cdot \left(c \cdot a\right)}{\left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b\right) \cdot \left(a \cdot -2\right)} \]
      associate-*r* [=>] 27.1	\[ \frac{\color{blue}{\left(-4 \cdot c\right) \cdot a}}{\left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b\right) \cdot \left(a \cdot -2\right)} \]
      *-commutative [<=] 27.1	\[ \frac{\color{blue}{\left(c \cdot -4\right)} \cdot a}{\left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b\right) \cdot \left(a \cdot -2\right)} \]
      *-commutative [<=] 27.1	\[ \frac{\color{blue}{a \cdot \left(c \cdot -4\right)}}{\left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b\right) \cdot \left(a \cdot -2\right)} \]
      *-commutative [=>] 27.1	\[ \frac{a \cdot \color{blue}{\left(-4 \cdot c\right)}}{\left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b\right) \cdot \left(a \cdot -2\right)} \]

   7. Applied egg-rr 54.4

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

   8. Simplified 13.8

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

      [Start] 54.4	\[ e^{\mathsf{log1p}\left(\frac{a \cdot \left(2 \cdot \frac{c}{a}\right)}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right) - b}\right)} - 1 \]
      expm1-def [=>] 35.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \left(2 \cdot \frac{c}{a}\right)}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right) - b}\right)\right)} \]
      expm1-log1p [=>] 27.9	\[ \color{blue}{\frac{a \cdot \left(2 \cdot \frac{c}{a}\right)}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right) - b}} \]
      *-commutative [=>] 27.9	\[ \frac{\color{blue}{\left(2 \cdot \frac{c}{a}\right) \cdot a}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right) - b} \]
      associate-*r/ [<=] 28.3	\[ \color{blue}{\left(2 \cdot \frac{c}{a}\right) \cdot \frac{a}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right) - b}} \]
      associate-*l* [=>] 28.3	\[ \color{blue}{2 \cdot \left(\frac{c}{a} \cdot \frac{a}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right) - b}\right)} \]
      times-frac [<=] 27.0	\[ 2 \cdot \color{blue}{\frac{c \cdot a}{a \cdot \left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right) - b\right)}} \]
      *-commutative [=>] 27.0	\[ 2 \cdot \frac{\color{blue}{a \cdot c}}{a \cdot \left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right) - b\right)} \]
      times-frac [=>] 13.8	\[ 2 \cdot \color{blue}{\left(\frac{a}{a} \cdot \frac{c}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right) - b}\right)} \]
      associate-*r/ [=>] 13.8	\[ 2 \cdot \color{blue}{\frac{\frac{a}{a} \cdot c}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right) - b}} \]
      *-inverses [=>] 13.8	\[ 2 \cdot \frac{\color{blue}{1} \cdot c}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right) - b} \]
      *-lft-identity [=>] 13.8	\[ 2 \cdot \frac{\color{blue}{c}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right) - b} \]
      associate-*r* [=>] 13.8	\[ 2 \cdot \frac{c}{\mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c}}\right) - b} \]
      *-commutative [=>] 13.8	\[ 2 \cdot \frac{c}{\mathsf{hypot}\left(b, \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}\right) - b} \]

   if 3.20000000000000004e-261 < b < 2.25e65

   1. Initial program 8.5

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

   if 2.25e65 < b

   1. Initial program 41.3

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

   2. Simplified 41.4

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

      [Start] 41.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 41.3	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 41.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 41.3	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 41.4	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 41.4	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 41.4	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 41.4	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around inf 4.9

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

   4. Simplified 4.9

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

      [Start] 4.9	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      +-commutative [=>] 4.9	\[ \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
      associate-*r/ [=>] 4.9	\[ \color{blue}{\frac{-1 \cdot b}{a}} + \frac{c}{b} \]
      mul-1-neg [=>] 4.9	\[ \frac{\color{blue}{-b}}{a} + \frac{c}{b} \]

3. Recombined 4 regimes into one program.

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \frac{c}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-261}:\\ \;\;\;\;2 \cdot \frac{c}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-88}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	10.3
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+65}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	13.9
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-132}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	13.9
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-132}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Error	23.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-258}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6
Error	39.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -29000000000000:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	23.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-228}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8
Error	62.3
Cost	192

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

Alternative 9
Error	56.7
Cost	192

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

Reproduce

herbie shell --seed 2023027 
(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)))