Average Error: 28.9 → 5.0
Time: 6.2s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.3163313487072055:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} - \frac{\mathsf{fma}\left(1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \mathsf{fma}\left(1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, 3.1640625 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3}\\ \end{array} \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* a c) (* b b))))
   (if (<= b 1.3163313487072055)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* a 3.0))
     (-
      (* -0.5 (/ c b))
      (/
       (fma
        1.125
        (/ (pow (* a c) 2.0) (pow b 3.0))
        (fma
         1.6875
         (/ (pow (* a c) 3.0) (pow b 5.0))
         (* 3.1640625 (/ (pow (* a c) 4.0) (pow b 7.0)))))
       (* a 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 t_0 = fma(-3.0, (a * c), (b * b));
	double tmp;
	if (b <= 1.3163313487072055) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) - (fma(1.125, (pow((a * c), 2.0) / pow(b, 3.0)), fma(1.6875, (pow((a * c), 3.0) / pow(b, 5.0)), (3.1640625 * (pow((a * c), 4.0) / pow(b, 7.0))))) / (a * 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)
	t_0 = fma(-3.0, Float64(a * c), Float64(b * b))
	tmp = 0.0
	if (b <= 1.3163313487072055)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) - Float64(fma(1.125, Float64((Float64(a * c) ^ 2.0) / (b ^ 3.0)), fma(1.6875, Float64((Float64(a * c) ^ 3.0) / (b ^ 5.0)), Float64(3.1640625 * Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0))))) / Float64(a * 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_] := Block[{t$95$0 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.3163313487072055], 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[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(1.125 * N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.6875 * N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(3.1640625 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\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(-3, a \cdot c, b \cdot b\right)\\
\mathbf{if}\;b \leq 1.3163313487072055:\\
\;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b} - \frac{\mathsf{fma}\left(1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \mathsf{fma}\left(1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, 3.1640625 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3}\\


\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 < 1.31633134870720547

    1. Initial program 11.6

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

    2. Applied egg-rr10.7

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

    if 1.31633134870720547 < b

    1. Initial program 32.1

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

    2. Taylor expanded in b around inf 4.2

      \[\leadsto \frac{\color{blue}{-\left(1.5 \cdot \frac{c \cdot a}{b} + \left(1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(3.1640625 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{7}} + 1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)\right)}}{3 \cdot a} \]

    3. Simplified4.2

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{c \cdot a}{b} - \mathsf{fma}\left(1.125, \frac{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}{{b}^{3}}, \mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}, 3.1640625 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{7}}\right)\right)}}{3 \cdot a} \]

    4. Applied egg-rr4.2

      \[\leadsto \color{blue}{\frac{-1.5 \cdot \frac{c \cdot a}{b}}{a \cdot 3} - \frac{\mathsf{fma}\left(1.125, \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}}, \mathsf{fma}\left(1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, 3.1640625 \cdot \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3}} \]

    5. Taylor expanded in c around 0 3.9

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} - \frac{\mathsf{fma}\left(1.125, \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}}, \mathsf{fma}\left(1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, 3.1640625 \cdot \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.3163313487072055:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} - \frac{\mathsf{fma}\left(1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \mathsf{fma}\left(1.6875, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, 3.1640625 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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)))