Average Error: 33.7 → 10.7
Time: 4.6m
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{1}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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{\frac{-b}{2}}{a} - \frac{1}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2}\\

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

\mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\
\;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r13416759 = b;
        double r13416760 = -r13416759;
        double r13416761 = r13416759 * r13416759;
        double r13416762 = 4.0;
        double r13416763 = a;
        double r13416764 = c;
        double r13416765 = r13416763 * r13416764;
        double r13416766 = r13416762 * r13416765;
        double r13416767 = r13416761 - r13416766;
        double r13416768 = sqrt(r13416767);
        double r13416769 = r13416760 - r13416768;
        double r13416770 = 2.0;
        double r13416771 = r13416770 * r13416763;
        double r13416772 = r13416769 / r13416771;
        return r13416772;
}
↓
double f(double a, double b, double c) {
        double r13416773 = b;
        double r13416774 = -7.363255598823911e-15;
        bool r13416775 = r13416773 <= r13416774;
        double r13416776 = c;
        double r13416777 = r13416776 / r13416773;
        double r13416778 = -r13416777;
        double r13416779 = -6.936587154412951e-28;
        bool r13416780 = r13416773 <= r13416779;
        double r13416781 = -r13416773;
        double r13416782 = 2.0;
        double r13416783 = r13416781 / r13416782;
        double r13416784 = a;
        double r13416785 = r13416783 / r13416784;
        double r13416786 = 1.0;
        double r13416787 = r13416786 / r13416784;
        double r13416788 = -4.0;
        double r13416789 = r13416784 * r13416788;
        double r13416790 = r13416773 * r13416773;
        double r13416791 = fma(r13416776, r13416789, r13416790);
        double r13416792 = sqrt(r13416791);
        double r13416793 = r13416792 / r13416782;
        double r13416794 = r13416787 * r13416793;
        double r13416795 = r13416785 - r13416794;
        double r13416796 = -2.3344326820285623e-123;
        bool r13416797 = r13416773 <= r13416796;
        double r13416798 = 1.6691257204922504e+85;
        bool r13416799 = r13416773 <= r13416798;
        double r13416800 = r13416784 / r13416793;
        double r13416801 = r13416786 / r13416800;
        double r13416802 = r13416785 - r13416801;
        double r13416803 = r13416773 / r13416784;
        double r13416804 = r13416777 - r13416803;
        double r13416805 = r13416799 ? r13416802 : r13416804;
        double r13416806 = r13416797 ? r13416778 : r13416805;
        double r13416807 = r13416780 ? r13416795 : r13416806;
        double r13416808 = r13416775 ? r13416778 : r13416807;
        return r13416808;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

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

Derivation

  1. Split input into 4 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. Simplified50.9

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

    3. Taylor expanded around -inf 10.6

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

    4. Simplified10.6

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

    if -7.363255598823911e-15 < b < -6.936587154412951e-28

    1. Initial program 35.8

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

    2. Simplified35.8

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

    4. Applied div-sub35.8

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

    5. Applied div-sub35.8

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

    7. Applied div-inv35.9

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

    if -2.3344326820285623e-123 < b < 1.6691257204922504e+85

    1. Initial program 12.6

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

    2. Simplified12.7

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

    4. Applied div-sub12.7

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

    5. Applied div-sub12.7

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

    7. Applied clear-num12.8

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

    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. Simplified42.9

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

    3. Taylor expanded around inf 3.7

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{1}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))