Average Error: 33.8 → 6.7
Time: 7.7s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.569045129100026 \cdot 10^{+109}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq -1.8264876192729505 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 7.787065085468061 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{\frac{-\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]
\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 -1.569045129100026 \cdot 10^{+109}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \leq -1.8264876192729505 \cdot 10^{-225}:\\
\;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\

\mathbf{elif}\;b \leq 7.787065085468061 \cdot 10^{+124}:\\
\;\;\;\;\frac{1}{\frac{-\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right)}{c}}\\

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

\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 -1.569045129100026e+109)
   (- (* 0.5 (/ c b)) (* 0.6666666666666666 (/ b a)))
   (if (<= b -1.8264876192729505e-225)
     (/ (- (sqrt (+ (* b b) (* a (* c -3.0)))) b) (* a 3.0))
     (if (<= b 7.787065085468061e+124)
       (/ 1.0 (/ (- (+ b (sqrt (+ (* b b) (* a (* c -3.0)))))) c))
       (* (/ c b) -0.5)))))
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 <= -1.569045129100026e+109) {
		tmp = (0.5 * (c / b)) - (0.6666666666666666 * (b / a));
	} else if (b <= -1.8264876192729505e-225) {
		tmp = (sqrt((b * b) + (a * (c * -3.0))) - b) / (a * 3.0);
	} else if (b <= 7.787065085468061e+124) {
		tmp = 1.0 / (-(b + sqrt((b * b) + (a * (c * -3.0)))) / c);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 4 regimes

  2. if b < -1.56904512910002603e109

    1. Initial program 49.1

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

    2. Simplified49.1

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

    3. Taylor expanded around -inf 3.9

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

    if -1.56904512910002603e109 < b < -1.82648761927295047e-225

    1. Initial program 7.6

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

    2. Simplified7.6

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

    4. Applied sub-neg_binary64_17767.6

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

    5. Simplified7.7

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

    if -1.82648761927295047e-225 < b < 7.7870650854680606e124

    1. Initial program 31.2

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

    2. Simplified31.2

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

    4. Applied flip--_binary64_175831.3

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

    5. Simplified16.6

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

    6. Simplified16.6

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

    8. Applied clear-num_binary64_178216.8

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\frac{a \cdot \left(c \cdot -3\right)}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}\]

    9. Simplified9.9

      \[\leadsto \frac{1}{\color{blue}{\left(b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{c}}}\]
    10. Using strategy rm

    11. Applied sub-neg_binary64_17769.9

      \[\leadsto \frac{1}{\left(b + \sqrt{\color{blue}{b \cdot b + \left(-\left(3 \cdot a\right) \cdot c\right)}}\right) \cdot \frac{-1}{c}}\]

    12. Simplified9.9

      \[\leadsto \frac{1}{\left(b + \sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -3\right)}}\right) \cdot \frac{-1}{c}}\]
    13. Using strategy rm

    14. Applied associate-*r/_binary64_17259.9

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right) \cdot -1}{c}}}\]

    15. Simplified9.9

      \[\leadsto \frac{1}{\frac{\color{blue}{-\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right)}}{c}}\]

    if 7.7870650854680606e124 < b

    1. Initial program 61.0

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

    2. Simplified61.0

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

    3. Taylor expanded around inf 1.4

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.569045129100026 \cdot 10^{+109}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq -1.8264876192729505 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 7.787065085468061 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{\frac{-\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2020298 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))