Average Error: 33.0 → 10.5
Time: 36.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.615151909502748 \cdot 10^{-87}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.5387363548079373 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\sqrt{2} \cdot \frac{a}{\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]
\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}:\\
\;\;\;\;\frac{1}{\sqrt{2} \cdot \frac{a}{\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3146837 = b;
        double r3146838 = -r3146837;
        double r3146839 = r3146837 * r3146837;
        double r3146840 = 4.0;
        double r3146841 = a;
        double r3146842 = c;
        double r3146843 = r3146841 * r3146842;
        double r3146844 = r3146840 * r3146843;
        double r3146845 = r3146839 - r3146844;
        double r3146846 = sqrt(r3146845);
        double r3146847 = r3146838 - r3146846;
        double r3146848 = 2.0;
        double r3146849 = r3146848 * r3146841;
        double r3146850 = r3146847 / r3146849;
        return r3146850;
}
↓
double f(double a, double b, double c) {
        double r3146851 = b;
        double r3146852 = -6.615151909502748e-87;
        bool r3146853 = r3146851 <= r3146852;
        double r3146854 = c;
        double r3146855 = r3146854 / r3146851;
        double r3146856 = -r3146855;
        double r3146857 = 3.5387363548079373e+99;
        bool r3146858 = r3146851 <= r3146857;
        double r3146859 = 1.0;
        double r3146860 = 2.0;
        double r3146861 = sqrt(r3146860);
        double r3146862 = a;
        double r3146863 = -r3146851;
        double r3146864 = -4.0;
        double r3146865 = r3146862 * r3146864;
        double r3146866 = r3146851 * r3146851;
        double r3146867 = fma(r3146865, r3146854, r3146866);
        double r3146868 = sqrt(r3146867);
        double r3146869 = r3146863 - r3146868;
        double r3146870 = sqrt(r3146861);
        double r3146871 = r3146869 / r3146870;
        double r3146872 = r3146871 / r3146870;
        double r3146873 = r3146862 / r3146872;
        double r3146874 = r3146861 * r3146873;
        double r3146875 = r3146859 / r3146874;
        double r3146876 = r3146851 / r3146862;
        double r3146877 = -r3146876;
        double r3146878 = r3146858 ? r3146875 : r3146877;
        double r3146879 = r3146853 ? r3146856 : r3146878;
        return r3146879;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.0
Target20.1
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

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. Simplified52.0

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

    3. Taylor expanded around -inf 10.0

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

    4. 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. Simplified12.8

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

    4. Applied *-un-lft-identity12.8

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

    5. Applied associate-/l*12.9

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

    7. Applied add-sqr-sqrt13.6

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

    8. Applied *-un-lft-identity13.6

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

    9. Applied *-un-lft-identity13.6

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

    10. Applied distribute-lft-out--13.6

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

    11. Applied times-frac13.4

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

    12. Applied *-un-lft-identity13.4

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

    13. Applied times-frac13.3

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

    14. Simplified13.3

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

    16. Applied add-sqr-sqrt13.3

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

    17. Applied sqrt-prod13.0

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

    18. Applied associate-/r*13.0

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

    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. Simplified44.4

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

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

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

    5. Applied associate-/l*44.5

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

    6. Taylor expanded around 0 3.9

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

    7. Simplified3.9

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.615151909502748 \cdot 10^{-87}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.5387363548079373 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\sqrt{2} \cdot \frac{a}{\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(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)))