Average Error: 33.7 → 10.3
Time: 22.8s
Precision: binary64
Cost: 7624
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{1}{a} \cdot -0.6666666666666666, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
(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 -6.6e+80)
   (fma b (* (/ 1.0 a) -0.6666666666666666) (* 0.5 (/ c b)))
   (if (<= b 8.5e-52)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ (* c -0.5) b))))
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 <= -6.6e+80) {
		tmp = fma(b, ((1.0 / a) * -0.6666666666666666), (0.5 * (c / b)));
	} else if (b <= 8.5e-52) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	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 <= -6.6e+80)
		tmp = fma(b, Float64(Float64(1.0 / a) * -0.6666666666666666), Float64(0.5 * Float64(c / b)));
	elseif (b <= 8.5e-52)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	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, -6.6e+80], N[(b * N[(N[(1.0 / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e-52], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $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 -6.6 \cdot 10^{+80}:\\
\;\;\;\;\mathsf{fma}\left(b, \frac{1}{a} \cdot -0.6666666666666666, 0.5 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot -0.5}{b}\\


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if b < -6.59999999999999982e80

    1. Initial program 41.6

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

    2. Taylor expanded in b around -inf 4.2

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

    3. Applied egg-rr4.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{1}{a} \cdot -0.6666666666666666, 0.5 \cdot \frac{c}{b}\right)} \]

    if -6.59999999999999982e80 < b < 8.50000000000000006e-52

    1. Initial program 14.4

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

    if 8.50000000000000006e-52 < b

    1. Initial program 54.3

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

    2. Taylor expanded in b around inf 8.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]

    3. Simplified8.7

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{b}{c}}} \]

    4. Applied egg-rr30.2

      \[\leadsto \color{blue}{e^{\log \left(-0.5 \cdot \frac{c}{b}\right)}} \]

    5. Applied egg-rr8.0

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{1}{a} \cdot -0.6666666666666666, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error10.4
Cost7624
\[\begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{1}{a} \cdot -0.6666666666666666, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
Alternative 2
Error13.8
Cost7368
\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} + -0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
Alternative 3
Error13.8
Cost7368
\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} + -0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
Alternative 4
Error13.8
Cost7368
\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{1}{a} \cdot -0.6666666666666666, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
Alternative 5
Error13.8
Cost7368
\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{1}{a} \cdot -0.6666666666666666, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
Alternative 6
Error23.1
Cost452
\[\begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{-278}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
Alternative 7
Error23.0
Cost452
\[\begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{-278}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
Alternative 8
Error45.4
Cost320
\[\frac{b \cdot -0.6666666666666666}{a} \]

Error

Reproduce

herbie shell --seed 2022284 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))