Average Error: 34.3 → 10.4
Time: 8.4s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.0131745716447322 \cdot 10^{+153}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.1010667460580007 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -2.0131745716447322 \cdot 10^{+153}:\\
\;\;\;\;\frac{b_2 \cdot -2}{a}\\

\mathbf{elif}\;b_2 \leq 1.1010667460580007 \cdot 10^{-45}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{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 -2.0131745716447322e+153)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 1.1010667460580007e-45)
     (- (/ (sqrt (- (* b_2 b_2) (* a c))) a) (/ b_2 a))
     (* -0.5 (/ c 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 <= -2.0131745716447322e+153) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.1010667460580007e-45) {
		tmp = (sqrt((b_2 * b_2) - (a * c)) / a) - (b_2 / a);
	} else {
		tmp = -0.5 * (c / 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

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -2.0131745716447322e153

    1. Initial program 63.8

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

    2. Simplified63.8

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

    3. Applied clear-num_binary6463.8

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

    4. Simplified35.9

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

    5. Applied *-un-lft-identity_binary6435.9

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

    6. Applied *-un-lft-identity_binary6435.9

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

    7. Applied times-frac_binary6435.9

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

    8. Applied add-cube-cbrt_binary6435.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{a}{\mathsf{hypot}\left(\sqrt{-c \cdot a}, b_2\right) - b_2}} \]

    9. Applied times-frac_binary6435.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\mathsf{hypot}\left(\sqrt{-c \cdot a}, b_2\right) - b_2}}} \]

    10. Simplified35.9

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\mathsf{hypot}\left(\sqrt{-c \cdot a}, b_2\right) - b_2}} \]

    11. Simplified35.8

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

    12. Taylor expanded in b_2 around -inf 2.0

      \[\leadsto 1 \cdot \frac{\color{blue}{-2 \cdot b_2}}{a} \]

    if -2.0131745716447322e153 < b_2 < 1.1010667460580007e-45

    1. Initial program 13.7

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

    2. Simplified13.7

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

    3. Applied div-sub_binary6413.7

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

    if 1.1010667460580007e-45 < b_2

    1. Initial program 54.1

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

    2. Simplified54.1

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

    3. Taylor expanded in b_2 around inf 8.2

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.0131745716447322 \cdot 10^{+153}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.1010667460580007 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))