Quadratic roots, full range

Average Error: 33.0 → 10.6

Time: 23.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.794505329565205 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.6194276288860963:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(a \cdot -4\right)\right)\right)} - b}{\frac{a}{\frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1208522 = b;
        double r1208523 = -r1208522;
        double r1208524 = r1208522 * r1208522;
        double r1208525 = 4.0;
        double r1208526 = a;
        double r1208527 = r1208525 * r1208526;
        double r1208528 = c;
        double r1208529 = r1208527 * r1208528;
        double r1208530 = r1208524 - r1208529;
        double r1208531 = sqrt(r1208530);
        double r1208532 = r1208523 + r1208531;
        double r1208533 = 2.0;
        double r1208534 = r1208533 * r1208526;
        double r1208535 = r1208532 / r1208534;
        return r1208535;
}

↓

double f(double a, double b, double c) {
        double r1208536 = b;
        double r1208537 = -3.794505329565205e+146;
        bool r1208538 = r1208536 <= r1208537;
        double r1208539 = c;
        double r1208540 = r1208539 / r1208536;
        double r1208541 = a;
        double r1208542 = r1208536 / r1208541;
        double r1208543 = r1208540 - r1208542;
        double r1208544 = 1.6194276288860963;
        bool r1208545 = r1208536 <= r1208544;
        double r1208546 = -4.0;
        double r1208547 = r1208541 * r1208546;
        double r1208548 = r1208539 * r1208547;
        double r1208549 = fma(r1208536, r1208536, r1208548);
        double r1208550 = sqrt(r1208549);
        double r1208551 = r1208550 - r1208536;
        double r1208552 = 0.5;
        double r1208553 = r1208541 / r1208552;
        double r1208554 = r1208551 / r1208553;
        double r1208555 = -r1208540;
        double r1208556 = r1208545 ? r1208554 : r1208555;
        double r1208557 = r1208538 ? r1208543 : r1208556;
        return r1208557;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -3.794505329565205e+146

   1. Initial program 58.0

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

   2. Simplified 58.0

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

   4. Applied div-inv 58.0

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

   5. Applied associate-/l* 58.0

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

   6. Simplified 58.0

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

   7. Taylor expanded around -inf 3.1

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

   if -3.794505329565205e+146 < b < 1.6194276288860963

   1. Initial program 15.0

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

   2. Simplified 15.0

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

   4. Applied div-inv 15.0

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

   5. Applied associate-/l* 15.0

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

   6. Simplified 15.0

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

   if 1.6194276288860963 < b

   1. Initial program 54.4

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

   2. Simplified 54.4

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

   4. Applied div-inv 54.4

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

   5. Applied associate-/l* 54.4

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

   6. Simplified 54.4

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

   7. Taylor expanded around inf 5.9

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

   8. Simplified 5.9

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

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.794505329565205 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.6194276288860963:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(a \cdot -4\right)\right)\right)} - b}{\frac{a}{\frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))