Average Error: 34.4 → 9.9
Time: 16.7s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -5.977464547809642 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\ \mathbf{elif}\;b \leq 2.048643829272188 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
\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 -5.977464547809642 \cdot 10^{+102}:\\
\;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\

\mathbf{elif}\;b \leq 2.048643829272188 \cdot 10^{-57}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - 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 -5.977464547809642e+102)
   (/ (/ (* b -2.0) 3.0) a)
   (if (<= b 2.048643829272188e-57)
     (/ (- (sqrt (fma b b (* c (* a -3.0)))) 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 <= -5.977464547809642e+102) {
		tmp = ((b * -2.0) / 3.0) / a;
	} else if (b <= 2.048643829272188e-57) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - 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

Derivation

  1. Split input into 3 regimes

  2. if b < -5.9774645478096418e102

    1. Initial program 47.1

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

    2. Applied associate-/r*_binary6447.1

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

    3. Simplified35.5

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

    4. Taylor expanded in b around -inf 3.7

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

    if -5.9774645478096418e102 < b < 2.0486438292721881e-57

    1. Initial program 13.6

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

    2. Applied fma-neg_binary6413.6

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

    3. Simplified13.6

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

    if 2.0486438292721881e-57 < b

    1. Initial program 54.0

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

    2. Taylor expanded in b around inf 8.0

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.977464547809642 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\ \mathbf{elif}\;b \leq 2.048643829272188 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))