jeff quadratic root 2

Average Error: 18.9 → 13.1

Time: 21.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 4.1813913858897907 \cdot 10^{+68}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r488439 = b;
        double r488440 = 0.0;
        bool r488441 = r488439 >= r488440;
        double r488442 = 2.0;
        double r488443 = c;
        double r488444 = r488442 * r488443;
        double r488445 = -r488439;
        double r488446 = r488439 * r488439;
        double r488447 = 4.0;
        double r488448 = a;
        double r488449 = r488447 * r488448;
        double r488450 = r488449 * r488443;
        double r488451 = r488446 - r488450;
        double r488452 = sqrt(r488451);
        double r488453 = r488445 - r488452;
        double r488454 = r488444 / r488453;
        double r488455 = r488445 + r488452;
        double r488456 = r488442 * r488448;
        double r488457 = r488455 / r488456;
        double r488458 = r488441 ? r488454 : r488457;
        return r488458;
}

↓

double f(double a, double b, double c) {
        double r488459 = b;
        double r488460 = 4.1813913858897907e+68;
        bool r488461 = r488459 <= r488460;
        double r488462 = 0.0;
        bool r488463 = r488459 >= r488462;
        double r488464 = 2.0;
        double r488465 = c;
        double r488466 = r488464 * r488465;
        double r488467 = -r488459;
        double r488468 = -4.0;
        double r488469 = a;
        double r488470 = r488469 * r488465;
        double r488471 = r488459 * r488459;
        double r488472 = fma(r488468, r488470, r488471);
        double r488473 = sqrt(r488472);
        double r488474 = sqrt(r488473);
        double r488475 = r488474 * r488474;
        double r488476 = r488467 - r488475;
        double r488477 = r488466 / r488476;
        double r488478 = r488473 - r488459;
        double r488479 = r488478 / r488464;
        double r488480 = r488479 / r488469;
        double r488481 = r488463 ? r488477 : r488480;
        double r488482 = r488467 - r488459;
        double r488483 = r488466 / r488482;
        double r488484 = r488463 ? r488483 : r488480;
        double r488485 = r488461 ? r488481 : r488484;
        return r488485;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 4.1813913858897907e+68

   1. Initial program 16.3

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

   2. Simplified 16.3

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 16.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

   if 4.1813913858897907e+68 < b

   1. Initial program 26.4

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

   2. Simplified 26.4

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}}\]

   3. Taylor expanded around 0 3.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]
3. Recombined 2 regimes into one program.

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 4.1813913858897907 \cdot 10^{+68}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))