Quadratic roots, full range

Percentage Accurate: 51.6% → 90.6%

Time: 19.1s

Precision: binary64

Cost: 14412

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+91}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a \cdot -2}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{a}{a \cdot -2} \cdot \frac{c \cdot 4}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.55e+91)
   (/ (- b) a)
   (if (<= b 3.4e-291)
     (/ (- b (sqrt (- (* b b) (* a (* c 4.0))))) (* a -2.0))
     (if (<= b 1.05e+103)
       (*
        (/ a (* a -2.0))
        (/ (* c 4.0) (+ b (sqrt (fma a (* c -4.0) (* b b))))))
       (/ (- 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 tmp;
	if (b <= -1.55e+91) {
		tmp = -b / a;
	} else if (b <= 3.4e-291) {
		tmp = (b - sqrt(((b * b) - (a * (c * 4.0))))) / (a * -2.0);
	} else if (b <= 1.05e+103) {
		tmp = (a / (a * -2.0)) * ((c * 4.0) / (b + sqrt(fma(a, (c * -4.0), (b * b)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.55e+91)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3.4e-291)
		tmp = Float64(Float64(b - sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 4.0))))) / Float64(a * -2.0));
	elseif (b <= 1.05e+103)
		tmp = Float64(Float64(a / Float64(a * -2.0)) * Float64(Float64(c * 4.0) / Float64(b + sqrt(fma(a, Float64(c * -4.0), Float64(b * b))))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := If[LessEqual[b, -1.55e+91], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3.4e-291], N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e+103], N[(N[(a / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(c * 4.0), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.54999999999999999e91

   1. Initial program 54.8%

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

   2. Simplified 54.7%

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

      [Start] 54.8%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 54.8%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 54.8%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 54.8%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 54.8%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 54.6%	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 54.6%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 54.6%	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 54.6%	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 54.6%	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Taylor expanded in b around -inf 93.0%

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

   4. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
      Step-by-step derivation

      [Start] 93.0%	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 93.0%	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 93.0%	\[ \frac{\color{blue}{-b}}{a} \]

   if -1.54999999999999999e91 < b < 3.40000000000000026e-291

   1. Initial program 85.6%

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

   2. Simplified 85.4%

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

      [Start] 85.6%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 85.6%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 85.6%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 85.6%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 85.6%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 85.4%	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 85.4%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 85.4%	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 85.4%	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 85.4%	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Applied egg-rr 85.4%

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

      [Start] 85.4%	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      fma-udef [=>] 85.4%	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 85.4%	\[ \left(b - \sqrt{a \cdot \color{blue}{\left(-4 \cdot c\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      associate-*r* [=>] 85.4%	\[ \left(b - \sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      metadata-eval [<=] 85.4%	\[ \left(b - \sqrt{\left(a \cdot \color{blue}{\left(-4\right)}\right) \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      distribute-rgt-neg-in [<=] 85.4%	\[ \left(b - \sqrt{\color{blue}{\left(-a \cdot 4\right)} \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [<=] 85.4%	\[ \left(b - \sqrt{\left(-\color{blue}{4 \cdot a}\right) \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      distribute-lft-neg-in [<=] 85.4%	\[ \left(b - \sqrt{\color{blue}{\left(-\left(4 \cdot a\right) \cdot c\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      +-commutative [=>] 85.4%	\[ \left(b - \sqrt{\color{blue}{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)}}\right) \cdot \frac{-0.5}{a} \]
      sub-neg [<=] 85.4%	\[ \left(b - \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 85.4%	\[ \left(b - \sqrt{b \cdot b - \color{blue}{\left(a \cdot 4\right)} \cdot c}\right) \cdot \frac{-0.5}{a} \]
      associate-*l* [=>] 85.4%	\[ \left(b - \sqrt{b \cdot b - \color{blue}{a \cdot \left(4 \cdot c\right)}}\right) \cdot \frac{-0.5}{a} \]

   4. Simplified 85.4%

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

      [Start] 85.4%	\[ \left(b - \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 85.4%	\[ \left(b - \sqrt{b \cdot b - a \cdot \color{blue}{\left(c \cdot 4\right)}}\right) \cdot \frac{-0.5}{a} \]

   5. Applied egg-rr 85.6%

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

      [Start] 85.4%	\[ \left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot \frac{-0.5}{a} \]
      clear-num [=>] 85.4%	\[ \left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot \color{blue}{\frac{1}{\frac{a}{-0.5}}} \]
      un-div-inv [=>] 85.6%	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{\frac{a}{-0.5}}} \]
      div-inv [=>] 85.6%	\[ \frac{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{\color{blue}{a \cdot \frac{1}{-0.5}}} \]
      metadata-eval [=>] 85.6%	\[ \frac{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a \cdot \color{blue}{-2}} \]

   if 3.40000000000000026e-291 < b < 1.0500000000000001e103

   1. Initial program 42.5%

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

   2. Simplified 42.5%

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

      [Start] 42.5%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 42.5%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 42.5%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 42.5%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 42.5%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 42.5%	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 42.5%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 42.5%	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 42.5%	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 42.5%	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Applied egg-rr 42.5%

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

      [Start] 42.5%	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      fma-udef [=>] 42.5%	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 42.5%	\[ \left(b - \sqrt{a \cdot \color{blue}{\left(-4 \cdot c\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      associate-*r* [=>] 42.5%	\[ \left(b - \sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      metadata-eval [<=] 42.5%	\[ \left(b - \sqrt{\left(a \cdot \color{blue}{\left(-4\right)}\right) \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      distribute-rgt-neg-in [<=] 42.5%	\[ \left(b - \sqrt{\color{blue}{\left(-a \cdot 4\right)} \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [<=] 42.5%	\[ \left(b - \sqrt{\left(-\color{blue}{4 \cdot a}\right) \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      distribute-lft-neg-in [<=] 42.5%	\[ \left(b - \sqrt{\color{blue}{\left(-\left(4 \cdot a\right) \cdot c\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      +-commutative [=>] 42.5%	\[ \left(b - \sqrt{\color{blue}{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)}}\right) \cdot \frac{-0.5}{a} \]
      sub-neg [<=] 42.5%	\[ \left(b - \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 42.5%	\[ \left(b - \sqrt{b \cdot b - \color{blue}{\left(a \cdot 4\right)} \cdot c}\right) \cdot \frac{-0.5}{a} \]
      associate-*l* [=>] 42.5%	\[ \left(b - \sqrt{b \cdot b - \color{blue}{a \cdot \left(4 \cdot c\right)}}\right) \cdot \frac{-0.5}{a} \]

   4. Simplified 42.5%

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

      [Start] 42.5%	\[ \left(b - \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 42.5%	\[ \left(b - \sqrt{b \cdot b - a \cdot \color{blue}{\left(c \cdot 4\right)}}\right) \cdot \frac{-0.5}{a} \]

   5. Applied egg-rr 36.1%

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

      [Start] 42.5%	\[ \left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 42.5%	\[ \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)} \]
      clear-num [=>] 42.5%	\[ \color{blue}{\frac{1}{\frac{a}{-0.5}}} \cdot \left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \]
      flip-- [=>] 42.1%	\[ \frac{1}{\frac{a}{-0.5}} \cdot \color{blue}{\frac{b \cdot b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} \cdot \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}} \]
      frac-times [=>] 36.0%	\[ \color{blue}{\frac{1 \cdot \left(b \cdot b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} \cdot \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)}{\frac{a}{-0.5} \cdot \left(b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)}} \]
      *-un-lft-identity [<=] 36.0%	\[ \frac{\color{blue}{b \cdot b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} \cdot \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}}{\frac{a}{-0.5} \cdot \left(b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)} \]
      add-sqr-sqrt [<=] 36.1%	\[ \frac{b \cdot b - \color{blue}{\left(b \cdot b - a \cdot \left(c \cdot 4\right)\right)}}{\frac{a}{-0.5} \cdot \left(b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)} \]
      div-inv [=>] 36.1%	\[ \frac{b \cdot b - \left(b \cdot b - a \cdot \left(c \cdot 4\right)\right)}{\color{blue}{\left(a \cdot \frac{1}{-0.5}\right)} \cdot \left(b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)} \]
      metadata-eval [=>] 36.1%	\[ \frac{b \cdot b - \left(b \cdot b - a \cdot \left(c \cdot 4\right)\right)}{\left(a \cdot \color{blue}{-2}\right) \cdot \left(b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)} \]

   6. Simplified 71.7%

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

      [Start] 36.1%	\[ \frac{b \cdot b - \left(b \cdot b - a \cdot \left(c \cdot 4\right)\right)}{\left(a \cdot -2\right) \cdot \left(b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)} \]
      *-commutative [<=] 36.1%	\[ \frac{b \cdot b - \left(b \cdot b - a \cdot \left(c \cdot 4\right)\right)}{\color{blue}{\left(b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot \left(a \cdot -2\right)}} \]
      associate-/r* [=>] 42.2%	\[ \color{blue}{\frac{\frac{b \cdot b - \left(b \cdot b - a \cdot \left(c \cdot 4\right)\right)}{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}}{a \cdot -2}} \]

   7. Applied egg-rr 25.6%

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

      [Start] 71.7%	\[ \frac{\frac{4 \cdot \left(c \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a \cdot -2} \]
      expm1-log1p-u [=>] 63.0%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{4 \cdot \left(c \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a \cdot -2}\right)\right)} \]
      expm1-udef [=>] 26.0%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{4 \cdot \left(c \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a \cdot -2}\right)} - 1} \]
      associate-/l/ [=>] 25.6%	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot -2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}}\right)} - 1 \]
      *-commutative [=>] 25.6%	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{\left(a \cdot -2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\right)} - 1 \]
      *-commutative [=>] 25.6%	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(a \cdot c\right)} \cdot 4}{\left(a \cdot -2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\right)} - 1 \]
      associate-*r* [<=] 25.6%	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{a \cdot \left(c \cdot 4\right)}}{\left(a \cdot -2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\right)} - 1 \]

   8. Simplified 86.3%

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

      [Start] 25.6%	\[ e^{\mathsf{log1p}\left(\frac{a \cdot \left(c \cdot 4\right)}{\left(a \cdot -2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\right)} - 1 \]
      expm1-def [=>] 56.2%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \left(c \cdot 4\right)}{\left(a \cdot -2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\right)\right)} \]
      expm1-log1p [=>] 64.7%	\[ \color{blue}{\frac{a \cdot \left(c \cdot 4\right)}{\left(a \cdot -2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}} \]
      times-frac [=>] 86.3%	\[ \color{blue}{\frac{a}{a \cdot -2} \cdot \frac{c \cdot 4}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}} \]

   if 1.0500000000000001e103 < b

   1. Initial program 4.5%

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

   2. Simplified 4.5%

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

      [Start] 4.5%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 4.5%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 4.5%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 4.5%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 4.5%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 4.5%	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 4.5%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 4.5%	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 4.5%	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 4.5%	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Taylor expanded in b around inf 93.2%

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

   4. Simplified 93.2%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
      Step-by-step derivation

      [Start] 93.2%	\[ -1 \cdot \frac{c}{b} \]
      mul-1-neg [=>] 93.2%	\[ \color{blue}{-\frac{c}{b}} \]
      distribute-neg-frac [=>] 93.2%	\[ \color{blue}{\frac{-c}{b}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+91}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a \cdot -2}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{a}{a \cdot -2} \cdot \frac{c \cdot 4}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.6%
Cost	14412

\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+91}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a \cdot -2}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{a}{a \cdot -2} \cdot \frac{c \cdot 4}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Accuracy	87.5%
Cost	14284

\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+91}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-233}:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a \cdot -2}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{-2 \cdot \frac{a \cdot c}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Accuracy	83.4%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+91}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 10^{-198}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Accuracy	83.5%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+91}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 10^{-198}:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Accuracy	78.6%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 10^{-198}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6
Accuracy	78.6%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 10^{-198}:\\ \;\;\;\;\frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7
Accuracy	68.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8
Accuracy	43.6%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+36}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 9
Accuracy	68.4%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 10
Accuracy	10.8%
Cost	192

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

Reproduce

herbie shell --seed 2023263 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))