Average Error: 34.6 → 9.9
Time: 16.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, b, \frac{a \cdot 1.5}{\frac{b}{c}}\right)}{3}}{a}\\ \mathbf{elif}\;b \le 8.085265835057349842233247168077451568119 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot \left(-c\right)\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, b, \frac{a \cdot 1.5}{\frac{b}{c}}\right)}{3}}{a}\\

\mathbf{elif}\;b \le 8.085265835057349842233247168077451568119 \cdot 10^{-63}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot \left(-c\right)\right)} - b}{3}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-0.5 \cdot c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r110069 = b;
        double r110070 = -r110069;
        double r110071 = r110069 * r110069;
        double r110072 = 3.0;
        double r110073 = a;
        double r110074 = r110072 * r110073;
        double r110075 = c;
        double r110076 = r110074 * r110075;
        double r110077 = r110071 - r110076;
        double r110078 = sqrt(r110077);
        double r110079 = r110070 + r110078;
        double r110080 = r110079 / r110074;
        return r110080;
}
↓
double f(double a, double b, double c) {
        double r110081 = b;
        double r110082 = -7.943482039519134e+75;
        bool r110083 = r110081 <= r110082;
        double r110084 = -2.0;
        double r110085 = a;
        double r110086 = 1.5;
        double r110087 = r110085 * r110086;
        double r110088 = c;
        double r110089 = r110081 / r110088;
        double r110090 = r110087 / r110089;
        double r110091 = fma(r110084, r110081, r110090);
        double r110092 = 3.0;
        double r110093 = r110091 / r110092;
        double r110094 = r110093 / r110085;
        double r110095 = 8.08526583505735e-63;
        bool r110096 = r110081 <= r110095;
        double r110097 = r110092 * r110085;
        double r110098 = -r110088;
        double r110099 = r110097 * r110098;
        double r110100 = fma(r110081, r110081, r110099);
        double r110101 = sqrt(r110100);
        double r110102 = r110101 - r110081;
        double r110103 = r110102 / r110092;
        double r110104 = r110103 / r110085;
        double r110105 = -0.5;
        double r110106 = r110105 * r110088;
        double r110107 = r110106 / r110081;
        double r110108 = r110096 ? r110104 : r110107;
        double r110109 = r110083 ? r110094 : r110108;
        return r110109;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -7.943482039519134e+75

    1. Initial program 42.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied associate-/r*42.8

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}}\]

    4. Simplified42.8

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)} - b}{3}}}{a}\]

    5. Taylor expanded around -inf 9.7

      \[\leadsto \frac{\frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3}}{a}\]

    6. Simplified4.5

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-2, b, \frac{a \cdot 1.5}{\frac{b}{c}}\right)}}{3}}{a}\]

    if -7.943482039519134e+75 < b < 8.08526583505735e-63

    1. Initial program 13.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied associate-/r*13.7

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}}\]

    4. Simplified13.7

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)} - b}{3}}}{a}\]

    if 8.08526583505735e-63 < b

    1. Initial program 53.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around inf 8.3

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]

    3. Simplified8.3

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, b, \frac{a \cdot 1.5}{\frac{b}{c}}\right)}{3}}{a}\\ \mathbf{elif}\;b \le 8.085265835057349842233247168077451568119 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot \left(-c\right)\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))