Average Error: 34.2 → 12.9
Time: 6.4s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.6422358087771794 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.4072850734806817 \cdot 10^{-122}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.100785156979165 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot -3\right)}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot -1.5}{a \cdot 3}\\ \end{array}\]
\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 -1.6422358087771794 \cdot 10^{+154}:\\
\;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{a \cdot 3}\\

\mathbf{elif}\;b \leq 4.4072850734806817 \cdot 10^{-122}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\

\mathbf{elif}\;b \leq 8.100785156979165 \cdot 10^{+130}:\\
\;\;\;\;\frac{\frac{a \cdot \left(c \cdot -3\right)}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{a \cdot c}{b} \cdot -1.5}{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
 (if (<= b -1.6422358087771794e+154)
   (/ (- (- (* 1.5 (/ (* a c) b)) b) b) (* a 3.0))
   (if (<= b 4.4072850734806817e-122)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (if (<= b 8.100785156979165e+130)
       (/
        (/ (* a (* c -3.0)) (+ b (sqrt (- (* b b) (* c (* a 3.0))))))
        (* a 3.0))
       (/ (* (/ (* a c) b) -1.5) (* 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 tmp;
	if (b <= -1.6422358087771794e+154) {
		tmp = (((1.5 * ((a * c) / b)) - b) - b) / (a * 3.0);
	} else if (b <= 4.4072850734806817e-122) {
		tmp = (sqrt((b * b) - (c * (a * 3.0))) - b) / (a * 3.0);
	} else if (b <= 8.100785156979165e+130) {
		tmp = ((a * (c * -3.0)) / (b + sqrt((b * b) - (c * (a * 3.0))))) / (a * 3.0);
	} else {
		tmp = (((a * c) / b) * -1.5) / (a * 3.0);
	}
	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 < -1.6422358087771794e154

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around -inf 11.3

      \[\leadsto \frac{\color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)} - b}{3 \cdot a}\]

    if -1.6422358087771794e154 < b < 4.4072850734806817e-122

    1. Initial program 11.3

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

    2. Simplified11.3

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

    if 4.4072850734806817e-122 < b < 8.10078515697916522e130

    1. Initial program 42.1

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

    2. Simplified42.1

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

    4. Applied flip--_binary64_141742.2

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

    5. Simplified15.3

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

    6. Simplified15.3

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

    if 8.10078515697916522e130 < b

    1. Initial program 61.7

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

    2. Simplified61.7

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

    3. Taylor expanded around inf 14.6

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]

    4. Simplified14.6

      \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{b} \cdot -1.5}}{3 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6422358087771794 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.4072850734806817 \cdot 10^{-122}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.100785156979165 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot -3\right)}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot -1.5}{a \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020295 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))