quadm (p42, negative)

Average Error: 33.9 → 6.9

Time: 17.7s

Precision: binary64

Cost: 13896

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+144}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

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 -1.15e+144)
   (/ (* c 2.0) (fma 2.0 (/ c (/ b a)) (* b -2.0)))
   (if (<= b 9.2e-281)
     (/ (* c 2.0) (- (sqrt (fma c (* a -4.0) (* b b))) b))
     (if (<= b 9e+61)
       (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* 2.0 a))
       (/ (- 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 <= -1.15e+144) {
		tmp = (c * 2.0) / fma(2.0, (c / (b / a)), (b * -2.0));
	} else if (b <= 9.2e-281) {
		tmp = (c * 2.0) / (sqrt(fma(c, (a * -4.0), (b * b))) - b);
	} else if (b <= 9e+61) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = -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 <= -1.15e+144)
		tmp = Float64(Float64(c * 2.0) / fma(2.0, Float64(c / Float64(b / a)), Float64(b * -2.0)));
	elseif (b <= 9.2e-281)
		tmp = Float64(Float64(c * 2.0) / Float64(sqrt(fma(c, Float64(a * -4.0), Float64(b * b))) - b));
	elseif (b <= 9e+61)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(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, -1.15e+144], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.2e-281], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9e+61], 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[((-b) / a), $MachinePrecision]]]]

Target

Original	33.9
Target	21.0
Herbie	6.9

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

Derivation

1. Split input into 4 regimes

2. if b < -1.1500000000000001e144

   1. Initial program 62.8

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

   2. Simplified 62.8

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

      [Start] 62.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 62.8	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 62.8	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 62.8	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 62.8	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 62.8	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 62.8	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 62.8	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 62.8	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 62.8	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 62.8	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 62.8	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 62.8	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 62.8	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 62.8	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Applied egg-rr 62.8

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

   4. Simplified 62.8

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

      [Start] 62.8	\[ \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot -2\right) \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right)} \]
      *-commutative [=>] 62.8	\[ \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right) \cdot \left(a \cdot -2\right)}} \]
      *-commutative [=>] 62.8	\[ \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right) \cdot \color{blue}{\left(-2 \cdot a\right)}} \]

   5. Applied egg-rr 62.8

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

   6. Simplified 36.0

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b}} \]
      Proof

      [Start] 62.8	\[ \left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b\right) \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}{a \cdot -2} \]
      associate-*r/ [=>] 62.8	\[ \color{blue}{\frac{\left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}{a \cdot -2}} \]
      associate-*l/ [<=] 62.8	\[ \color{blue}{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a \cdot -2} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}} \]
      associate-*r/ [=>] 62.8	\[ \color{blue}{\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a \cdot -2} \cdot 1}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}} \]
      *-commutative [<=] 62.8	\[ \frac{\color{blue}{1 \cdot \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a \cdot -2}}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      fma-udef [=>] 62.8	\[ \frac{1 \cdot \frac{\color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)} - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      associate-+r- [<=] 60.7	\[ \frac{1 \cdot \frac{\color{blue}{c \cdot \left(a \cdot -4\right) + \left(b \cdot b - b \cdot b\right)}}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      fma-def [=>] 60.7	\[ \frac{1 \cdot \frac{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b - b \cdot b\right)}}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      *-commutative [=>] 60.7	\[ \frac{1 \cdot \frac{\mathsf{fma}\left(c, \color{blue}{-4 \cdot a}, b \cdot b - b \cdot b\right)}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      +-inverses [=>] 36.0	\[ \frac{1 \cdot \frac{\mathsf{fma}\left(c, -4 \cdot a, \color{blue}{0}\right)}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      *-commutative [=>] 36.0	\[ \frac{1 \cdot \frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, \color{blue}{-4 \cdot a}, b \cdot b\right)} - b} \]

   7. Taylor expanded in c around 0 35.8

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

   8. Simplified 35.8

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

      [Start] 35.8	\[ \frac{1 \cdot \left(2 \cdot c\right)}{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b} \]
      *-commutative [=>] 35.8	\[ \frac{1 \cdot \color{blue}{\left(c \cdot 2\right)}}{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b} \]

   9. Taylor expanded in b around -inf 6.2

      \[\leadsto \frac{1 \cdot \left(c \cdot 2\right)}{\color{blue}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}} \]

   10. Simplified 1.2

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

       [Start] 6.2	\[ \frac{1 \cdot \left(c \cdot 2\right)}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b} \]
       fma-def [=>] 6.2	\[ \frac{1 \cdot \left(c \cdot 2\right)}{\color{blue}{\mathsf{fma}\left(2, \frac{c \cdot a}{b}, -2 \cdot b\right)}} \]
       associate-/l* [=>] 1.2	\[ \frac{1 \cdot \left(c \cdot 2\right)}{\mathsf{fma}\left(2, \color{blue}{\frac{c}{\frac{b}{a}}}, -2 \cdot b\right)} \]
       *-commutative [=>] 1.2	\[ \frac{1 \cdot \left(c \cdot 2\right)}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot -2}\right)} \]

   if -1.1500000000000001e144 < b < 9.19999999999999956e-281

   1. Initial program 33.3

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

   2. Simplified 33.3

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

      [Start] 33.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 33.3	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 33.3	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 33.3	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 33.3	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 33.3	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 33.3	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 33.3	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 33.3	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 33.3	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 33.3	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 33.3	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 33.3	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 33.3	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 33.3	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Applied egg-rr 37.6

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

   4. Simplified 37.6

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

      [Start] 37.6	\[ \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot -2\right) \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right)} \]
      *-commutative [=>] 37.6	\[ \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right) \cdot \left(a \cdot -2\right)}} \]
      *-commutative [=>] 37.6	\[ \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right) \cdot \color{blue}{\left(-2 \cdot a\right)}} \]

   5. Applied egg-rr 37.6

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

   6. Simplified 14.7

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b}} \]
      Proof

      [Start] 37.6	\[ \left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b\right) \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}{a \cdot -2} \]
      associate-*r/ [=>] 33.3	\[ \color{blue}{\frac{\left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}{a \cdot -2}} \]
      associate-*l/ [<=] 33.3	\[ \color{blue}{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a \cdot -2} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}} \]
      associate-*r/ [=>] 33.2	\[ \color{blue}{\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a \cdot -2} \cdot 1}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}} \]
      *-commutative [<=] 33.2	\[ \frac{\color{blue}{1 \cdot \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a \cdot -2}}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      fma-udef [=>] 33.2	\[ \frac{1 \cdot \frac{\color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)} - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      associate-+r- [<=] 14.7	\[ \frac{1 \cdot \frac{\color{blue}{c \cdot \left(a \cdot -4\right) + \left(b \cdot b - b \cdot b\right)}}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      fma-def [=>] 14.7	\[ \frac{1 \cdot \frac{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b - b \cdot b\right)}}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      *-commutative [=>] 14.7	\[ \frac{1 \cdot \frac{\mathsf{fma}\left(c, \color{blue}{-4 \cdot a}, b \cdot b - b \cdot b\right)}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      +-inverses [=>] 14.7	\[ \frac{1 \cdot \frac{\mathsf{fma}\left(c, -4 \cdot a, \color{blue}{0}\right)}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      *-commutative [=>] 14.7	\[ \frac{1 \cdot \frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, \color{blue}{-4 \cdot a}, b \cdot b\right)} - b} \]

   7. Taylor expanded in c around 0 8.7

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

   8. Simplified 8.7

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

      [Start] 8.7	\[ \frac{1 \cdot \left(2 \cdot c\right)}{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b} \]
      *-commutative [=>] 8.7	\[ \frac{1 \cdot \color{blue}{\left(c \cdot 2\right)}}{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b} \]

   9. Applied egg-rr 49.7

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

   10. Simplified 8.7

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

       [Start] 49.7	\[ e^{\mathsf{log1p}\left(c \cdot \frac{2}{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b}\right)} - 1 \]
       expm1-def [=>] 20.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{2}{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b}\right)\right)} \]
       expm1-log1p [=>] 8.8	\[ \color{blue}{c \cdot \frac{2}{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b}} \]
       associate-*r/ [=>] 8.7	\[ \color{blue}{\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b}} \]
       *-commutative [=>] 8.7	\[ \frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot -4}, b \cdot b\right)} - b} \]

   if 9.19999999999999956e-281 < b < 9e61

   1. Initial program 9.3

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

   if 9e61 < b

   1. Initial program 39.7

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

   2. Simplified 39.8

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

      [Start] 39.7	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 39.7	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 39.7	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 39.7	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 39.8	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 39.8	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 39.8	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 39.8	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 39.8	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 39.8	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 39.8	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 39.8	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 39.8	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 39.8	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 39.8	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Taylor expanded in a around 0 5.8

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

   4. Simplified 5.8

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

      [Start] 5.8	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 5.8	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 5.8	\[ \frac{\color{blue}{-b}}{a} \]

3. Recombined 4 regimes into one program.

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+144}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.0
Cost	13896

\[\begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-280}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2
Error	10.3
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-77}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3
Error	13.7
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-77}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-110}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	13.6
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-110}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Error	40.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+71}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	23.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-279}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	62.3
Cost	192

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

Alternative 8
Error	56.8
Cost	192

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

Reproduce

herbie shell --seed 2023012 
(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)))