Average Error: 28.7 → 15.9
Time: 6.0s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq 2007.413650500031:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot \left(b \cdot 2\right)}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq 2007.413650500031:\\
\;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot \left(b \cdot 2\right)}\\

\end{array}
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))) / ((double) (2.0 * a)));
}
↓
double code(double a, double b, double c) {
	double VAR;
	if ((b <= 2007.413650500031)) {
		VAR = ((((double) (((double) (b * b)) - ((double) (((double) (b * b)) + ((double) (4.0 * ((double) (a * c)))))))) / ((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (a * c))))))))))) / ((double) (a * 2.0)));
	} else {
		VAR = (((double) (((double) (a * c)) * -2.0)) / ((double) (a * ((double) (b * 2.0)))));
	}
	return VAR;
}

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 < 2007.4136505000311

    1. Initial program Error: 17.3 bits

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. SimplifiedError: 17.3 bits

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}}\]
    3. Using strategy rm

    4. Applied flip--Error: 17.3 bits

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b \cdot b}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + b}}}{a \cdot 2}\]

    5. SimplifiedError: 16.3 bits

      \[\leadsto \frac{\frac{\color{blue}{b \cdot b - \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + b}}{a \cdot 2}\]

    6. SimplifiedError: 16.3 bits

      \[\leadsto \frac{\frac{b \cdot b - \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2}\]

    if 2007.4136505000311 < b

    1. Initial program Error: 37.3 bits

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. SimplifiedError: 37.3 bits

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}}\]

    3. Taylor expanded around inf Error: 15.6 bits

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

    4. SimplifiedError: 15.6 bits

      \[\leadsto \frac{\color{blue}{-2 \cdot \left(\frac{a}{b} \cdot c\right)}}{a \cdot 2}\]
    5. Using strategy rm

    6. Applied associate-*l/Error: 15.6 bits

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

    7. Applied associate-*r/Error: 15.6 bits

      \[\leadsto \frac{\color{blue}{\frac{-2 \cdot \left(a \cdot c\right)}{b}}}{a \cdot 2}\]

    8. Applied associate-/l/Error: 15.6 bits

      \[\leadsto \color{blue}{\frac{-2 \cdot \left(a \cdot c\right)}{\left(a \cdot 2\right) \cdot b}}\]

    9. SimplifiedError: 15.6 bits

      \[\leadsto \frac{-2 \cdot \left(a \cdot c\right)}{\color{blue}{a \cdot \left(b \cdot 2\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 15.9 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2007.413650500031:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot \left(b \cdot 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))