The quadratic formula (r2)

Average Error: 33.7 → 6.7

Time: 21.8s

Precision: binary64

Cost: 7820

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-244}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b} - b}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+93}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+153)
   (/ (* c 2.0) (fma 2.0 (/ c (/ b a)) (* b -2.0)))
   (if (<= b 1.35e-244)
     (/ (* c 2.0) (- (sqrt (+ (* c (* a -4.0)) (* b b))) b))
     (if (<= b 3e+93)
       (/ (- (- 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 tmp;
	if (b <= -5e+153) {
		tmp = (c * 2.0) / fma(2.0, (c / (b / a)), (b * -2.0));
	} else if (b <= 1.35e-244) {
		tmp = (c * 2.0) / (sqrt(((c * (a * -4.0)) + (b * b))) - b);
	} else if (b <= 3e+93) {
		tmp = (-b - 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)
	tmp = 0.0
	if (b <= -5e+153)
		tmp = Float64(Float64(c * 2.0) / fma(2.0, Float64(c / Float64(b / a)), Float64(b * -2.0)));
	elseif (b <= 1.35e-244)
		tmp = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(Float64(c * Float64(a * -4.0)) + Float64(b * b))) - b));
	elseif (b <= 3e+93)
		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

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, -5e+153], 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, 1.35e-244], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3e+93], 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	33.7
Target	20.3
Herbie	6.7

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

Derivation

1. Split input into 4 regimes

2. if b < -5.00000000000000018e153

   1. Initial program 63.9

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

   2. Simplified 63.9

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

      [Start] 63.9	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 63.9	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 63.9	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 63.9	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 63.9	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 63.9	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 63.9	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 63.9	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 63.9	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 63.9	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 63.9	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 63.9	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 63.9	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 63.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 63.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]
      associate-*r* [=>] 63.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\left(-\color{blue}{\left(4 \cdot a\right) \cdot c}\right) + b \cdot b}\right) \]
      distribute-lft-neg-in [=>] 63.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c} + b \cdot b}\right) \]
      *-commutative [=>] 63.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(-4 \cdot a\right)} + b \cdot b}\right) \]
      fma-def [=>] 63.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right) \]
      *-commutative [=>] 63.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, -\color{blue}{a \cdot 4}, b \cdot b\right)}\right) \]
      distribute-rgt-neg-in [=>] 63.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(-4\right)}, b \cdot b\right)}\right) \]
      metadata-eval [=>] 63.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-4}, b \cdot b\right)}\right) \]

   3. Applied egg-rr 63.9

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

   4. Simplified 63.9

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

      [Start] 63.9	\[ \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot -2\right) \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right)} \]
      associate-/r* [=>] 63.9	\[ \color{blue}{\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}} \]
      fma-def [<=] 63.9	\[ \frac{\frac{\color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)} - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      +-commutative [<=] 63.9	\[ \frac{\frac{\color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)} - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      fma-def [=>] 63.9	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      *-commutative [=>] 63.9	\[ \frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b}{\color{blue}{-2 \cdot a}}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      fma-def [<=] 63.9	\[ \frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b}{-2 \cdot a}}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}} - b} \]
      +-commutative [<=] 63.9	\[ \frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b}{-2 \cdot a}}{\sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}} - b} \]
      fma-def [=>] 63.9	\[ \frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b}{-2 \cdot a}}{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} - b} \]

   5. Taylor expanded in b around 0 36.4

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

   6. Simplified 36.4

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

      [Start] 36.4	\[ \frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b} \]
      *-commutative [=>] 36.4	\[ \frac{\color{blue}{c \cdot 2}}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b} \]

   7. Taylor expanded in b around -inf 6.8

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

   8. Simplified 1.7

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

      [Start] 6.8	\[ \frac{c \cdot 2}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b} \]
      fma-def [=>] 6.8	\[ \frac{c \cdot 2}{\color{blue}{\mathsf{fma}\left(2, \frac{c \cdot a}{b}, -2 \cdot b\right)}} \]
      associate-/l* [=>] 1.7	\[ \frac{c \cdot 2}{\mathsf{fma}\left(2, \color{blue}{\frac{c}{\frac{b}{a}}}, -2 \cdot b\right)} \]
      *-commutative [=>] 1.7	\[ \frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot -2}\right)} \]

   if -5.00000000000000018e153 < b < 1.35e-244

   1. Initial program 31.9

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

   2. Simplified 31.9

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

      [Start] 31.9	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 31.9	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 31.9	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 31.9	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 31.9	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 31.9	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 31.9	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 31.9	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 31.9	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 31.9	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 31.9	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 31.9	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 31.9	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 31.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 31.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]
      associate-*r* [=>] 31.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\left(-\color{blue}{\left(4 \cdot a\right) \cdot c}\right) + b \cdot b}\right) \]
      distribute-lft-neg-in [=>] 31.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c} + b \cdot b}\right) \]
      *-commutative [=>] 31.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(-4 \cdot a\right)} + b \cdot b}\right) \]
      fma-def [=>] 31.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right) \]
      *-commutative [=>] 31.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, -\color{blue}{a \cdot 4}, b \cdot b\right)}\right) \]
      distribute-rgt-neg-in [=>] 31.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(-4\right)}, b \cdot b\right)}\right) \]
      metadata-eval [=>] 31.9	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-4}, b \cdot b\right)}\right) \]

   3. Applied egg-rr 36.3

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

   4. Simplified 31.9

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

      [Start] 36.3	\[ \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot -2\right) \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right)} \]
      associate-/r* [=>] 31.9	\[ \color{blue}{\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}} \]
      fma-def [<=] 31.9	\[ \frac{\frac{\color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)} - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      +-commutative [<=] 31.9	\[ \frac{\frac{\color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)} - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      fma-def [=>] 31.9	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      *-commutative [=>] 31.9	\[ \frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b}{\color{blue}{-2 \cdot a}}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      fma-def [<=] 31.9	\[ \frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b}{-2 \cdot a}}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}} - b} \]
      +-commutative [<=] 31.9	\[ \frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b}{-2 \cdot a}}{\sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}} - b} \]
      fma-def [=>] 31.9	\[ \frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b}{-2 \cdot a}}{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} - b} \]

   5. Taylor expanded in b around 0 8.4

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

   6. Simplified 8.4

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

      [Start] 8.4	\[ \frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b} \]
      *-commutative [=>] 8.4	\[ \frac{\color{blue}{c \cdot 2}}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b} \]

   7. Applied egg-rr 8.5

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

   if 1.35e-244 < b < 2.99999999999999978e93

   1. Initial program 8.4

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

   if 2.99999999999999978e93 < b

   1. Initial program 46.3

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

   2. Simplified 46.4

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

      [Start] 46.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 46.3	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 46.3	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 46.3	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 46.4	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 46.4	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 46.4	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 46.4	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 46.4	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 46.4	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 46.4	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 46.4	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 46.4	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 46.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 46.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]
      associate-*r* [=>] 46.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\left(-\color{blue}{\left(4 \cdot a\right) \cdot c}\right) + b \cdot b}\right) \]
      distribute-lft-neg-in [=>] 46.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c} + b \cdot b}\right) \]
      *-commutative [=>] 46.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(-4 \cdot a\right)} + b \cdot b}\right) \]
      fma-def [=>] 46.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right) \]
      *-commutative [=>] 46.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, -\color{blue}{a \cdot 4}, b \cdot b\right)}\right) \]
      distribute-rgt-neg-in [=>] 46.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(-4\right)}, b \cdot b\right)}\right) \]
      metadata-eval [=>] 46.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-4}, b \cdot b\right)}\right) \]

   3. Applied egg-rr 63.5

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

   4. Simplified 63.3

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

      [Start] 63.5	\[ \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot -2\right) \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right)} \]
      associate-/r* [=>] 63.3	\[ \color{blue}{\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}} \]
      fma-def [<=] 63.3	\[ \frac{\frac{\color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)} - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      +-commutative [<=] 63.3	\[ \frac{\frac{\color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)} - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      fma-def [=>] 63.3	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b \cdot b}{a \cdot -2}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      *-commutative [=>] 63.3	\[ \frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b}{\color{blue}{-2 \cdot a}}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \]
      fma-def [<=] 63.3	\[ \frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b}{-2 \cdot a}}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}} - b} \]
      +-commutative [<=] 63.3	\[ \frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b}{-2 \cdot a}}{\sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}} - b} \]
      fma-def [=>] 63.3	\[ \frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b}{-2 \cdot a}}{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} - b} \]

   5. Taylor expanded in b around 0 62.2

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

   6. Simplified 62.2

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

      [Start] 62.2	\[ \frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b} \]
      *-commutative [=>] 62.2	\[ \frac{\color{blue}{c \cdot 2}}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b} \]

   7. Taylor expanded in c around 0 4.7

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

   8. Simplified 4.7

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

      [Start] 4.7	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 4.7	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 4.7	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-244}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b} - b}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+93}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.7
Cost	7756

\[\begin{array}{l} t_0 := \sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-231}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \mathbf{elif}\;b \leq 3.45 \cdot 10^{+93}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	10.5
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	13.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	13.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Error	13.6
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-46}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6
Error	23.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 7
Error	39.7
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -135000:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8
Error	23.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-249}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 9
Error	62.3
Cost	192

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

Alternative 10
Error	56.3
Cost	192

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

Reproduce

herbie shell --seed 2022364 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))