Average Error: 34.4 → 6.4
Time: 1.7min
Precision: binary64
Cost: 8131
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.1862957928694407 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + -2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq -2.9200144778223044 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.5125984377002414 \cdot 10^{+71}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -2.1862957928694407 \cdot 10^{+151}:\\
\;\;\;\;0.5 \cdot \frac{c}{b_2} + -2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \leq -2.9200144778223044 \cdot 10^{-303}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

\mathbf{elif}\;b_2 \leq 3.5125984377002414 \cdot 10^{+71}:\\
\;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}\\

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

\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 -2.1862957928694407e+151)
   (+ (* 0.5 (/ c b_2)) (* -2.0 (/ b_2 a)))
   (if (<= b_2 -2.9200144778223044e-303)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (if (<= b_2 3.5125984377002414e+71)
       (/ c (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))))
       (/ (* c -0.5) b_2)))))
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 <= -2.1862957928694407e+151) {
		tmp = (0.5 * (c / b_2)) + (-2.0 * (b_2 / a));
	} else if (b_2 <= -2.9200144778223044e-303) {
		tmp = (sqrt((b_2 * b_2) - (c * a)) - b_2) / a;
	} else if (b_2 <= 3.5125984377002414e+71) {
		tmp = c / (-b_2 - sqrt((b_2 * b_2) - (c * a)));
	} else {
		tmp = (c * -0.5) / b_2;
	}
	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

Alternatives

Alternative 1
Error10.1
Cost7746
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.0217367461644927 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + -2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 8.404990259364683 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array}\]
Alternative 2
Error13.3
Cost7618
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.7726378306444592 \cdot 10^{-85}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + -2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 5.5095413459467915 \cdot 10^{-12}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{-c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array}\]
Alternative 3
Error13.4
Cost8132
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.6649790089458955 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + -2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.9552053462202354 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{-c \cdot a} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.3852823591340843 \cdot 10^{-66}:\\ \;\;\;\;\frac{c}{0.5 \cdot \frac{c \cdot a}{b_2} + b_2 \cdot -2}\\ \mathbf{elif}\;b_2 \leq 3.3790711201759446 \cdot 10^{-21}:\\ \;\;\;\;\frac{-c}{\sqrt{-c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array}\]
Alternative 4
Error13.6
Cost7490
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -8.785183345378378 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + -2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.3790711201759446 \cdot 10^{-21}:\\ \;\;\;\;\frac{-c}{\sqrt{-c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array}\]
Alternative 5
Error13.6
Cost7426
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.3641434901267597 \cdot 10^{-86}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + -2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 4.029481774360446 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array}\]
Alternative 6
Error22.2
Cost641
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.5364996268637023 \cdot 10^{-221}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array}\]
Alternative 7
Error36.1
Cost641
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.2785931220637834 \cdot 10^{-223}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array}\]
Alternative 8
Error36.2
Cost641
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.2785931220637834 \cdot 10^{-223}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]
Alternative 9
Error52.8
Cost577
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.6479611445196 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 10
Error55.8
Cost64
\[0\]
Alternative 11
Error61.6
Cost64
\[1\]

Error

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -2.18629579286944072e151

    1. Initial program 62.8

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

    2. Simplified62.8

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

    3. Taylor expanded around -inf 2.3

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

    4. Simplified2.3

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

    5. Simplified2.3

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

    if -2.18629579286944072e151 < b_2 < -2.92001447782230439e-303

    1. Initial program 8.6

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

    2. Simplified8.6

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

    4. Applied *-un-lft-identity_binary648.6

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

    5. Simplified8.6

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

    if -2.92001447782230439e-303 < b_2 < 3.5125984377002414e71

    1. Initial program 30.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv_binary6430.5

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

    5. Applied flip-+_binary6430.5

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

    6. Applied associate-*l/_binary6430.5

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

    7. Simplified14.8

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

    8. Taylor expanded around 0 8.8

      \[\leadsto \frac{\color{blue}{c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\]
    9. Using strategy rm

    10. Applied sub-neg_binary648.8

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

    11. Simplified8.8

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

    if 3.5125984377002414e71 < b_2

    1. Initial program 58.2

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

    2. Simplified58.2

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

    3. Taylor expanded around inf 3.1

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}}\]
    4. Using strategy rm

    5. Applied associate-*r/_binary643.0

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

    6. Simplified3.0

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.1862957928694407 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + -2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq -2.9200144778223044 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.5125984377002414 \cdot 10^{+71}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))