Cubic critical, wide range

Average Error: 52.4 → 1.2

Time: 21.1s

Precision: binary64

Cost: 40964

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

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot 3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t_0} - b}{a \cdot 3} \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;{\left({\left(\frac{t_0 + \left(b \cdot b - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)}^{3}\right)}^{0.3333333333333333} \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{{b}^{3}}\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 3.0))))
   (if (<= (/ (- (sqrt (- (* b b) t_0)) b) (* a 3.0)) -1.25e-5)
     (*
      (pow
       (pow
        (/ (+ t_0 (- (* b b) (* b b))) (+ b (sqrt (fma a (* c -3.0) (* b b)))))
        3.0)
       0.3333333333333333)
      (/ -0.3333333333333333 a))
     (fma
      -0.5625
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (fma -0.5 (/ c b) (/ (* -0.375 (* (* c c) a)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	double tmp;
	if (((sqrt(((b * b) - t_0)) - b) / (a * 3.0)) <= -1.25e-5) {
		tmp = pow(pow(((t_0 + ((b * b) - (b * b))) / (b + sqrt(fma(a, (c * -3.0), (b * b))))), 3.0), 0.3333333333333333) * (-0.3333333333333333 / a);
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.5, (c / b), ((-0.375 * ((c * c) * a)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

↓

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 3.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(a * 3.0)) <= -1.25e-5)
		tmp = Float64(((Float64(Float64(t_0 + Float64(Float64(b * b) - Float64(b * b))) / Float64(b + sqrt(fma(a, Float64(c * -3.0), Float64(b * b))))) ^ 3.0) ^ 0.3333333333333333) * Float64(-0.3333333333333333 / a));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(Float64(c * c) * a)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -1.25e-5], N[(N[Power[N[Power[N[(N[(t$95$0 + N[(N[(b * b), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $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 3 a) c)))) (*.f64 3 a)) < -1.25000000000000006e-5

   1. Initial program 22.5

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

   2. Simplified 22.6

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}} \]
      Proof
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b)))) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 3))) (*.f64 b b)))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (*.f64 c (neg.f64 3))) (*.f64 b b))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 c (neg.f64 3)) a)) (*.f64 b b)))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 c (*.f64 (neg.f64 3) a))) (*.f64 b b)))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (+.f64 (*.f64 c (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 3 a)))) (*.f64 b b)))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 3 a)) c)) (*.f64 b b)))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 (neg.f64 (*.f64 3 a)) c))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (Rewrite=> sub-neg_binary64 (+.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 b 1)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (/.f64 b (Rewrite<= metadata-eval (/.f64 -1 -1))) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 b -1) -1)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 b)) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (+.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 b)) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (/.f64 (neg.f64 b) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 (Rewrite<= metadata-eval (/.f64 -1 3)) a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (/.f64 (neg.f64 b) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 3 a)))): 0 points increase in error, 31 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 -1 (*.f64 3 a)) (+.f64 (/.f64 (neg.f64 b) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))))): 31 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (/.f64 (neg.f64 b) -1)) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))))): 0 points increase in error, 31 points decrease in error
      (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 b)) -1)) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 31 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 b -1)) -1)) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 31 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (Rewrite=> associate-/l*_binary64 (/.f64 b (/.f64 -1 -1)))) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 0 points increase in error, 31 points decrease in error
      (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (/.f64 b (Rewrite=> metadata-eval 1))) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 0 points increase in error, 31 points decrease in error
      (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (Rewrite=> /-rgt-identity_binary64 b)) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 31 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-in_binary64 (*.f64 (/.f64 -1 (*.f64 3 a)) (+.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))))): 31 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 3 a)) (Rewrite<= sub-neg_binary64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 31 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 3 a))): 0 points increase in error, 31 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 31 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 31 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 3 a)): 0 points increase in error, 31 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)): 31 points increase in error, 0 points decrease in error

   3. Applied egg-rr 22.7

      \[\leadsto \color{blue}{{\left({\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \frac{-0.3333333333333333}{a} \]

   4. Applied egg-rr 21.7

      \[\leadsto {\left({\color{blue}{\left(\left(b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right) \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)}}^{3}\right)}^{0.3333333333333333} \cdot \frac{-0.3333333333333333}{a} \]

   5. Simplified 21.7

      \[\leadsto {\left({\color{blue}{\left(\frac{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)}}^{3}\right)}^{0.3333333333333333} \cdot \frac{-0.3333333333333333}{a} \]
      Proof
      (*.f64 (pow.f64 (pow.f64 (/.f64 (-.f64 (*.f64 b b) (fma.f64 a (*.f64 c -3) (*.f64 b b))) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b))))) 3) 1/3) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (pow.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (-.f64 (*.f64 b b) (fma.f64 a (*.f64 c -3) (*.f64 b b))) 1)) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b))))) 3) 1/3) (/.f64 -1/3 a)): 0 points increase in error, 3 points decrease in error
      (*.f64 (pow.f64 (pow.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 (*.f64 b b) (fma.f64 a (*.f64 c -3) (*.f64 b b))) (/.f64 1 (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b))))))) 3) 1/3) (/.f64 -1/3 a)): 3 points increase in error, 0 points decrease in error

   6. Applied egg-rr 2.9

      \[\leadsto {\left({\left(\frac{\color{blue}{\left(-\left(a \cdot -3\right) \cdot c\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)}^{3}\right)}^{0.3333333333333333} \cdot \frac{-0.3333333333333333}{a} \]

   if -1.25000000000000006e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 57.0

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

   2. Simplified 57.0

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}} \]
      Proof
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b)))) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 3))) (*.f64 b b)))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (*.f64 c (neg.f64 3))) (*.f64 b b))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 c (neg.f64 3)) a)) (*.f64 b b)))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 c (*.f64 (neg.f64 3) a))) (*.f64 b b)))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (+.f64 (*.f64 c (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 3 a)))) (*.f64 b b)))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 3 a)) c)) (*.f64 b b)))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 (neg.f64 (*.f64 3 a)) c))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (Rewrite=> sub-neg_binary64 (+.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 b 1)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (/.f64 b (Rewrite<= metadata-eval (/.f64 -1 -1))) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 b -1) -1)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 b)) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
      (*.f64 (+.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 b)) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (/.f64 (neg.f64 b) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 (Rewrite<= metadata-eval (/.f64 -1 3)) a)): 31 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (/.f64 (neg.f64 b) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 3 a)))): 0 points increase in error, 31 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 -1 (*.f64 3 a)) (+.f64 (/.f64 (neg.f64 b) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))))): 31 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (/.f64 (neg.f64 b) -1)) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))))): 0 points increase in error, 31 points decrease in error
      (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 b)) -1)) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 31 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 b -1)) -1)) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 31 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (Rewrite=> associate-/l*_binary64 (/.f64 b (/.f64 -1 -1)))) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 0 points increase in error, 31 points decrease in error
      (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (/.f64 b (Rewrite=> metadata-eval 1))) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 0 points increase in error, 31 points decrease in error
      (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (Rewrite=> /-rgt-identity_binary64 b)) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 31 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-in_binary64 (*.f64 (/.f64 -1 (*.f64 3 a)) (+.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))))): 31 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 3 a)) (Rewrite<= sub-neg_binary64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 31 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 3 a))): 0 points increase in error, 31 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 31 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 31 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 3 a)): 0 points increase in error, 31 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)): 31 points increase in error, 0 points decrease in error

   3. Taylor expanded in b around inf 0.9

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

   4. Simplified 0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)} \]
      Proof
      (*.f64 (pow.f64 (pow.f64 (/.f64 (-.f64 (*.f64 b b) (fma.f64 a (*.f64 c -3) (*.f64 b b))) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b))))) 3) 1/3) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (pow.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (-.f64 (*.f64 b b) (fma.f64 a (*.f64 c -3) (*.f64 b b))) 1)) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b))))) 3) 1/3) (/.f64 -1/3 a)): 0 points increase in error, 3 points decrease in error
      (*.f64 (pow.f64 (pow.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 (*.f64 b b) (fma.f64 a (*.f64 c -3) (*.f64 b b))) (/.f64 1 (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b))))))) 3) 1/3) (/.f64 -1/3 a)): 3 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;{\left({\left(\frac{c \cdot \left(a \cdot 3\right) + \left(b \cdot b - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)}^{3}\right)}^{0.3333333333333333} \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	41600

\[\begin{array}{l} t_0 := c \cdot \left(c \cdot a\right)\\ \mathsf{fma}\left(-0.5625, \left(c \cdot c\right) \cdot \left(c \cdot \frac{a \cdot a}{{b}^{5}}\right), \mathsf{fma}\left(-0.16666666666666666, \frac{\left(a \cdot a\right) \cdot \left(t_0 \cdot t_0\right)}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{{b}^{3}}\right)\right)\right) \end{array} \]

Alternative 2
Error	1.2
Cost	34884

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot 3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t_0} - b}{a \cdot 3} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;{\left({\left(\frac{t_0 + \left(b \cdot b - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)}^{3}\right)}^{0.3333333333333333} \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 3
Error	1.2
Cost	34372

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot {\left({\left(\frac{\left(c \cdot a\right) \cdot 3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 4
Error	2.1
Cost	20608

\[\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \left(a \cdot a\right) \cdot \left(\frac{c}{{b}^{3}} \cdot 0.375\right), \frac{a \cdot 1.5}{b}\right)\right)} \]

Alternative 5
Error	2.4
Cost	14464

\[\frac{1}{\left(a \cdot 3\right) \cdot \left(\frac{0.5}{b} - \mathsf{fma}\left(-1, \frac{c \cdot \left(a \cdot 0.375\right)}{{b}^{3}}, \frac{b \cdot 0.6666666666666666}{c \cdot a}\right)\right)} \]

Alternative 6
Error	2.4
Cost	8576

\[\frac{1}{\left(a \cdot -3\right) \cdot \left(\left(0.6666666666666666 \cdot \frac{b}{c \cdot a} - \frac{\left(c \cdot a\right) \cdot 0.75 - \left(c \cdot a\right) \cdot 0.375}{{b}^{3}}\right) + -0.5 \cdot \frac{1}{b}\right)} \]

Alternative 7
Error	3.1
Cost	832

\[\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \]

Alternative 8
Error	6.5
Cost	320

\[c \cdot \frac{-0.5}{b} \]

Alternative 9
Error	6.5
Cost	320

\[\frac{-0.5}{\frac{b}{c}} \]

Alternative 10
Error	6.3
Cost	320

\[\frac{c \cdot -0.5}{b} \]

Reproduce

herbie shell --seed 2022343 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))