Average Error: 34.5 → 6.5
Time: 16.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.517290934480198379594061663675559980392 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 1.098350897361241909514821132660621002261 \cdot 10^{141}:\\ \;\;\;\;\frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2} \cdot \left(c \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{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 -8.517290934480198379594061663675559980392 \cdot 10^{-304}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\

\mathbf{elif}\;b \le 1.098350897361241909514821132660621002261 \cdot 10^{141}:\\
\;\;\;\;\frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2} \cdot \left(c \cdot 4\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r54043 = b;
        double r54044 = -r54043;
        double r54045 = r54043 * r54043;
        double r54046 = 4.0;
        double r54047 = a;
        double r54048 = c;
        double r54049 = r54047 * r54048;
        double r54050 = r54046 * r54049;
        double r54051 = r54045 - r54050;
        double r54052 = sqrt(r54051);
        double r54053 = r54044 + r54052;
        double r54054 = 2.0;
        double r54055 = r54054 * r54047;
        double r54056 = r54053 / r54055;
        return r54056;
}
↓
double f(double a, double b, double c) {
        double r54057 = b;
        double r54058 = -1.8572382657132166e+109;
        bool r54059 = r54057 <= r54058;
        double r54060 = 1.0;
        double r54061 = c;
        double r54062 = r54061 / r54057;
        double r54063 = a;
        double r54064 = r54057 / r54063;
        double r54065 = r54062 - r54064;
        double r54066 = r54060 * r54065;
        double r54067 = -8.517290934480198e-304;
        bool r54068 = r54057 <= r54067;
        double r54069 = 1.0;
        double r54070 = 2.0;
        double r54071 = r54070 * r54063;
        double r54072 = r54057 * r54057;
        double r54073 = 4.0;
        double r54074 = r54063 * r54061;
        double r54075 = r54073 * r54074;
        double r54076 = r54072 - r54075;
        double r54077 = sqrt(r54076);
        double r54078 = r54077 - r54057;
        double r54079 = r54071 / r54078;
        double r54080 = r54069 / r54079;
        double r54081 = 1.0983508973612419e+141;
        bool r54082 = r54057 <= r54081;
        double r54083 = -r54057;
        double r54084 = r54083 - r54077;
        double r54085 = r54069 / r54084;
        double r54086 = r54085 / r54070;
        double r54087 = r54061 * r54073;
        double r54088 = r54086 * r54087;
        double r54089 = -1.0;
        double r54090 = r54089 * r54062;
        double r54091 = r54082 ? r54088 : r54090;
        double r54092 = r54068 ? r54080 : r54091;
        double r54093 = r54059 ? r54066 : r54092;
        return r54093;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -1.8572382657132166e+109

    1. Initial program 50.1

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

    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 < -8.517290934480198e-304

    1. Initial program 9.0

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

    3. Applied clear-num9.1

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

    4. Simplified9.1

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

    if -8.517290934480198e-304 < b < 1.0983508973612419e+141

    1. Initial program 34.5

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

    3. Applied flip-+34.5

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

    4. Simplified15.8

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

    6. Applied clear-num16.0

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

    7. Simplified16.0

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

    9. Applied div-inv16.4

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

    10. Applied add-cube-cbrt16.4

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

    11. Applied times-frac16.3

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

    12. Applied times-frac15.2

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

    13. Simplified15.2

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

    14. Simplified8.2

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

    if 1.0983508973612419e+141 < b

    1. Initial program 62.7

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.517290934480198379594061663675559980392 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 1.098350897361241909514821132660621002261 \cdot 10^{141}:\\ \;\;\;\;\frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2} \cdot \left(c \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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)))