Average Error: 34.5 → 10.4
Time: 16.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -63362873442066488610789523456:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, \frac{c}{b} \cdot 2\right)}{2}\\ \mathbf{elif}\;b \le 6.484072051994263737451444554171174935457 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)} \cdot \sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)}}, \sqrt{\sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)}}, -b\right)}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -63362873442066488610789523456:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, \frac{c}{b} \cdot 2\right)}{2}\\

\mathbf{elif}\;b \le 6.484072051994263737451444554171174935457 \cdot 10^{-107}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)} \cdot \sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)}}, \sqrt{\sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)}}, -b\right)}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r3255025 = b;
        double r3255026 = -r3255025;
        double r3255027 = r3255025 * r3255025;
        double r3255028 = 4.0;
        double r3255029 = a;
        double r3255030 = c;
        double r3255031 = r3255029 * r3255030;
        double r3255032 = r3255028 * r3255031;
        double r3255033 = r3255027 - r3255032;
        double r3255034 = sqrt(r3255033);
        double r3255035 = r3255026 + r3255034;
        double r3255036 = 2.0;
        double r3255037 = r3255036 * r3255029;
        double r3255038 = r3255035 / r3255037;
        return r3255038;
}
↓
double f(double a, double b, double c) {
        double r3255039 = b;
        double r3255040 = -6.336287344206649e+28;
        bool r3255041 = r3255039 <= r3255040;
        double r3255042 = a;
        double r3255043 = r3255039 / r3255042;
        double r3255044 = -2.0;
        double r3255045 = c;
        double r3255046 = r3255045 / r3255039;
        double r3255047 = 2.0;
        double r3255048 = r3255046 * r3255047;
        double r3255049 = fma(r3255043, r3255044, r3255048);
        double r3255050 = r3255049 / r3255047;
        double r3255051 = 6.484072051994264e-107;
        bool r3255052 = r3255039 <= r3255051;
        double r3255053 = r3255039 * r3255039;
        double r3255054 = 4.0;
        double r3255055 = r3255054 * r3255045;
        double r3255056 = r3255042 * r3255055;
        double r3255057 = r3255053 - r3255056;
        double r3255058 = cbrt(r3255057);
        double r3255059 = r3255058 * r3255058;
        double r3255060 = sqrt(r3255059);
        double r3255061 = sqrt(r3255058);
        double r3255062 = -r3255039;
        double r3255063 = fma(r3255060, r3255061, r3255062);
        double r3255064 = r3255063 / r3255042;
        double r3255065 = r3255064 / r3255047;
        double r3255066 = -2.0;
        double r3255067 = r3255046 * r3255066;
        double r3255068 = r3255067 / r3255047;
        double r3255069 = r3255052 ? r3255065 : r3255068;
        double r3255070 = r3255041 ? r3255050 : r3255069;
        return r3255070;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.5
Target21.0
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.336287344206649e+28

    1. Initial program 34.8

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

    2. Simplified34.9

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

    3. Taylor expanded around -inf 7.0

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

    4. Simplified7.0

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

    if -6.336287344206649e+28 < b < 6.484072051994264e-107

    1. Initial program 12.9

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

    2. Simplified12.9

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

    4. Applied add-cube-cbrt13.4

      \[\leadsto \frac{\frac{\sqrt{\color{blue}{\left(\sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)} \cdot \sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)}\right) \cdot \sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)}}} - b}{a}}{2}\]

    5. Applied sqrt-prod13.4

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{\sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)} \cdot \sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)}} \cdot \sqrt{\sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)}}} - b}{a}}{2}\]

    6. Applied fma-neg13.4

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)} \cdot \sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)}}, \sqrt{\sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)}}, -b\right)}}{a}}{2}\]

    if 6.484072051994264e-107 < b

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around inf 9.7

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -63362873442066488610789523456:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, \frac{c}{b} \cdot 2\right)}{2}\\ \mathbf{elif}\;b \le 6.484072051994263737451444554171174935457 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)} \cdot \sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)}}, \sqrt{\sqrt[3]{b \cdot b - a \cdot \left(4 \cdot c\right)}}, -b\right)}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

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

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