Average Error: 33.8 → 9.7
Time: 15.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6257289292067596 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.739098950628615 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}}}{2}\\ \mathbf{elif}\;b \le 1.8656332031849816 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \mathbf{elif}\;b \le 5.297236684235463 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b \cdot b}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.6257289292067596 \cdot 10^{+144}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.739098950628615 \cdot 10^{-79}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}}}{2}\\

\mathbf{elif}\;b \le 1.8656332031849816 \cdot 10^{-25}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\mathbf{elif}\;b \le 5.297236684235463 \cdot 10^{-16}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b \cdot b}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + b}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1136644 = b;
        double r1136645 = -r1136644;
        double r1136646 = r1136644 * r1136644;
        double r1136647 = 4.0;
        double r1136648 = a;
        double r1136649 = r1136647 * r1136648;
        double r1136650 = c;
        double r1136651 = r1136649 * r1136650;
        double r1136652 = r1136646 - r1136651;
        double r1136653 = sqrt(r1136652);
        double r1136654 = r1136645 + r1136653;
        double r1136655 = 2.0;
        double r1136656 = r1136655 * r1136648;
        double r1136657 = r1136654 / r1136656;
        return r1136657;
}
↓
double f(double a, double b, double c) {
        double r1136658 = b;
        double r1136659 = -1.6257289292067596e+144;
        bool r1136660 = r1136658 <= r1136659;
        double r1136661 = c;
        double r1136662 = r1136661 / r1136658;
        double r1136663 = a;
        double r1136664 = r1136658 / r1136663;
        double r1136665 = r1136662 - r1136664;
        double r1136666 = 2.0;
        double r1136667 = r1136665 * r1136666;
        double r1136668 = r1136667 / r1136666;
        double r1136669 = 1.739098950628615e-79;
        bool r1136670 = r1136658 <= r1136669;
        double r1136671 = 1.0;
        double r1136672 = -4.0;
        double r1136673 = r1136661 * r1136663;
        double r1136674 = r1136672 * r1136673;
        double r1136675 = fma(r1136658, r1136658, r1136674);
        double r1136676 = sqrt(r1136675);
        double r1136677 = r1136676 - r1136658;
        double r1136678 = r1136663 / r1136677;
        double r1136679 = r1136671 / r1136678;
        double r1136680 = r1136679 / r1136666;
        double r1136681 = 1.8656332031849816e-25;
        bool r1136682 = r1136658 <= r1136681;
        double r1136683 = -2.0;
        double r1136684 = r1136662 * r1136683;
        double r1136685 = r1136684 / r1136666;
        double r1136686 = 5.297236684235463e-16;
        bool r1136687 = r1136658 <= r1136686;
        double r1136688 = r1136676 * r1136676;
        double r1136689 = r1136658 * r1136658;
        double r1136690 = r1136688 - r1136689;
        double r1136691 = r1136663 / r1136690;
        double r1136692 = r1136671 / r1136691;
        double r1136693 = r1136676 + r1136658;
        double r1136694 = r1136671 / r1136693;
        double r1136695 = r1136692 * r1136694;
        double r1136696 = r1136695 / r1136666;
        double r1136697 = r1136687 ? r1136696 : r1136685;
        double r1136698 = r1136682 ? r1136685 : r1136697;
        double r1136699 = r1136670 ? r1136680 : r1136698;
        double r1136700 = r1136660 ? r1136668 : r1136699;
        return r1136700;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Target20.3
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6257289292067596e+144

    1. Initial program 58.0

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

    2. Simplified58.0

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

    3. Taylor expanded around -inf 2.5

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

    4. Simplified2.5

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

    if -1.6257289292067596e+144 < b < 1.739098950628615e-79

    1. Initial program 11.7

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

    2. Simplified11.8

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

    4. Applied clear-num11.9

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

    5. Taylor expanded around 0 11.9

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

    6. Simplified11.9

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

    if 1.739098950628615e-79 < b < 1.8656332031849816e-25 or 5.297236684235463e-16 < b

    1. Initial program 53.2

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

    2. Simplified53.2

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

    3. Taylor expanded around inf 8.2

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

    if 1.8656332031849816e-25 < b < 5.297236684235463e-16

    1. Initial program 42.7

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

    2. Simplified42.7

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

    4. Applied clear-num42.8

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

    5. Taylor expanded around 0 42.8

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

    6. Simplified42.8

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

    8. Applied flip--42.9

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

    9. Applied associate-/r/43.0

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

    10. Applied *-un-lft-identity43.0

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

    11. Applied times-frac43.0

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b \cdot b}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} + b}}}{2}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6257289292067596 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.739098950628615 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}}}{2}\\ \mathbf{elif}\;b \le 1.8656332031849816 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \mathbf{elif}\;b \le 5.297236684235463 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b \cdot b}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))