Average Error: 34.2 → 11.9
Time: 2.2m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.547666603636537260513437138645901028344 \cdot 10^{50}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.547666603636537260513437138645901028344 \cdot 10^{50}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r82102 = b;
        double r82103 = -r82102;
        double r82104 = r82102 * r82102;
        double r82105 = 4.0;
        double r82106 = a;
        double r82107 = c;
        double r82108 = r82106 * r82107;
        double r82109 = r82105 * r82108;
        double r82110 = r82104 - r82109;
        double r82111 = sqrt(r82110);
        double r82112 = r82103 + r82111;
        double r82113 = 2.0;
        double r82114 = r82113 * r82106;
        double r82115 = r82112 / r82114;
        return r82115;
}
↓
double f(double a, double b, double c) {
        double r82116 = b;
        double r82117 = -1.5476666036365373e+50;
        bool r82118 = r82116 <= r82117;
        double r82119 = 1.0;
        double r82120 = c;
        double r82121 = r82120 / r82116;
        double r82122 = a;
        double r82123 = r82116 / r82122;
        double r82124 = r82121 - r82123;
        double r82125 = r82119 * r82124;
        double r82126 = 7.455592343308264e-170;
        bool r82127 = r82116 <= r82126;
        double r82128 = 1.0;
        double r82129 = 2.0;
        double r82130 = r82129 * r82122;
        double r82131 = -r82116;
        double r82132 = r82116 * r82116;
        double r82133 = 4.0;
        double r82134 = r82122 * r82120;
        double r82135 = r82133 * r82134;
        double r82136 = r82132 - r82135;
        double r82137 = sqrt(r82136);
        double r82138 = r82131 + r82137;
        double r82139 = r82130 / r82138;
        double r82140 = r82128 / r82139;
        double r82141 = -1.0;
        double r82142 = r82141 * r82121;
        double r82143 = r82127 ? r82140 : r82142;
        double r82144 = r82118 ? r82125 : r82143;
        return r82144;
}

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
Target20.8
Herbie11.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5476666036365373e+50

    1. Initial program 37.8

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

    2. Taylor expanded around -inf 5.8

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

    3. Simplified5.8

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

    if -1.5476666036365373e+50 < b < 7.455592343308264e-170

    1. Initial program 12.4

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

    3. Applied clear-num12.5

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

    if 7.455592343308264e-170 < b

    1. Initial program 48.9

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

    2. Taylor expanded around inf 14.1

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

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.547666603636537260513437138645901028344 \cdot 10^{50}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))