Average Error: 28.6 → 16.1
Time: 6.3s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq 594.5104231633945:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array}\]
\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 594.5104231633945:\\
\;\;\;\;\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}:\\
\;\;\;\;c \cdot \frac{-0.5}{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 594.5104231633945)
   (/
    (/
     (- (- (* b b) (* (* 3.0 a) c)) (* b b))
     (+ b (sqrt (- (* b b) (* (* 3.0 a) c)))))
    (* 3.0 a))
   (* c (/ -0.5 b))))
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (3.0 * a)) * c)))))))) / ((double) (3.0 * a)));
}
↓
double code(double a, double b, double c) {
	double tmp;
	if ((b <= 594.5104231633945)) {
		tmp = ((((double) (((double) (((double) (b * b)) - ((double) (((double) (3.0 * a)) * c)))) - ((double) (b * b)))) / ((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (3.0 * a)) * c))))))))) / ((double) (3.0 * a)));
	} else {
		tmp = ((double) (c * (-0.5 / 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 2 regimes

  2. if b < 594.51042316339453

    1. Initial program 16.6

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

    2. Simplified16.6

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

    4. Applied flip--_binary6416.6

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

    5. Simplified15.6

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

    6. Simplified15.6

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

    if 594.51042316339453 < b

    1. Initial program 36.1

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

    2. Simplified36.1

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

    3. Taylor expanded around inf 16.5

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

    5. Applied associate-/l*_binary6416.5

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

    6. Simplified16.4

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

    8. Applied div-inv_binary6416.5

      \[\leadsto \frac{-1.5}{\frac{3}{\color{blue}{c \cdot \frac{1}{b}}}}\]

    9. Applied *-un-lft-identity_binary6416.5

      \[\leadsto \frac{-1.5}{\frac{\color{blue}{1 \cdot 3}}{c \cdot \frac{1}{b}}}\]

    10. Applied times-frac_binary6416.5

      \[\leadsto \frac{-1.5}{\color{blue}{\frac{1}{c} \cdot \frac{3}{\frac{1}{b}}}}\]

    11. Applied *-un-lft-identity_binary6416.5

      \[\leadsto \frac{\color{blue}{1 \cdot -1.5}}{\frac{1}{c} \cdot \frac{3}{\frac{1}{b}}}\]

    12. Applied times-frac_binary6416.5

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{c}} \cdot \frac{-1.5}{\frac{3}{\frac{1}{b}}}}\]

    13. Simplified16.5

      \[\leadsto \color{blue}{c} \cdot \frac{-1.5}{\frac{3}{\frac{1}{b}}}\]

    14. Simplified16.5

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

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 594.5104231633945:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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)))