Average Error: 34.1 → 9.3
Time: 6.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.3044033969831823 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 8.9305277508569929 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{elif}\;b \le 4.01993084419163312 \cdot 10^{109}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\frac{1}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{\frac{a}{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.3044033969831823 \cdot 10^{153}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 8.9305277508569929 \cdot 10^{-82}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\

\mathbf{elif}\;b \le 4.01993084419163312 \cdot 10^{109}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\frac{1}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{\frac{a}{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r182002 = b;
        double r182003 = -r182002;
        double r182004 = r182002 * r182002;
        double r182005 = 4.0;
        double r182006 = a;
        double r182007 = r182005 * r182006;
        double r182008 = c;
        double r182009 = r182007 * r182008;
        double r182010 = r182004 - r182009;
        double r182011 = sqrt(r182010);
        double r182012 = r182003 + r182011;
        double r182013 = 2.0;
        double r182014 = r182013 * r182006;
        double r182015 = r182012 / r182014;
        return r182015;
}
↓
double f(double a, double b, double c) {
        double r182016 = b;
        double r182017 = -2.3044033969831823e+153;
        bool r182018 = r182016 <= r182017;
        double r182019 = 1.0;
        double r182020 = c;
        double r182021 = r182020 / r182016;
        double r182022 = a;
        double r182023 = r182016 / r182022;
        double r182024 = r182021 - r182023;
        double r182025 = r182019 * r182024;
        double r182026 = 8.930527750856993e-82;
        bool r182027 = r182016 <= r182026;
        double r182028 = 1.0;
        double r182029 = 2.0;
        double r182030 = r182028 / r182029;
        double r182031 = -r182016;
        double r182032 = r182016 * r182016;
        double r182033 = 4.0;
        double r182034 = r182033 * r182022;
        double r182035 = r182034 * r182020;
        double r182036 = r182032 - r182035;
        double r182037 = sqrt(r182036);
        double r182038 = r182031 + r182037;
        double r182039 = r182038 / r182022;
        double r182040 = r182030 * r182039;
        double r182041 = 4.019930844191633e+109;
        bool r182042 = r182016 <= r182041;
        double r182043 = r182031 - r182037;
        double r182044 = cbrt(r182043);
        double r182045 = r182044 * r182044;
        double r182046 = r182028 / r182045;
        double r182047 = r182022 * r182020;
        double r182048 = r182033 * r182047;
        double r182049 = r182022 / r182048;
        double r182050 = r182049 * r182044;
        double r182051 = r182046 / r182050;
        double r182052 = r182030 * r182051;
        double r182053 = -1.0;
        double r182054 = r182053 * r182021;
        double r182055 = r182042 ? r182052 : r182054;
        double r182056 = r182027 ? r182040 : r182055;
        double r182057 = r182018 ? r182025 : r182056;
        return r182057;
}

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

Target

Original34.1
Target20.7
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.3044033969831823e+153

    1. Initial program 63.5

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

    2. Taylor expanded around -inf 2.0

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

    3. Simplified2.0

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

    if -2.3044033969831823e+153 < b < 8.930527750856993e-82

    1. Initial program 12.5

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

    3. Applied *-un-lft-identity12.5

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

    4. Applied times-frac12.5

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}}\]

    if 8.930527750856993e-82 < b < 4.019930844191633e+109

    1. Initial program 43.0

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

    3. Applied *-un-lft-identity43.0

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

    4. Applied times-frac43.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}}\]
    5. Using strategy rm

    6. Applied flip-+43.0

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a}\]

    7. Simplified14.9

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt15.6

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\left(\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{a}\]

    10. Applied *-un-lft-identity15.6

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{\left(\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a}\]

    11. Applied times-frac15.6

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{\frac{1}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{a}\]

    12. Applied associate-/l*14.3

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\frac{1}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{\frac{a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}}\]

    13. Simplified13.8

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{1}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{\color{blue}{\frac{a}{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\]

    if 4.019930844191633e+109 < b

    1. Initial program 59.9

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

    2. Taylor expanded around inf 2.4

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.3044033969831823 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 8.9305277508569929 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{elif}\;b \le 4.01993084419163312 \cdot 10^{109}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\frac{1}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{\frac{a}{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))