Quadratic roots, medium range

Average Error: 43.5 → 3.1

Time: 20.2s

Precision: binary64

Cost: 66752

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{{c}^{4}}{{b}^{6}}\\ \left(\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \frac{{a}^{3} \cdot -0.25}{\frac{b}{\mathsf{fma}\left(16, t_0, 4 \cdot t_0\right)}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \end{array} \]

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 (/ (pow c 4.0) (pow b 6.0))))
   (-
    (-
     (fma
      -2.0
      (/ (* (pow c 3.0) (* a a)) (pow b 5.0))
      (/ (* (pow a 3.0) -0.25) (/ b (fma 16.0 t_0 (* 4.0 t_0)))))
     (/ c b))
    (/ (* c (* c a)) (pow b 3.0)))))

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 = pow(c, 4.0) / pow(b, 6.0);
	return (fma(-2.0, ((pow(c, 3.0) * (a * a)) / pow(b, 5.0)), ((pow(a, 3.0) * -0.25) / (b / fma(16.0, t_0, (4.0 * t_0))))) - (c / b)) - ((c * (c * a)) / pow(b, 3.0));
}

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 ^ 4.0) / (b ^ 6.0))
	return Float64(Float64(fma(-2.0, Float64(Float64((c ^ 3.0) * Float64(a * a)) / (b ^ 5.0)), Float64(Float64((a ^ 3.0) * -0.25) / Float64(b / fma(16.0, t_0, Float64(4.0 * t_0))))) - Float64(c / b)) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)))
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[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $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[(N[(N[Power[a, 3.0], $MachinePrecision] * -0.25), $MachinePrecision] / N[(b / N[(16.0 * t$95$0 + N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 43.5

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

2. Simplified 43.5

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   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)): 9 points increase in error, 8 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 3.1

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

4. Simplified 3.1

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \frac{{a}^{3} \cdot -0.25}{\frac{b}{\mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}} \]
   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))): 1 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. Final simplification 3.1

   \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \frac{{a}^{3} \cdot -0.25}{\frac{b}{\mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

Alternatives

Alternative 1
Error	3.1
Cost	60544

\[\begin{array}{l} t_0 := \frac{{c}^{4}}{{b}^{6}}\\ \left(\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \frac{{a}^{3} \cdot -0.25}{\frac{b}{\mathsf{fma}\left(16, t_0, 4 \cdot t_0\right)}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{b \cdot b} \cdot \frac{a}{b} \end{array} \]

Alternative 2
Error	3.1
Cost	40704

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

Alternative 3
Error	4.1
Cost	20736

\[\left(\frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}} \]

Alternative 4
Error	4.1
Cost	20736

\[\left(\frac{{c}^{3} \cdot \left(a \cdot \left(-2 \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} \]

Alternative 5
Error	6.1
Cost	7232

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

Alternative 6
Error	12.3
Cost	256

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

Alternative 7
Error	63.0
Cost	192

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

Reproduce

herbie shell --seed 2022329 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))