Average Error: 33.3 → 6.4
Time: 2.5m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.263941314600607 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -4.687918346756617 \cdot 10^{-254}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}{2}}{a}\\ \mathbf{elif}\;b \le 3.463606471108268 \cdot 10^{+121}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -3.263941314600607 \cdot 10^{+152}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le -4.687918346756617 \cdot 10^{-254}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}{2}}{a}\\

\mathbf{elif}\;b \le 3.463606471108268 \cdot 10^{+121}:\\
\;\;\;\;\frac{c \cdot -2}{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)} + b}\\

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

\end{array}
double f(double a, double b, double c) {
        double r29381556 = b;
        double r29381557 = -r29381556;
        double r29381558 = r29381556 * r29381556;
        double r29381559 = 4.0;
        double r29381560 = a;
        double r29381561 = r29381559 * r29381560;
        double r29381562 = c;
        double r29381563 = r29381561 * r29381562;
        double r29381564 = r29381558 - r29381563;
        double r29381565 = sqrt(r29381564);
        double r29381566 = r29381557 + r29381565;
        double r29381567 = 2.0;
        double r29381568 = r29381567 * r29381560;
        double r29381569 = r29381566 / r29381568;
        return r29381569;
}
↓
double f(double a, double b, double c) {
        double r29381570 = b;
        double r29381571 = -3.263941314600607e+152;
        bool r29381572 = r29381570 <= r29381571;
        double r29381573 = c;
        double r29381574 = r29381573 / r29381570;
        double r29381575 = a;
        double r29381576 = r29381570 / r29381575;
        double r29381577 = r29381574 - r29381576;
        double r29381578 = -4.687918346756617e-254;
        bool r29381579 = r29381570 <= r29381578;
        double r29381580 = r29381573 * r29381575;
        double r29381581 = -4.0;
        double r29381582 = r29381570 * r29381570;
        double r29381583 = fma(r29381580, r29381581, r29381582);
        double r29381584 = sqrt(r29381583);
        double r29381585 = r29381584 - r29381570;
        double r29381586 = 2.0;
        double r29381587 = r29381585 / r29381586;
        double r29381588 = r29381587 / r29381575;
        double r29381589 = 3.463606471108268e+121;
        bool r29381590 = r29381570 <= r29381589;
        double r29381591 = -2.0;
        double r29381592 = r29381573 * r29381591;
        double r29381593 = r29381575 * r29381581;
        double r29381594 = r29381593 * r29381573;
        double r29381595 = fma(r29381570, r29381570, r29381594);
        double r29381596 = sqrt(r29381595);
        double r29381597 = r29381596 + r29381570;
        double r29381598 = r29381592 / r29381597;
        double r29381599 = -r29381574;
        double r29381600 = r29381590 ? r29381598 : r29381599;
        double r29381601 = r29381579 ? r29381588 : r29381600;
        double r29381602 = r29381572 ? r29381577 : r29381601;
        return r29381602;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target20.3
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.263941314600607e+152

    1. Initial program 60.1

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

    2. Simplified60.1

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

    3. Taylor expanded around -inf 2.3

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

    if -3.263941314600607e+152 < b < -4.687918346756617e-254

    1. Initial program 7.8

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

    2. Simplified7.8

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

    3. Taylor expanded around -inf 7.8

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

    4. Simplified7.8

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

    if -4.687918346756617e-254 < b < 3.463606471108268e+121

    1. Initial program 31.6

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

    2. Simplified31.6

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

    4. Applied flip--31.8

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

    5. Simplified15.5

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

    7. Applied *-un-lft-identity15.5

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

    8. Applied *-un-lft-identity15.5

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

    9. Applied times-frac15.5

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

    10. Simplified15.5

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

    11. Simplified8.7

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

    if 3.463606471108268e+121 < b

    1. Initial program 59.8

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

    2. Simplified59.8

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

    3. Taylor expanded around inf 2.3

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

    4. Simplified2.3

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.263941314600607 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -4.687918346756617 \cdot 10^{-254}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}{2}}{a}\\ \mathbf{elif}\;b \le 3.463606471108268 \cdot 10^{+121}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +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)))