Average Error: 34.2 → 9.1
Time: 18.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -763129212434271441067123993682640896:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.580019013081130749755184029236910886016 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -763129212434271441067123993682640896:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 9.580019013081130749755184029236910886016 \cdot 10^{-278}:\\
\;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r64043 = b;
        double r64044 = -r64043;
        double r64045 = r64043 * r64043;
        double r64046 = 4.0;
        double r64047 = a;
        double r64048 = c;
        double r64049 = r64047 * r64048;
        double r64050 = r64046 * r64049;
        double r64051 = r64045 - r64050;
        double r64052 = sqrt(r64051);
        double r64053 = r64044 - r64052;
        double r64054 = 2.0;
        double r64055 = r64054 * r64047;
        double r64056 = r64053 / r64055;
        return r64056;
}
↓
double f(double a, double b, double c) {
        double r64057 = b;
        double r64058 = -7.631292124342714e+35;
        bool r64059 = r64057 <= r64058;
        double r64060 = -1.0;
        double r64061 = c;
        double r64062 = r64061 / r64057;
        double r64063 = r64060 * r64062;
        double r64064 = 9.580019013081131e-278;
        bool r64065 = r64057 <= r64064;
        double r64066 = 4.0;
        double r64067 = a;
        double r64068 = r64066 * r64067;
        double r64069 = r64061 * r64068;
        double r64070 = 2.0;
        double r64071 = pow(r64057, r64070);
        double r64072 = r64067 * r64061;
        double r64073 = r64066 * r64072;
        double r64074 = r64071 - r64073;
        double r64075 = sqrt(r64074);
        double r64076 = r64075 - r64057;
        double r64077 = r64069 / r64076;
        double r64078 = 2.0;
        double r64079 = r64078 * r64067;
        double r64080 = r64077 / r64079;
        double r64081 = 5.031608061939103e+53;
        bool r64082 = r64057 <= r64081;
        double r64083 = -r64057;
        double r64084 = r64057 * r64057;
        double r64085 = r64084 - r64073;
        double r64086 = sqrt(r64085);
        double r64087 = r64083 - r64086;
        double r64088 = 1.0;
        double r64089 = r64088 / r64079;
        double r64090 = r64087 * r64089;
        double r64091 = 1.0;
        double r64092 = r64057 / r64067;
        double r64093 = r64062 - r64092;
        double r64094 = r64091 * r64093;
        double r64095 = r64082 ? r64090 : r64094;
        double r64096 = r64065 ? r64080 : r64095;
        double r64097 = r64059 ? r64063 : r64096;
        return r64097;
}

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
Target21.3
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.631292124342714e+35

    1. Initial program 56.2

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

    2. Taylor expanded around -inf 4.5

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

    if -7.631292124342714e+35 < b < 9.580019013081131e-278

    1. Initial program 27.7

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

    3. Applied flip--27.7

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

    4. Simplified16.7

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

    5. Simplified16.7

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

    if 9.580019013081131e-278 < b < 5.031608061939103e+53

    1. Initial program 9.4

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

    3. Applied div-inv9.6

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

    if 5.031608061939103e+53 < b

    1. Initial program 39.6

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

    2. Taylor expanded around inf 5.7

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

    3. Simplified5.7

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -763129212434271441067123993682640896:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.580019013081130749755184029236910886016 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

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

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