Quadratic roots, narrow range

Average Error: 28.7 → 4.9

Time: 16.0s

Precision: binary64

Cost: 73796

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{a}{\frac{e^{\mathsf{log1p}\left({b}^{5} \cdot {c}^{-3}\right)} + -1}{a}}, \frac{{a}^{3} \cdot -0.25}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\\ \end{array} \]

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 (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -10.0)
   (/ (- (sqrt (fma b b (* -4.0 (* a c)))) b) (* a 2.0))
   (-
    (fma
     -2.0
     (/ a (/ (+ (exp (log1p (* (pow b 5.0) (pow c -3.0)))) -1.0) a))
     (* (/ (* (pow a 3.0) -0.25) b) (* (/ (pow c 4.0) (pow b 6.0)) 20.0)))
    (fma (/ (* c c) (pow b 3.0)) 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 (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -10.0) {
		tmp = (sqrt(fma(b, b, (-4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = fma(-2.0, (a / ((exp(log1p((pow(b, 5.0) * pow(c, -3.0)))) + -1.0) / a)), (((pow(a, 3.0) * -0.25) / b) * ((pow(c, 4.0) / pow(b, 6.0)) * 20.0))) - fma(((c * c) / pow(b, 3.0)), 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 (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -10.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-2.0, Float64(a / Float64(Float64(exp(log1p(Float64((b ^ 5.0) * (c ^ -3.0)))) + -1.0) / a)), Float64(Float64(Float64((a ^ 3.0) * -0.25) / b) * Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 20.0))) - fma(Float64(Float64(c * c) / (b ^ 3.0)), 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[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -10.0], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(a / N[(N[(N[Exp[N[Log[1 + N[(N[Power[b, 5.0], $MachinePrecision] * N[Power[c, -3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] * -0.25), $MachinePrecision] / b), $MachinePrecision] * N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * a + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -10

   1. Initial program 9.8

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

   2. Applied egg-rr 9.8

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

   if -10 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 30.7

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

   2. Taylor expanded in a around 0 4.4

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

   3. Simplified 4.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{a}{\frac{\frac{{b}^{5}}{{c}^{3}}}{a}}, \frac{{a}^{3} \cdot -0.25}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)} \]
      Proof
      (-.f64 (fma.f64 -2 (/.f64 a (/.f64 (/.f64 (pow.f64 b 5) (pow.f64 c 3)) a)) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 20))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a a) (/.f64 (pow.f64 b 5) (pow.f64 c 3)))) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 20))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (/.f64 (pow.f64 b 5) (pow.f64 c 3))) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 20))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 c 3)) (pow.f64 b 5))) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 20))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 c 3) (pow.f64 a 2))) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 20))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) (Rewrite<= metadata-eval (+.f64 16 4))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 4 (/.f64 (pow.f64 c 4) (pow.f64 b 6))))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (Rewrite<= metadata-eval (*.f64 -2 -2)) (/.f64 (pow.f64 c 4) (pow.f64 b 6)))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -2 -2) (/.f64 (pow.f64 c (Rewrite<= metadata-eval (*.f64 2 2))) (pow.f64 b 6)))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -2 -2) (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 c 2) (pow.f64 c 2))) (pow.f64 b 6)))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -2 -2) (/.f64 (*.f64 (pow.f64 c 2) (pow.f64 c 2)) (pow.f64 b (Rewrite<= metadata-eval (*.f64 2 3)))))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -2 -2) (/.f64 (*.f64 (pow.f64 c 2) (pow.f64 c 2)) (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 b 3) (pow.f64 b 3)))))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -2 -2) (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (pow.f64 c 2) (pow.f64 b 3)) (/.f64 (pow.f64 c 2) (pow.f64 b 3)))))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3)))))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (Rewrite<= unpow2_binary64 (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 (pow.f64 a 3) -1/4) (/.f64 b (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2)))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (pow.f64 a 3) (/.f64 b (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2)))) -1/4))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) -1/4)) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (Rewrite<= *-commutative_binary64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (fma.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (pow.f64 b 3)) a (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (pow.f64 c 2) (pow.f64 b 3)) a) (/.f64 c b)))): 0 points increase in error, 1 points decrease in error
      (-.f64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (/.f64 c b))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (neg.f64 (+.f64 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 c b))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 c b))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (*.f64 -1 (/.f64 c b))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (*.f64 -1 (/.f64 c b))) (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (+.f64 (*.f64 -1 (/.f64 c b)) (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))))))): 1 points increase in error, 3 points decrease in error

   4. Applied egg-rr 4.4

      \[\leadsto \mathsf{fma}\left(-2, \frac{a}{\frac{\color{blue}{e^{\mathsf{log1p}\left({b}^{5} \cdot {c}^{-3}\right)} - 1}}{a}}, \frac{{a}^{3} \cdot -0.25}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{a}{\frac{e^{\mathsf{log1p}\left({b}^{5} \cdot {c}^{-3}\right)} + -1}{a}}, \frac{{a}^{3} \cdot -0.25}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	4.9
Cost	61508

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{a}{\frac{e^{\mathsf{log1p}\left({b}^{5} \cdot {c}^{-3}\right)} + -1}{a}}, \frac{{a}^{3} \cdot -0.25}{b} \cdot \left(20 \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right)\right)\right) - \mathsf{fma}\left(\frac{c}{b} \cdot \frac{c}{b \cdot b}, a, \frac{c}{b}\right)\\ \end{array} \]

Alternative 2
Error	4.9
Cost	60868

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{a}{\frac{\frac{{b}^{5}}{{c}^{3}}}{a}}, \frac{{a}^{3} \cdot -0.25}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\\ \end{array} \]

Alternative 3
Error	4.9
Cost	42500

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{a}{\frac{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}{a}}, \frac{{a}^{3} \cdot -0.25}{b} \cdot \left(20 \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right)\right)\right) - \mathsf{fma}\left(\frac{c}{b} \cdot \frac{c}{b \cdot b}, a, \frac{c}{b}\right)\\ \end{array} \]

Alternative 4
Error	6.5
Cost	34436

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{c}^{3} \cdot \left(-2 \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\\ \end{array} \]

Alternative 5
Error	9.5
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.007:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\\ \end{array} \]

Alternative 6
Error	9.5
Cost	20932

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.007:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\\ \end{array} \]

Alternative 7
Error	9.6
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.007:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-1}{b} - c \cdot \left(c \cdot \frac{a}{{b}^{3}}\right)\\ \end{array} \]

Alternative 8
Error	9.6
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 3.45:\\ \;\;\;\;\left(\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-1}{b} - c \cdot \left(c \cdot \frac{a}{{b}^{3}}\right)\\ \end{array} \]

Alternative 9
Error	11.7
Cost	7296

\[c \cdot \frac{-1}{b} - c \cdot \left(c \cdot \frac{a}{{b}^{3}}\right) \]

Alternative 10
Error	11.7
Cost	960

\[c \cdot \left(\frac{-1}{b} - \frac{c}{\left(b \cdot b\right) \cdot \frac{b}{a}}\right) \]

Alternative 11
Error	22.7
Cost	256

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

Reproduce

herbie shell --seed 2022318 
(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)))