Average Error: 33.2 → 9.8
Time: 16.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\frac{-\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.8774910265390396 \cdot 10^{-73}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\
\;\;\;\;\frac{-\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r1596827 = b;
        double r1596828 = -r1596827;
        double r1596829 = r1596827 * r1596827;
        double r1596830 = 4.0;
        double r1596831 = a;
        double r1596832 = c;
        double r1596833 = r1596831 * r1596832;
        double r1596834 = r1596830 * r1596833;
        double r1596835 = r1596829 - r1596834;
        double r1596836 = sqrt(r1596835);
        double r1596837 = r1596828 - r1596836;
        double r1596838 = 2.0;
        double r1596839 = r1596838 * r1596831;
        double r1596840 = r1596837 / r1596839;
        return r1596840;
}
↓
double f(double a, double b, double c) {
        double r1596841 = b;
        double r1596842 = -1.8774910265390396e-73;
        bool r1596843 = r1596841 <= r1596842;
        double r1596844 = c;
        double r1596845 = r1596844 / r1596841;
        double r1596846 = -r1596845;
        double r1596847 = 2.5703497435733685e+102;
        bool r1596848 = r1596841 <= r1596847;
        double r1596849 = r1596841 * r1596841;
        double r1596850 = a;
        double r1596851 = r1596844 * r1596850;
        double r1596852 = 4.0;
        double r1596853 = r1596851 * r1596852;
        double r1596854 = r1596849 - r1596853;
        double r1596855 = sqrt(r1596854);
        double r1596856 = r1596855 + r1596841;
        double r1596857 = -r1596856;
        double r1596858 = 2.0;
        double r1596859 = r1596850 * r1596858;
        double r1596860 = r1596857 / r1596859;
        double r1596861 = r1596841 / r1596850;
        double r1596862 = r1596845 - r1596861;
        double r1596863 = r1596848 ? r1596860 : r1596862;
        double r1596864 = r1596843 ? r1596846 : r1596863;
        return r1596864;
}

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

Original33.2
Target20.4
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.8774910265390396e-73

    1. Initial program 52.5

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

    2. Taylor expanded around -inf 8.6

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

    3. Simplified8.6

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

    if -1.8774910265390396e-73 < b < 2.5703497435733685e+102

    1. Initial program 13.1

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

    3. Applied add-sqr-sqrt13.1

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

    4. Applied sqrt-prod13.3

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

    6. Applied neg-sub013.3

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

    7. Applied associate--l-13.3

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

    8. Simplified13.1

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

    if 2.5703497435733685e+102 < b

    1. Initial program 43.9

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\frac{-\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 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)))