Average Error: 34.2 → 13.8
Time: 25.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\frac{\frac{b}{2} \cdot -2}{a}\\ \mathbf{elif}\;b \le 1.627160326743933989296751920246101845745 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\sqrt{\sqrt{2}}}}{\sqrt{2}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \end{array}\]
\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 -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\
\;\;\;\;\frac{\frac{b}{2} \cdot -2}{a}\\

\mathbf{elif}\;b \le 1.627160326743933989296751920246101845745 \cdot 10^{-46}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\sqrt{\sqrt{2}}}}{\sqrt{2}}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4943995 = b;
        double r4943996 = -r4943995;
        double r4943997 = r4943995 * r4943995;
        double r4943998 = 4.0;
        double r4943999 = a;
        double r4944000 = r4943998 * r4943999;
        double r4944001 = c;
        double r4944002 = r4944000 * r4944001;
        double r4944003 = r4943997 - r4944002;
        double r4944004 = sqrt(r4944003);
        double r4944005 = r4943996 + r4944004;
        double r4944006 = 2.0;
        double r4944007 = r4944006 * r4943999;
        double r4944008 = r4944005 / r4944007;
        return r4944008;
}
↓
double f(double a, double b, double c) {
        double r4944009 = b;
        double r4944010 = -3.7108875578650606e+138;
        bool r4944011 = r4944009 <= r4944010;
        double r4944012 = 2.0;
        double r4944013 = r4944009 / r4944012;
        double r4944014 = -2.0;
        double r4944015 = r4944013 * r4944014;
        double r4944016 = a;
        double r4944017 = r4944015 / r4944016;
        double r4944018 = 1.627160326743934e-46;
        bool r4944019 = r4944009 <= r4944018;
        double r4944020 = 1.0;
        double r4944021 = sqrt(r4944012);
        double r4944022 = sqrt(r4944021);
        double r4944023 = r4944020 / r4944022;
        double r4944024 = r4944009 * r4944009;
        double r4944025 = 4.0;
        double r4944026 = r4944025 * r4944016;
        double r4944027 = c;
        double r4944028 = r4944026 * r4944027;
        double r4944029 = r4944024 - r4944028;
        double r4944030 = sqrt(r4944029);
        double r4944031 = r4944030 - r4944009;
        double r4944032 = r4944031 / r4944022;
        double r4944033 = r4944032 / r4944021;
        double r4944034 = r4944023 * r4944033;
        double r4944035 = r4944034 / r4944016;
        double r4944036 = -1.0;
        double r4944037 = r4944016 * r4944027;
        double r4944038 = r4944037 / r4944009;
        double r4944039 = r4944036 * r4944038;
        double r4944040 = r4944039 / r4944016;
        double r4944041 = r4944019 ? r4944035 : r4944040;
        double r4944042 = r4944011 ? r4944017 : r4944041;
        return r4944042;
}

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

Original34.2
Target21.0
Herbie13.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.7108875578650606e+138

    1. Initial program 58.5

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

    2. Simplified58.5

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

    4. Applied add-sqr-sqrt58.6

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

    5. Applied associate-/r*58.6

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

    6. Taylor expanded around -inf 3.3

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

    7. Simplified2.1

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

    if -3.7108875578650606e+138 < b < 1.627160326743934e-46

    1. Initial program 13.0

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

    2. Simplified13.0

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

    4. Applied add-sqr-sqrt13.8

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

    5. Applied associate-/r*13.5

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

    7. Applied *-un-lft-identity13.5

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

    8. Applied sqrt-prod13.5

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

    9. Applied add-sqr-sqrt13.5

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

    10. Applied sqrt-prod13.1

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

    11. Applied *-un-lft-identity13.1

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

    12. Applied times-frac13.2

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

    13. Applied times-frac13.2

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

    14. Simplified13.2

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

    if 1.627160326743934e-46 < b

    1. Initial program 54.2

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

    2. Simplified54.2

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

    3. Taylor expanded around inf 18.5

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

  4. Final simplification13.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\frac{\frac{b}{2} \cdot -2}{a}\\ \mathbf{elif}\;b \le 1.627160326743933989296751920246101845745 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\sqrt{\sqrt{2}}}}{\sqrt{2}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))