Average Error: 33.8 → 9.7
Time: 8.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.76334344440099942 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 8.4035980910933213 \cdot 10^{118}:\\ \;\;\;\;\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \left(\sqrt[3]{1} \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.76334344440099942 \cdot 10^{-73}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 8.4035980910933213 \cdot 10^{118}:\\
\;\;\;\;\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \left(\sqrt[3]{1} \cdot c\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
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 temp;
	if ((b_2 <= -2.7633434444009994e-73)) {
		temp = ((0.5 * (c / b_2)) - (2.0 * (b_2 / a)));
	} else {
		double temp_1;
		if ((b_2 <= 8.403598091093321e+118)) {
			temp_1 = ((1.0 / (-b_2 - sqrt(((b_2 * b_2) - (a * c))))) * (cbrt(1.0) * c));
		} else {
			temp_1 = (-0.5 * (c / b_2));
		}
		temp = temp_1;
	}
	return temp;
}

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 < -2.7633434444009994e-73

    1. Initial program 26.7

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

    2. Taylor expanded around -inf 10.8

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

    if -2.7633434444009994e-73 < b_2 < 8.403598091093321e+118

    1. Initial program 26.8

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

    3. Applied flip-+29.6

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

    4. Simplified17.9

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity17.9

      \[\leadsto \frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\color{blue}{1 \cdot a}}\]

    7. Applied associate-/r*17.9

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{1}}{a}}\]

    8. Simplified16.7

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

    10. Applied add-cube-cbrt17.4

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

    11. Applied div-inv17.4

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

    12. Applied add-cube-cbrt16.7

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

    13. Applied times-frac16.9

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

    14. Applied times-frac13.7

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

    15. Simplified13.7

      \[\leadsto \color{blue}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}} \cdot \frac{\frac{\sqrt[3]{a}}{\frac{1}{c}}}{\sqrt[3]{a}}\]
    16. Using strategy rm

    17. Applied *-un-lft-identity13.7

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

    18. Applied *-un-lft-identity13.7

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

    19. Applied add-sqr-sqrt13.7

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

    20. Applied times-frac13.7

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

    21. Applied *-un-lft-identity13.7

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

    22. Applied cbrt-prod13.7

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

    23. Applied times-frac13.7

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

    24. Applied times-frac13.7

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

    25. Simplified13.7

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

    26. Simplified12.4

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

    if 8.403598091093321e+118 < b_2

    1. Initial program 60.8

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

    2. Taylor expanded around inf 1.6

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.76334344440099942 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 8.4035980910933213 \cdot 10^{118}:\\ \;\;\;\;\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \left(\sqrt[3]{1} \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))