Average Error: 28.8 → 16.3
Time: 5.9s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq 7113.538462842543:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot \frac{2}{-2}}\\ \end{array}\]
\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 7113.538462842543:\\
\;\;\;\;\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{1}{\frac{b}{c} \cdot \frac{2}{-2}}\\

\end{array}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b 7113.538462842543)
   (/
    (/
     (- (* b b) (+ (* b b) (* 4.0 (* a c))))
     (+ b (sqrt (- (* b b) (* 4.0 (* a c))))))
    (* a 2.0))
   (/ 1.0 (* (/ b c) (/ 2.0 -2.0)))))
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 tmp;
	if ((b <= 7113.538462842543)) {
		tmp = ((((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 {
		tmp = (1.0 / ((double) ((b / c) * (2.0 / -2.0))));
	}
	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 2 regimes

  2. if b < 7113.53846284254269

    1. Initial program Error: 19.0 bits

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

    2. SimplifiedError: 19.0 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: 19.0 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: 17.9 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: 17.9 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 7113.53846284254269 < b

    1. Initial program Error: 38.3 bits

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

    2. SimplifiedError: 38.3 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: 14.9 bits

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

    4. SimplifiedError: 14.8 bits

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

    6. Applied clear-numError: 14.9 bits

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{-2 \cdot \left(\frac{a}{b} \cdot c\right)}}}\]

    7. SimplifiedError: 14.8 bits

      \[\leadsto \frac{1}{\color{blue}{\frac{b}{c} \cdot \frac{2}{-2}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 16.3 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7113.538462842543:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot \frac{2}{-2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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)))