Average Error: 32.9 → 29.0
Time: 17.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1.017935821261199 \cdot 10^{+105}:\\ \;\;\;\;\left(\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b} \cdot \frac{\frac{1}{2}}{a}\right) \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 1.017935821261199 \cdot 10^{+105}:\\
\;\;\;\;\left(\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b} \cdot \frac{\frac{1}{2}}{a}\right) \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double a, double b, double c) {
        double r1357948 = b;
        double r1357949 = -r1357948;
        double r1357950 = r1357948 * r1357948;
        double r1357951 = 4.0;
        double r1357952 = a;
        double r1357953 = c;
        double r1357954 = r1357952 * r1357953;
        double r1357955 = r1357951 * r1357954;
        double r1357956 = r1357950 - r1357955;
        double r1357957 = sqrt(r1357956);
        double r1357958 = r1357949 + r1357957;
        double r1357959 = 2.0;
        double r1357960 = r1357959 * r1357952;
        double r1357961 = r1357958 / r1357960;
        return r1357961;
}
↓
double f(double a, double b, double c) {
        double r1357962 = b;
        double r1357963 = 1.017935821261199e+105;
        bool r1357964 = r1357962 <= r1357963;
        double r1357965 = -4.0;
        double r1357966 = a;
        double r1357967 = c;
        double r1357968 = r1357966 * r1357967;
        double r1357969 = r1357962 * r1357962;
        double r1357970 = fma(r1357965, r1357968, r1357969);
        double r1357971 = sqrt(r1357970);
        double r1357972 = r1357971 - r1357962;
        double r1357973 = sqrt(r1357972);
        double r1357974 = 0.5;
        double r1357975 = r1357974 / r1357966;
        double r1357976 = r1357973 * r1357975;
        double r1357977 = r1357976 * r1357973;
        double r1357978 = 0.0;
        double r1357979 = r1357964 ? r1357977 : r1357978;
        return r1357979;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.9
Target20.3
Herbie29.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < 1.017935821261199e+105

    1. Initial program 25.4

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

    2. Simplified25.4

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

    4. Applied *-un-lft-identity25.4

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

    5. Applied div-inv25.5

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

    6. Applied times-frac25.5

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

    7. Simplified25.5

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

    8. Simplified25.5

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt25.8

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

    11. Applied associate-*l*25.8

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

    if 1.017935821261199e+105 < b

    1. Initial program 58.4

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

    2. Simplified58.4

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

    3. Taylor expanded around 0 39.7

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification29.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.017935821261199 \cdot 10^{+105}:\\ \;\;\;\;\left(\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b} \cdot \frac{\frac{1}{2}}{a}\right) \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))