Average Error: 34.5 → 6.5
Time: 14.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.631041364691662867016708422572785521166 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{elif}\;b \le 1.292564275165585001505551266323067045785 \cdot 10^{140}:\\ \;\;\;\;\frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{2}{4 \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 4.631041364691662867016708422572785521166 \cdot 10^{-308}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\

\mathbf{elif}\;b \le 1.292564275165585001505551266323067045785 \cdot 10^{140}:\\
\;\;\;\;\frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{2}{4 \cdot c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r75005 = b;
        double r75006 = -r75005;
        double r75007 = r75005 * r75005;
        double r75008 = 4.0;
        double r75009 = a;
        double r75010 = r75008 * r75009;
        double r75011 = c;
        double r75012 = r75010 * r75011;
        double r75013 = r75007 - r75012;
        double r75014 = sqrt(r75013);
        double r75015 = r75006 + r75014;
        double r75016 = 2.0;
        double r75017 = r75016 * r75009;
        double r75018 = r75015 / r75017;
        return r75018;
}
↓
double f(double a, double b, double c) {
        double r75019 = b;
        double r75020 = -1.8572382657132166e+109;
        bool r75021 = r75019 <= r75020;
        double r75022 = 1.0;
        double r75023 = c;
        double r75024 = r75023 / r75019;
        double r75025 = a;
        double r75026 = r75019 / r75025;
        double r75027 = r75024 - r75026;
        double r75028 = r75022 * r75027;
        double r75029 = 4.631041364691663e-308;
        bool r75030 = r75019 <= r75029;
        double r75031 = 1.0;
        double r75032 = 2.0;
        double r75033 = r75032 * r75025;
        double r75034 = r75019 * r75019;
        double r75035 = 4.0;
        double r75036 = r75035 * r75025;
        double r75037 = r75036 * r75023;
        double r75038 = r75034 - r75037;
        double r75039 = sqrt(r75038);
        double r75040 = r75039 - r75019;
        double r75041 = r75033 / r75040;
        double r75042 = r75031 / r75041;
        double r75043 = 1.292564275165585e+140;
        bool r75044 = r75019 <= r75043;
        double r75045 = -r75019;
        double r75046 = r75045 - r75039;
        double r75047 = r75031 / r75046;
        double r75048 = r75035 * r75023;
        double r75049 = r75032 / r75048;
        double r75050 = r75047 / r75049;
        double r75051 = -1.0;
        double r75052 = r75051 * r75024;
        double r75053 = r75044 ? r75050 : r75052;
        double r75054 = r75030 ? r75042 : r75053;
        double r75055 = r75021 ? r75028 : r75054;
        return r75055;
}

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

Original34.5
Target20.8
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.8572382657132166e+109

    1. Initial program 50.1

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

    2. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -1.8572382657132166e+109 < b < 4.631041364691663e-308

    1. Initial program 9.0

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

    3. Applied clear-num9.2

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

    4. Simplified9.2

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

    if 4.631041364691663e-308 < b < 1.292564275165585e+140

    1. Initial program 34.6

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

    3. Applied flip-+34.7

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

    4. Simplified15.8

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

    6. Applied clear-num16.0

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

    7. Simplified16.0

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

    9. Applied div-inv16.3

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

    10. Applied add-sqr-sqrt16.3

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

    11. Applied times-frac16.2

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

    12. Applied associate-/l*15.1

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

    13. Simplified14.7

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

    15. Applied *-un-lft-identity14.7

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

    16. Applied times-frac14.7

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

    17. Simplified14.7

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

    18. Simplified8.2

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

    if 1.292564275165585e+140 < b

    1. Initial program 62.6

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

    2. Taylor expanded around inf 1.4

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.631041364691662867016708422572785521166 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{elif}\;b \le 1.292564275165585001505551266323067045785 \cdot 10^{140}:\\ \;\;\;\;\frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{2}{4 \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(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)))