Average Error: 28.4 → 17.2
Time: 4.8s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq 34.79216116694219:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{a \cdot c}{b}}{\sqrt{a}}}{\sqrt{a}}\\ \end{array}\]
\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 34.79216116694219:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

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

\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 34.79216116694219)
   (/
    (/
     (- (- (* b b) (* (* 3.0 a) c)) (* b b))
     (+ b (sqrt (- (* b b) (* (* 3.0 a) c)))))
    (* 3.0 a))
   (/ (* -0.5 (/ (/ (* a c) b) (sqrt a))) (sqrt a))))
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 <= 34.79216116694219) {
		tmp = ((((b * b) - ((3.0 * a) * c)) - (b * b)) / (b + sqrt((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = (-0.5 * (((a * c) / b) / sqrt(a))) / sqrt(a);
	}
	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 2 regimes

  2. if b < 34.792161166942194

    1. Initial program 14.2

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

    2. Simplified14.2

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

    4. Applied flip--_binary6414.2

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

    5. Simplified13.3

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

    6. Simplified13.3

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

    if 34.792161166942194 < b

    1. Initial program 33.5

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

    2. Simplified33.5

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

    3. Taylor expanded around inf 18.6

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied associate-/r*_binary6418.6

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

    6. Simplified18.6

      \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{b} \cdot -0.5}}{a}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt_binary6418.6

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

    9. Applied associate-/r*_binary6418.6

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

    10. Simplified18.6

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{\frac{a \cdot c}{b}}{\sqrt{a}}}}{\sqrt{a}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 34.79216116694219:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\frac{a \cdot c}{b}}{\sqrt{a}}}{\sqrt{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))