Average Error: 33.7 → 10.6
Time: 17.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -7.363255598823911 \cdot 10^{-15}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\

\mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3192042 = b;
        double r3192043 = -r3192042;
        double r3192044 = r3192042 * r3192042;
        double r3192045 = 4.0;
        double r3192046 = a;
        double r3192047 = c;
        double r3192048 = r3192046 * r3192047;
        double r3192049 = r3192045 * r3192048;
        double r3192050 = r3192044 - r3192049;
        double r3192051 = sqrt(r3192050);
        double r3192052 = r3192043 - r3192051;
        double r3192053 = 2.0;
        double r3192054 = r3192053 * r3192046;
        double r3192055 = r3192052 / r3192054;
        return r3192055;
}
↓
double f(double a, double b, double c) {
        double r3192056 = b;
        double r3192057 = -7.363255598823911e-15;
        bool r3192058 = r3192056 <= r3192057;
        double r3192059 = c;
        double r3192060 = r3192059 / r3192056;
        double r3192061 = -r3192060;
        double r3192062 = -6.936587154412951e-28;
        bool r3192063 = r3192056 <= r3192062;
        double r3192064 = -r3192056;
        double r3192065 = 2.0;
        double r3192066 = a;
        double r3192067 = r3192065 * r3192066;
        double r3192068 = r3192064 / r3192067;
        double r3192069 = r3192056 * r3192056;
        double r3192070 = r3192066 * r3192059;
        double r3192071 = 4.0;
        double r3192072 = r3192070 * r3192071;
        double r3192073 = r3192069 - r3192072;
        double r3192074 = sqrt(r3192073);
        double r3192075 = r3192074 / r3192067;
        double r3192076 = r3192068 - r3192075;
        double r3192077 = -2.3344326820285623e-123;
        bool r3192078 = r3192056 <= r3192077;
        double r3192079 = 1.6691257204922504e+85;
        bool r3192080 = r3192056 <= r3192079;
        double r3192081 = r3192056 / r3192066;
        double r3192082 = r3192060 - r3192081;
        double r3192083 = r3192080 ? r3192076 : r3192082;
        double r3192084 = r3192078 ? r3192061 : r3192083;
        double r3192085 = r3192063 ? r3192076 : r3192084;
        double r3192086 = r3192058 ? r3192061 : r3192085;
        return r3192086;
}

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

Original33.7
Target21.0
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -7.363255598823911e-15 or -6.936587154412951e-28 < b < -2.3344326820285623e-123

    1. Initial program 50.9

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

    2. Taylor expanded around -inf 10.6

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

    3. Simplified10.6

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

    if -7.363255598823911e-15 < b < -6.936587154412951e-28 or -2.3344326820285623e-123 < b < 1.6691257204922504e+85

    1. Initial program 13.4

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

    3. Applied div-sub13.4

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

    if 1.6691257204922504e+85 < b

    1. Initial program 42.9

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))