Average Error: 34.4 → 9.1
Time: 18.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.741777288939024183924384840560245543701 \cdot 10^{81}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.118633551268419604580976770096584661265 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}{a \cdot 2}\\ \mathbf{elif}\;b \le 4.547489674828777234150499172138545356488 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -7.741777288939024183924384840560245543701 \cdot 10^{81}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.118633551268419604580976770096584661265 \cdot 10^{-302}:\\
\;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}{a \cdot 2}\\

\mathbf{elif}\;b \le 4.547489674828777234150499172138545356488 \cdot 10^{101}:\\
\;\;\;\;\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r3344662 = b;
        double r3344663 = -r3344662;
        double r3344664 = r3344662 * r3344662;
        double r3344665 = 4.0;
        double r3344666 = a;
        double r3344667 = c;
        double r3344668 = r3344666 * r3344667;
        double r3344669 = r3344665 * r3344668;
        double r3344670 = r3344664 - r3344669;
        double r3344671 = sqrt(r3344670);
        double r3344672 = r3344663 - r3344671;
        double r3344673 = 2.0;
        double r3344674 = r3344673 * r3344666;
        double r3344675 = r3344672 / r3344674;
        return r3344675;
}
↓
double f(double a, double b, double c) {
        double r3344676 = b;
        double r3344677 = -7.741777288939024e+81;
        bool r3344678 = r3344676 <= r3344677;
        double r3344679 = -1.0;
        double r3344680 = c;
        double r3344681 = r3344680 / r3344676;
        double r3344682 = r3344679 * r3344681;
        double r3344683 = -2.1186335512684196e-302;
        bool r3344684 = r3344676 <= r3344683;
        double r3344685 = a;
        double r3344686 = r3344685 * r3344680;
        double r3344687 = 4.0;
        double r3344688 = r3344686 * r3344687;
        double r3344689 = r3344676 * r3344676;
        double r3344690 = r3344689 - r3344688;
        double r3344691 = sqrt(r3344690);
        double r3344692 = r3344691 - r3344676;
        double r3344693 = r3344688 / r3344692;
        double r3344694 = 2.0;
        double r3344695 = r3344685 * r3344694;
        double r3344696 = r3344693 / r3344695;
        double r3344697 = 4.547489674828777e+101;
        bool r3344698 = r3344676 <= r3344697;
        double r3344699 = 1.0;
        double r3344700 = r3344699 / r3344695;
        double r3344701 = -r3344676;
        double r3344702 = r3344701 - r3344691;
        double r3344703 = r3344700 * r3344702;
        double r3344704 = r3344676 / r3344685;
        double r3344705 = r3344681 - r3344704;
        double r3344706 = 1.0;
        double r3344707 = r3344705 * r3344706;
        double r3344708 = r3344698 ? r3344703 : r3344707;
        double r3344709 = r3344684 ? r3344696 : r3344708;
        double r3344710 = r3344678 ? r3344682 : r3344709;
        return r3344710;
}

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

Original34.4
Target20.9
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 < -7.741777288939024e+81

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 2.9

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

    if -7.741777288939024e+81 < b < -2.1186335512684196e-302

    1. Initial program 31.7

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

    3. Applied flip--31.7

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

    4. Simplified16.5

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

    5. Simplified16.5

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

    if -2.1186335512684196e-302 < b < 4.547489674828777e+101

    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 4.547489674828777e+101 < b

    1. Initial program 46.8

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

    2. Taylor expanded around inf 4.4

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

    3. Simplified4.4

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.741777288939024183924384840560245543701 \cdot 10^{81}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.118633551268419604580976770096584661265 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}{a \cdot 2}\\ \mathbf{elif}\;b \le 4.547489674828777234150499172138545356488 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a b c)
  :name "quadm (p42, negative)"

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

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