Average Error: 33.0 → 10.4
Time: 19.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.615151909502748 \cdot 10^{-87}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.5387363548079373 \cdot 10^{+99}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\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 -6.615151909502748 \cdot 10^{-87}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 3.5387363548079373 \cdot 10^{+99}:\\
\;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3018016 = b;
        double r3018017 = -r3018016;
        double r3018018 = r3018016 * r3018016;
        double r3018019 = 4.0;
        double r3018020 = a;
        double r3018021 = c;
        double r3018022 = r3018020 * r3018021;
        double r3018023 = r3018019 * r3018022;
        double r3018024 = r3018018 - r3018023;
        double r3018025 = sqrt(r3018024);
        double r3018026 = r3018017 - r3018025;
        double r3018027 = 2.0;
        double r3018028 = r3018027 * r3018020;
        double r3018029 = r3018026 / r3018028;
        return r3018029;
}
↓
double f(double a, double b, double c) {
        double r3018030 = b;
        double r3018031 = -6.615151909502748e-87;
        bool r3018032 = r3018030 <= r3018031;
        double r3018033 = c;
        double r3018034 = r3018033 / r3018030;
        double r3018035 = -r3018034;
        double r3018036 = 3.5387363548079373e+99;
        bool r3018037 = r3018030 <= r3018036;
        double r3018038 = 2.0;
        double r3018039 = a;
        double r3018040 = r3018038 * r3018039;
        double r3018041 = r3018030 / r3018040;
        double r3018042 = -r3018041;
        double r3018043 = r3018030 * r3018030;
        double r3018044 = r3018039 * r3018033;
        double r3018045 = 4.0;
        double r3018046 = r3018044 * r3018045;
        double r3018047 = r3018043 - r3018046;
        double r3018048 = sqrt(r3018047);
        double r3018049 = r3018048 / r3018040;
        double r3018050 = r3018042 - r3018049;
        double r3018051 = -r3018030;
        double r3018052 = r3018051 / r3018039;
        double r3018053 = r3018037 ? r3018050 : r3018052;
        double r3018054 = r3018032 ? r3018035 : r3018053;
        return r3018054;
}

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

Original33.0
Target20.1
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.615151909502748e-87

    1. Initial program 51.9

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

    3. Applied div-sub52.5

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

    4. Taylor expanded around -inf 10.0

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

    5. Simplified10.0

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

    if -6.615151909502748e-87 < b < 3.5387363548079373e+99

    1. Initial program 12.8

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

    3. Applied div-sub12.8

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

    if 3.5387363548079373e+99 < b

    1. Initial program 44.4

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

    3. Applied *-un-lft-identity44.4

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

    4. Applied associate-/l*44.5

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

    5. Taylor expanded around 0 3.9

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

    6. Simplified3.9

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.615151909502748 \cdot 10^{-87}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.5387363548079373 \cdot 10^{+99}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(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)))