Cubic critical, narrow range

Average Accuracy: 55.1% → 92.0%

Time: 26.5s

Precision: binary64

Cost: 67204

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.0086:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(a \cdot a\right) \cdot \left(-1.125 \cdot \left(c \cdot c\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\right)\\ \end{array} \]

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 (fma a (* c -3.0) (* b b))))
   (if (<= b 0.0086)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* a 3.0))
     (fma
      -0.5625
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (fma
       -0.16666666666666666
       (/
        (+
         (pow (* (* a a) (* -1.125 (* c c))) 2.0)
         (* 5.0625 (* (pow c 4.0) (pow a 4.0))))
        (* a (pow b 7.0)))
       (fma -0.5 (/ c b) (* -0.375 (/ (* a (* c c)) (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 = fma(a, (c * -3.0), (b * b));
	double tmp;
	if (b <= 0.0086) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.16666666666666666, ((pow(((a * a) * (-1.125 * (c * c))), 2.0) + (5.0625 * (pow(c, 4.0) * pow(a, 4.0)))) / (a * pow(b, 7.0))), fma(-0.5, (c / b), (-0.375 * ((a * (c * c)) / 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 = fma(a, Float64(c * -3.0), Float64(b * b))
	tmp = 0.0
	if (b <= 0.0086)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.16666666666666666, Float64(Float64((Float64(Float64(a * a) * Float64(-1.125 * Float64(c * c))) ^ 2.0) + Float64(5.0625 * Float64((c ^ 4.0) * (a ^ 4.0)))) / Float64(a * (b ^ 7.0))), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / (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[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0086], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $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[(-0.16666666666666666 * N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] * N[(-1.125 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(5.0625 * N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0086

   1. Initial program 87.4%

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

   2. Simplified 87.4%

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

      [Start] 87.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      neg-sub0 [=>] 87.4	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-+l- [=>] 87.4	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      sub0-neg [=>] 87.4	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      neg-mul-1 [=>] 87.4	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      associate-*r/ [<=] 87.4	\[ \color{blue}{-1 \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [<=] 87.4	\[ \color{blue}{\frac{1}{-1}} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      metadata-eval [<=] 87.4	\[ \frac{\color{blue}{--1}}{-1} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 87.4	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      *-commutative [=>] 87.4	\[ \frac{\left(--1\right) \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{\left(3 \cdot a\right) \cdot -1}} \]
      times-frac [=>] 87.4	\[ \color{blue}{\frac{--1}{3 \cdot a} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}} \]
      associate-*l/ [=>] 87.4	\[ \color{blue}{\frac{\left(--1\right) \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}}{3 \cdot a}} \]

   3. Applied egg-rr 87.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot -3}}}}{3 \cdot a} \]
      Proof

      [Start] 87.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a} \]
      sub-neg [=>] 87.4	\[ \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      flip-+ [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}}{3 \cdot a} \]
      pow2 [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) - \left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      pow2 [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{2} \cdot \color{blue}{{b}^{2}} - \left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      pow-prod-up [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{\color{blue}{{b}^{\left(2 + 2\right)}} - \left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      metadata-eval [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{\color{blue}{4}} - \left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      *-commutative [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(-\color{blue}{\left(a \cdot c\right) \cdot 3}\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      distribute-rgt-neg-in [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \color{blue}{\left(\left(a \cdot c\right) \cdot \left(-3\right)\right)} \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      metadata-eval [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot \color{blue}{-3}\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      *-commutative [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(-\color{blue}{\left(a \cdot c\right) \cdot 3}\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      distribute-rgt-neg-in [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \left(-3\right)\right)}}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      metadata-eval [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot \color{blue}{-3}\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      *-commutative [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot 3}\right)}}}{3 \cdot a} \]
      distribute-rgt-neg-in [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot \left(-3\right)}}}}{3 \cdot a} \]
      metadata-eval [=>] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot \color{blue}{-3}}}}{3 \cdot a} \]

   4. Applied egg-rr 88.2%

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

      [Start] 87.0	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot -3}}}{3 \cdot a} \]
      +-commutative [=>] 87.0	\[ \frac{\color{blue}{\sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot -3}} + \left(-b\right)}}{3 \cdot a} \]
      flip-+ [=>] 86.5	\[ \frac{\color{blue}{\frac{\sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot -3}} \cdot \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot -3}} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot -3}} - \left(-b\right)}}}{3 \cdot a} \]

   5. Simplified 88.7%

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

      [Start] 88.2	\[ \frac{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{3 \cdot a} \]
      fma-def [<=] 88.7	\[ \frac{\frac{\color{blue}{\left(b \cdot b + a \cdot \left(c \cdot -3\right)\right)} - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{3 \cdot a} \]
      +-commutative [<=] 88.7	\[ \frac{\frac{\color{blue}{\left(a \cdot \left(c \cdot -3\right) + b \cdot b\right)} - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{3 \cdot a} \]
      fma-def [=>] 88.7	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{3 \cdot a} \]
      fma-def [<=] 88.7	\[ \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) - b \cdot b}{b + \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -3\right)}}}}{3 \cdot a} \]
      +-commutative [<=] 88.7	\[ \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) - b \cdot b}{b + \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right) + b \cdot b}}}}{3 \cdot a} \]
      fma-def [=>] 88.7	\[ \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) - b \cdot b}{b + \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}}{3 \cdot a} \]

   if 0.0086 < b

   1. Initial program 53.0%

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

   2. Simplified 53.0%

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

      [Start] 53.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      /-rgt-identity [<=] 53.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{3 \cdot a}{1}}} \]
      metadata-eval [<=] 53.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\frac{3 \cdot a}{\color{blue}{--1}}} \]
      associate-/r/ [=>] 53.0	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \left(--1\right)} \]
      metadata-eval [=>] 53.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{1} \]
      metadata-eval [<=] 53.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{\frac{-1}{-1}} \]
      times-frac [<=] 53.0	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\left(3 \cdot a\right) \cdot -1}} \]
      *-commutative [<=] 53.0	\[ \frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\color{blue}{-1 \cdot \left(3 \cdot a\right)}} \]
      times-frac [=>] 53.0	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      *-commutative [<=] 53.0	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}} \]
      associate-/r* [=>] 53.0	\[ \color{blue}{\frac{\frac{-1}{3}}{a}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \]
      associate-*l/ [=>] 53.0	\[ \color{blue}{\frac{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}}{a}} \]

   3. Taylor expanded in b around inf 92.2%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

   4. Simplified 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(-1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\right)} \]
      Proof

      [Start] 92.2	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
      fma-def [=>] 92.2	\[ \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]
      associate-/l* [=>] 92.2	\[ \mathsf{fma}\left(-0.5625, \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{2}}}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
      unpow2 [=>] 92.2	\[ \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{\color{blue}{a \cdot a}}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
      fma-def [=>] 92.2	\[ \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0086:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(a \cdot a\right) \cdot \left(-1.125 \cdot \left(c \cdot c\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.1%
Cost	47428

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.015:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}, -0.16666666666666666, \mathsf{fma}\left(-0.375, c \cdot \left(c \cdot \frac{a}{{b}^{3}}\right), \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right), -0.5 \cdot \frac{c}{b}\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	89.5%
Cost	33668

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.015:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	85.8%
Cost	28228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.04:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 4
Accuracy	85.5%
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.04:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 5
Accuracy	85.2%
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.04:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \frac{c \cdot c}{\frac{b \cdot b}{a} \cdot \frac{b}{a}}, -1.5 \cdot \frac{c}{\frac{b}{a}}\right)}{a \cdot 3}\\ \end{array} \]

Alternative 6
Accuracy	85.2%
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.04:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \frac{c \cdot c}{\frac{b \cdot b}{a} \cdot \frac{b}{a}}, -1.5 \cdot \frac{c}{\frac{b}{a}}\right)}{a \cdot 3}\\ \end{array} \]

Alternative 7
Accuracy	85.2%
Cost	15428

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.04:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \frac{c \cdot c}{\frac{b \cdot b}{a} \cdot \frac{b}{a}}, -1.5 \cdot \frac{c}{\frac{b}{a}}\right)}{a \cdot 3}\\ \end{array} \]

Alternative 8
Accuracy	76.8%
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 9
Accuracy	73.1%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 15.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10
Accuracy	64.6%
Cost	320

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

Alternative 11
Accuracy	64.6%
Cost	320

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

Alternative 12
Accuracy	64.6%
Cost	320

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

Reproduce

herbie shell --seed 2023152 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))