Average Error: 34.3 → 10.1
Time: 8.4s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -6.124802208226831 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\ \mathbf{elif}\;b \leq 5.537390450507026 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} - \frac{b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
\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 -6.124802208226831 \cdot 10^{+93}:\\
\;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\

\mathbf{elif}\;b \leq 5.537390450507026 \cdot 10^{-101}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} - \frac{b}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{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 -6.124802208226831e+93)
   (/ (/ (* b -2.0) 3.0) a)
   (if (<= b 5.537390450507026e-101)
     (- (/ (sqrt (- (* b b) (* (* 3.0 a) c))) (* 3.0 a)) (/ b (* 3.0 a)))
     (* -0.5 (/ c 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 <= -6.124802208226831e+93) {
		tmp = ((b * -2.0) / 3.0) / a;
	} else if (b <= 5.537390450507026e-101) {
		tmp = (sqrt((b * b) - ((3.0 * a) * c)) / (3.0 * a)) - (b / (3.0 * a));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -6.12480220822683113e93

    1. Initial program 45.5

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

    2. Simplified45.5

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}} \]
    3. Using strategy rm

    4. Applied associate-/r*_binary6445.5

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

    5. Simplified45.5

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

    6. Taylor expanded around -inf 3.9

      \[\leadsto \frac{\frac{\color{blue}{-2 \cdot b}}{3}}{a} \]

    7. Simplified3.9

      \[\leadsto \frac{\frac{\color{blue}{b \cdot -2}}{3}}{a} \]

    if -6.12480220822683113e93 < b < 5.5373904505070259e-101

    1. Initial program 12.4

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

    2. Simplified12.4

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}} \]
    3. Using strategy rm

    4. Applied div-sub_binary6412.4

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

    if 5.5373904505070259e-101 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 10.0

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.124802208226831 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\ \mathbf{elif}\;b \leq 5.537390450507026 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} - \frac{b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))