Average Error: 28.5 → 5.4
Time: 4.8s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq 0.41327613984999245:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \left(0.3333333333333333 \cdot {a}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}\right) \cdot -0.5625 - \mathsf{fma}\left(0.5, \frac{c}{b}, \mathsf{fma}\left(1.0546875, \frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}, 0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right)\\ \end{array} \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b 0.41327613984999245)
   (*
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (* 0.3333333333333333 (pow a -1.0)))
   (-
    (* (* a (/ (pow c 3.0) (/ (pow b 5.0) a))) -0.5625)
    (fma
     0.5
     (/ c b)
     (fma
      1.0546875
      (* (/ (pow c 4.0) (pow b 7.0)) (pow a 3.0))
      (* 0.375 (* a (/ (* c c) (pow b 3.0)))))))))
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 tmp;
	if (b <= 0.41327613984999245) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * (0.3333333333333333 * pow(a, -1.0));
	} else {
		tmp = ((a * (pow(c, 3.0) / (pow(b, 5.0) / a))) * -0.5625) - fma(0.5, (c / b), fma(1.0546875, ((pow(c, 4.0) / pow(b, 7.0)) * pow(a, 3.0)), (0.375 * (a * ((c * c) / pow(b, 3.0))))));
	}
	return tmp;
}

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)
	tmp = 0.0
	if (b <= 0.41327613984999245)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * Float64(0.3333333333333333 * (a ^ -1.0)));
	else
		tmp = Float64(Float64(Float64(a * Float64((c ^ 3.0) / Float64((b ^ 5.0) / a))) * -0.5625) - fma(0.5, Float64(c / b), fma(1.0546875, Float64(Float64((c ^ 4.0) / (b ^ 7.0)) * (a ^ 3.0)), Float64(0.375 * Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))))));
	end
	return tmp
end

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_] := If[LessEqual[b, 0.41327613984999245], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 * N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision] - N[(0.5 * N[(c / b), $MachinePrecision] + N[(1.0546875 * N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq 0.41327613984999245:\\
\;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \left(0.3333333333333333 \cdot {a}^{-1}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}\right) \cdot -0.5625 - \mathsf{fma}\left(0.5, \frac{c}{b}, \mathsf{fma}\left(1.0546875, \frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}, 0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right)\\


\end{array}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.41327613984999245

    1. Initial program 11.4

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

    2. Simplified11.3

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

    3. Applied egg-rr11.3

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

    if 0.41327613984999245 < b

    1. Initial program 31.1

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

    2. Simplified31.0

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

    3. Taylor expanded in b around inf 4.5

      \[\leadsto \color{blue}{-\left(0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 0.5 \cdot \frac{c}{b}\right)\right)\right)} \]

    4. Simplified4.5

      \[\leadsto \color{blue}{\left(\frac{{c}^{3}}{\frac{{b}^{5}}{a}} \cdot a\right) \cdot -0.5625 - \mathsf{fma}\left(0.5, \frac{c}{b}, \mathsf{fma}\left(1.0546875, \frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}, 0.375 \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot a\right)\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.41327613984999245:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \left(0.3333333333333333 \cdot {a}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}\right) \cdot -0.5625 - \mathsf{fma}\left(0.5, \frac{c}{b}, \mathsf{fma}\left(1.0546875, \frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}, 0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))