Quadratic roots, narrow range

Average Error: 28.8 → 5.3

Time: 16.1s

Precision: binary64

Cost: 41924

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

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, -5 \cdot \left(a \cdot \left(\left(a \cdot a\right) \cdot \left({c}^{4} \cdot {b}^{-7}\right)\right)\right)\right) - \frac{c}{b}\right) - \frac{c \cdot c}{b \cdot b} \cdot \frac{a}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a -4.0))))
   (if (<= (/ (- (sqrt (+ (* b b) t_0)) b) (* a 2.0)) -5.0)
     (/ (- (sqrt (fma b b t_0)) b) (* a 2.0))
     (-
      (-
       (fma
        -2.0
        (/ (* (pow c 3.0) (* a a)) (pow b 5.0))
        (* -5.0 (* a (* (* a a) (* (pow c 4.0) (pow b -7.0))))))
       (/ c b))
      (* (/ (* c c) (* b b)) (/ a 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 t_0 = c * (a * -4.0);
	double tmp;
	if (((sqrt(((b * b) + t_0)) - b) / (a * 2.0)) <= -5.0) {
		tmp = (sqrt(fma(b, b, t_0)) - b) / (a * 2.0);
	} else {
		tmp = (fma(-2.0, ((pow(c, 3.0) * (a * a)) / pow(b, 5.0)), (-5.0 * (a * ((a * a) * (pow(c, 4.0) * pow(b, -7.0)))))) - (c / b)) - (((c * c) / (b * b)) * (a / 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)
	t_0 = Float64(c * Float64(a * -4.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) + t_0)) - b) / Float64(a * 2.0)) <= -5.0)
		tmp = Float64(Float64(sqrt(fma(b, b, t_0)) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-2.0, Float64(Float64((c ^ 3.0) * Float64(a * a)) / (b ^ 5.0)), Float64(-5.0 * Float64(a * Float64(Float64(a * a) * Float64((c ^ 4.0) * (b ^ -7.0)))))) - Float64(c / b)) - Float64(Float64(Float64(c * c) / Float64(b * b)) * Float64(a / 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_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -5.0], N[(N[(N[Sqrt[N[(b * b + t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(a * N[(N[(a * a), $MachinePrecision] * N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[b, -7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(a / 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)) < -5

   1. Initial program 10.6

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

   2. Simplified 10.5

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -4)))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a (Rewrite<= metadata-eval (neg.f64 4)))))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4)))))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 a)))))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c (*.f64 4 a)))))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 4 a) c))))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) b) (*.f64 a 2)): 22 points increase in error, 10 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) (neg.f64 b))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= *-commutative_binary64 (*.f64 2 a))): 0 points increase in error, 0 points decrease in error

   if -5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 31.2

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

   2. Simplified 31.1

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -4)))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a (Rewrite<= metadata-eval (neg.f64 4)))))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4)))))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 a)))))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c (*.f64 4 a)))))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 4 a) c))))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) b) (*.f64 a 2)): 22 points increase in error, 10 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) (neg.f64 b))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= *-commutative_binary64 (*.f64 2 a))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in a around 0 4.6

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

   4. Simplified 4.6

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

   5. Taylor expanded in c around 0 4.6

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

   6. Simplified 4.6

      \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \color{blue}{-5 \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]
      Proof
      (*.f64 -5 (/.f64 (pow.f64 c 4) (/.f64 (pow.f64 b 7) (pow.f64 a 3)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -5 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 3)) (pow.f64 b 7)))): 32 points increase in error, 41 points decrease in error

   7. Applied egg-rr 4.6

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

   8. Applied egg-rr 4.6

      \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, -5 \cdot \color{blue}{\left(\left(\left({c}^{4} \cdot {b}^{-7}\right) \cdot \left(a \cdot a\right)\right) \cdot a\right)}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{b \cdot b} \cdot \frac{a}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.3

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

Alternatives

Alternative 1
Error	6.8
Cost	28164

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 2
Error	9.3
Cost	21060

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -0.18:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 3
Error	9.3
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.18:\\ \;\;\;\;\left(\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 4
Error	11.8
Cost	7232

\[\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}} \]

Alternative 5
Error	11.9
Cost	1600

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

Alternative 6
Error	11.9
Cost	1600

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

Alternative 7
Error	22.6
Cost	256

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

Alternative 8
Error	63.0
Cost	192

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

Reproduce

herbie shell --seed 2022331 
(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)))