Quadratic roots, narrow range

Average Accuracy: 55.0% → 92.1%

Time: 17.4s

Precision: binary64

Cost: 40900

• Unsound rule application detected in e-graph. Results may not be sound.

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq 0.07:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(a \cdot c\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right)\\ \end{array} \]

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 0.07)
   (* (- (sqrt (fma b b (* -4.0 (* a c)))) b) (/ 0.5 a))
   (fma
    (/ -0.25 a)
    (/ (* (pow (* a c) 4.0) 20.0) (pow b 7.0))
    (-
     (* -2.0 (* (* (* c c) (* c (pow b -5.0))) (* a a)))
     (fma (/ c (/ (pow b 3.0) c)) a (/ 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 <= 0.07) {
		tmp = (sqrt(fma(b, b, (-4.0 * (a * c)))) - b) * (0.5 / a);
	} else {
		tmp = fma((-0.25 / a), ((pow((a * c), 4.0) * 20.0) / pow(b, 7.0)), ((-2.0 * (((c * c) * (c * pow(b, -5.0))) * (a * a))) - fma((c / (pow(b, 3.0) / c)), a, (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 <= 0.07)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(a * c)))) - b) * Float64(0.5 / a));
	else
		tmp = fma(Float64(-0.25 / a), Float64(Float64((Float64(a * c) ^ 4.0) * 20.0) / (b ^ 7.0)), Float64(Float64(-2.0 * Float64(Float64(Float64(c * c) * Float64(c * (b ^ -5.0))) * Float64(a * a))) - fma(Float64(c / Float64((b ^ 3.0) / c)), a, 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, 0.07], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 / a), $MachinePrecision] * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[(N[(c * c), $MachinePrecision] * N[(c * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * a + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.070000000000000007

   1. Initial program 90.0%

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

   2. Simplified 90.2%

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

      [Start] 90.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 90.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 90.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [<=] 90.0	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 90.0	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]
      +-commutative [=>] 90.0	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]
      unsub-neg [=>] 90.0	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]
      fma-neg [=>] 90.2	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      associate-*l* [=>] 90.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      *-commutative [=>] 90.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      distribute-rgt-neg-in [=>] 90.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      metadata-eval [=>] 90.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      associate-/r* [=>] 90.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]
      metadata-eval [=>] 90.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]
      metadata-eval [=>] 90.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   if 0.070000000000000007 < b

   1. Initial program 53.2%

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

   2. Simplified 53.3%

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

      [Start] 53.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 53.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]
      +-commutative [=>] 53.2	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]
      unsub-neg [=>] 53.2	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]
      fma-neg [=>] 53.3	\[ \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]
      associate-*l* [=>] 53.3	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b}{a \cdot 2} \]
      *-commutative [=>] 53.3	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b}{a \cdot 2} \]
      distribute-rgt-neg-in [=>] 53.3	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b}{a \cdot 2} \]
      metadata-eval [=>] 53.3	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b}{a \cdot 2} \]

   3. Taylor expanded in b around inf 91.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

   4. Simplified 91.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right)} \]
      Proof

      [Start] 91.1	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
      +-commutative [=>] 91.1	\[ \color{blue}{\left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
      associate-+l+ [=>] 91.1	\[ \color{blue}{-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
      associate-*r/ [=>] 91.1	\[ \color{blue}{\frac{-0.25 \cdot \left({\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)\right)}{a \cdot {b}^{7}}} + \left(\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) \]
      times-frac [=>] 91.1	\[ \color{blue}{\frac{-0.25}{a} \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}} + \left(\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) \]
      +-commutative [<=] 91.1	\[ \frac{-0.25}{a} \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}} + \color{blue}{\left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]
      fma-def [=>] 91.1	\[ \color{blue}{\mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}, -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

   5. Applied egg-rr 91.1%

      \[\leadsto \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\color{blue}{\left(0 + {\left(c \cdot a\right)}^{4}\right)} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      Proof

      [Start] 91.1	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      add-log-exp [=>] 68.0	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\color{blue}{\log \left(e^{{c}^{4} \cdot {a}^{4}}\right)} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      *-un-lft-identity [=>] 68.0	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\log \color{blue}{\left(1 \cdot e^{{c}^{4} \cdot {a}^{4}}\right)} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      log-prod [=>] 68.0	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\color{blue}{\left(\log 1 + \log \left(e^{{c}^{4} \cdot {a}^{4}}\right)\right)} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      metadata-eval [=>] 68.0	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left(\color{blue}{0} + \log \left(e^{{c}^{4} \cdot {a}^{4}}\right)\right) \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      add-log-exp [<=] 91.1	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left(0 + \color{blue}{{c}^{4} \cdot {a}^{4}}\right) \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      pow-prod-down [=>] 91.1	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left(0 + \color{blue}{{\left(c \cdot a\right)}^{4}}\right) \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]

   6. Simplified 91.1%

      \[\leadsto \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\color{blue}{{\left(c \cdot a\right)}^{4}} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      Proof

      [Start] 91.1	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left(0 + {\left(c \cdot a\right)}^{4}\right) \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      +-lft-identity [=>] 91.1	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\color{blue}{{\left(c \cdot a\right)}^{4}} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]

   7. Applied egg-rr 91.1%

      \[\leadsto \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right)} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      Proof

      [Start] 91.1	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      div-inv [=>] 91.1	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\color{blue}{\left({c}^{3} \cdot \frac{1}{{b}^{5}}\right)} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      unpow3 [=>] 91.1	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\left(\color{blue}{\left(\left(c \cdot c\right) \cdot c\right)} \cdot \frac{1}{{b}^{5}}\right) \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      associate-*l* [=>] 91.1	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot \frac{1}{{b}^{5}}\right)\right)} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      pow-flip [=>] 91.1	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot \color{blue}{{b}^{\left(-5\right)}}\right)\right) \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
      metadata-eval [=>] 91.1	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{\color{blue}{-5}}\right)\right) \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.07:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(a \cdot c\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	88.9%
Cost	20868

\[\begin{array}{l} \mathbf{if}\;b \leq 18:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 2
Accuracy	85.1%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 31:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c}{\left(b \cdot b\right) \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 3
Accuracy	85.0%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 31:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c}{\left(b \cdot b\right) \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 4
Accuracy	85.0%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 31:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c}{\left(b \cdot b\right) \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 5
Accuracy	82.0%
Cost	1024

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

Alternative 6
Accuracy	64.8%
Cost	256

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

Reproduce

herbie shell --seed 2023160 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))