Average Error: 33.9 → 7.7
Time: 6.4s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.1639527502413392 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq -7.947315750646704 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a}} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.696534500200679 \cdot 10^{+77}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -1.1639527502413392 \cdot 10^{+40}:\\
\;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \leq -7.947315750646704 \cdot 10^{-83}:\\
\;\;\;\;\frac{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a}} - b_2}{a}\\

\mathbf{elif}\;b_2 \leq 1.696534500200679 \cdot 10^{+77}:\\
\;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b_2} \cdot -0.5\\

\end{array}
(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))
↓
(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.1639527502413392e+40)
   (- (* 0.5 (/ c b_2)) (* 2.0 (/ b_2 a)))
   (if (<= b_2 -7.947315750646704e-83)
     (/
      (-
       (*
        (sqrt (sqrt (- (* b_2 b_2) (* c a))))
        (sqrt (sqrt (- (* b_2 b_2) (* c a)))))
       b_2)
      a)
     (if (<= b_2 1.696534500200679e+77)
       (/ (- c) (+ b_2 (sqrt (- (* b_2 b_2) (* c a)))))
       (* (/ c b_2) -0.5)))))
double code(double a, double b_2, double c) {
	return (-b_2 + sqrt((b_2 * b_2) - (a * c))) / a;
}
↓
double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.1639527502413392e+40) {
		tmp = (0.5 * (c / b_2)) - (2.0 * (b_2 / a));
	} else if (b_2 <= -7.947315750646704e-83) {
		tmp = ((sqrt(sqrt((b_2 * b_2) - (c * a))) * sqrt(sqrt((b_2 * b_2) - (c * a)))) - b_2) / a;
	} else if (b_2 <= 1.696534500200679e+77) {
		tmp = -c / (b_2 + sqrt((b_2 * b_2) - (c * a)));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b_2

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_2 < -1.16395275024133921e40

    1. Initial program 37.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified37.4

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around -inf 6.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    if -1.16395275024133921e40 < b_2 < -7.94731575064670391e-83

    1. Initial program 4.9

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified4.9

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt_binary645.1

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a}\]

    if -7.94731575064670391e-83 < b_2 < 1.6965345002006789e77

    1. Initial program 24.5

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified24.5

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied flip--_binary6427.0

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - b_2 \cdot b_2}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}}{a}\]

    5. Simplified16.8

      \[\leadsto \frac{\frac{\color{blue}{-a \cdot c}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}{a}\]

    6. Simplified16.8

      \[\leadsto \frac{\frac{-a \cdot c}{\color{blue}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity_binary6416.8

      \[\leadsto \frac{\frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\color{blue}{1 \cdot a}}\]

    9. Applied *-un-lft-identity_binary6416.8

      \[\leadsto \frac{\frac{-a \cdot c}{\color{blue}{1 \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}}{1 \cdot a}\]

    10. Applied *-un-lft-identity_binary6416.8

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(-a \cdot c\right)}}{1 \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{1 \cdot a}\]

    11. Applied times-frac_binary6416.8

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{1 \cdot a}\]

    12. Applied times-frac_binary6416.8

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{1} \cdot \frac{\frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}}\]

    13. Simplified16.8

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    14. Simplified11.9

      \[\leadsto 1 \cdot \color{blue}{\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    if 1.6965345002006789e77 < b_2

    1. Initial program 58.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified58.4

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around inf 2.9

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.1639527502413392 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq -7.947315750646704 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a}} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.696534500200679 \cdot 10^{+77}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2020270 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))