Cubic critical, narrow range

Average Error: 28.7 → 14.9

Time: 14.6s

Precision: 64

\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}{3 \cdot a} \le -1.5091074929644428 \cdot 10^{-08}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} \cdot \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right) - b \cdot \left(b \cdot b\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right) + b \cdot \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r3907587 = b;
        double r3907588 = -r3907587;
        double r3907589 = r3907587 * r3907587;
        double r3907590 = 3.0;
        double r3907591 = a;
        double r3907592 = r3907590 * r3907591;
        double r3907593 = c;
        double r3907594 = r3907592 * r3907593;
        double r3907595 = r3907589 - r3907594;
        double r3907596 = sqrt(r3907595);
        double r3907597 = r3907588 + r3907596;
        double r3907598 = r3907597 / r3907592;
        return r3907598;
}

↓

double f(double a, double b, double c) {
        double r3907599 = b;
        double r3907600 = r3907599 * r3907599;
        double r3907601 = 3.0;
        double r3907602 = a;
        double r3907603 = r3907601 * r3907602;
        double r3907604 = c;
        double r3907605 = r3907603 * r3907604;
        double r3907606 = r3907600 - r3907605;
        double r3907607 = sqrt(r3907606);
        double r3907608 = -r3907599;
        double r3907609 = r3907607 + r3907608;
        double r3907610 = r3907609 / r3907603;
        double r3907611 = -1.5091074929644428e-08;
        bool r3907612 = r3907610 <= r3907611;
        double r3907613 = r3907604 * r3907602;
        double r3907614 = -3.0;
        double r3907615 = r3907613 * r3907614;
        double r3907616 = fma(r3907599, r3907599, r3907615);
        double r3907617 = sqrt(r3907616);
        double r3907618 = r3907617 * r3907616;
        double r3907619 = r3907599 * r3907600;
        double r3907620 = r3907618 - r3907619;
        double r3907621 = r3907599 * r3907617;
        double r3907622 = r3907616 + r3907621;
        double r3907623 = fma(r3907599, r3907599, r3907622);
        double r3907624 = r3907620 / r3907623;
        double r3907625 = r3907624 / r3907603;
        double r3907626 = -0.5;
        double r3907627 = r3907604 / r3907599;
        double r3907628 = r3907626 * r3907627;
        double r3907629 = r3907612 ? r3907625 : r3907628;
        return r3907629;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)) < -1.5091074929644428e-08

   1. Initial program 19.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
   2. Using strategy rm

   3. Applied flip3-+ 19.2

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}^{3}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a}\]

   4. Simplified 18.5

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} \cdot \mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right) - \left(b \cdot b\right) \cdot b}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\]

   5. Simplified 18.5

      \[\leadsto \frac{\frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} \cdot \mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right) - \left(b \cdot b\right) \cdot b}{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} \cdot b\right)}}}{3 \cdot a}\]

   if -1.5091074929644428e-08 < (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a))

   1. Initial program 48.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Taylor expanded around inf 7.2

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 14.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}{3 \cdot a} \le -1.5091074929644428 \cdot 10^{-08}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} \cdot \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right) - b \cdot \left(b \cdot b\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right) + b \cdot \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))