Quadratic roots, wide range

Average Error: 52.1 → 1.5

Time: 17.0s

Precision: binary64

Cost: 48256

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (-
  (-
   (fma
    -0.25
    (*
     (/ (pow a 3.0) b)
     (fma
      16.0
      (/ (pow c 4.0) (pow b 6.0))
      (* 4.0 (* (* c c) (/ (* c c) (pow b 6.0))))))
    (* c (/ (* a (* a -2.0)) (/ (pow b 5.0) (* c c)))))
   (/ c b))
  (* a (* (/ c b) (/ c (* b 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) {
	return (fma(-0.25, ((pow(a, 3.0) / b) * fma(16.0, (pow(c, 4.0) / pow(b, 6.0)), (4.0 * ((c * c) * ((c * c) / pow(b, 6.0)))))), (c * ((a * (a * -2.0)) / (pow(b, 5.0) / (c * c))))) - (c / b)) - (a * ((c / b) * (c / (b * b))));
}

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)
	return Float64(Float64(fma(-0.25, Float64(Float64((a ^ 3.0) / b) * fma(16.0, Float64((c ^ 4.0) / (b ^ 6.0)), Float64(4.0 * Float64(Float64(c * c) * Float64(Float64(c * c) / (b ^ 6.0)))))), Float64(c * Float64(Float64(a * Float64(a * -2.0)) / Float64((b ^ 5.0) / Float64(c * c))))) - Float64(c / b)) - Float64(a * Float64(Float64(c / b) * Float64(c / Float64(b * b)))))
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_] := N[(N[(N[(-0.25 * N[(N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision] * N[(16.0 * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(a * N[(a * -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1

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

2. Simplified 52.1

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   Proof
   (/.f64 (+.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, 2 points decrease in error

3. Taylor expanded in a around 0 1.5

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

4. Simplified 1.5

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \frac{\left(\left(a \cdot -2\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a} \]
   Proof
   (-.f64 (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) 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 (*.f64 (*.f64 (*.f64 a -2) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) 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 (*.f64 (*.f64 (*.f64 a -2) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) 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 (*.f64 (*.f64 (*.f64 a -2) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) 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 (*.f64 (*.f64 (*.f64 a -2) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) 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 (*.f64 (*.f64 (*.f64 a -2) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) 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 (*.f64 (*.f64 (*.f64 a -2) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) 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 (*.f64 (*.f64 (*.f64 a -2) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) 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 (*.f64 (*.f64 (*.f64 a -2) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) 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 (*.f64 (*.f64 (*.f64 a -2) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) 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 (*.f64 (*.f64 (*.f64 a -2) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.f64 -1/4 (Rewrite<= associate-/r/_binary64 (/.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))))) (/.f64 (*.f64 (*.f64 (*.f64 a -2) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 28 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.f64 -1/4 (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)) (/.f64 (*.f64 (*.f64 (*.f64 a -2) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 28 points decrease in error
   (-.f64 (-.f64 (fma.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 (*.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 -2 a)) a) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.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 (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -2 (*.f64 a a))) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.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 (*.f64 (*.f64 -2 (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (pow.f64 c 3)) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.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 (Rewrite<= associate-*r*_binary64 (*.f64 -2 (*.f64 (pow.f64 a 2) (pow.f64 c 3)))) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 28 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (fma.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 (*.f64 -2 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)))) (pow.f64 b 5))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 28 points decrease in error
   (-.f64 (-.f64 (fma.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) (Rewrite<= associate-*r/_binary64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))))) (/.f64 c b)) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (Rewrite<= fma-def_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 (*.f64 c c) (pow.f64 b 3)) a)): 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 (*.f64 c c) (pow.f64 b 3)) a)): 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 (*.f64 c c) (pow.f64 b 3)) a)): 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 (*.f64 c c) (pow.f64 b 3)) a)): 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)) (pow.f64 b 3)) a)): 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<= associate-/r/_binary64 (/.f64 (pow.f64 c 2) (/.f64 (pow.f64 b 3) a)))): 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<= associate-/l*_binary64 (/.f64 (*.f64 (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))))): 28 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, 28 points decrease in error

5. Applied egg-rr 1.5

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

6. Applied egg-rr 1.5

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

7. Applied egg-rr 1.5

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

8. Final simplification 1.5

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

Alternatives

Alternative 1
Error	2.0
Cost	14528

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

Alternative 2
Error	2.1
Cost	7552

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

Alternative 3
Error	3.1
Cost	7232

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

Alternative 4
Error	3.2
Cost	576

\[\frac{1}{\frac{a}{b} - \frac{b}{c}} \]

Alternative 5
Error	6.5
Cost	256

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

Alternative 6
Error	61.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022340 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))