quad2m (problem 3.2.1, negative)

Average Error: 34.2 → 10.5

Time: 3.7s

Precision: 64

Math

\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.763265898390888870746272508253147533908 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r8672 = b_2;
        double r8673 = -r8672;
        double r8674 = r8672 * r8672;
        double r8675 = a;
        double r8676 = c;
        double r8677 = r8675 * r8676;
        double r8678 = r8674 - r8677;
        double r8679 = sqrt(r8678);
        double r8680 = r8673 - r8679;
        double r8681 = r8680 / r8675;
        return r8681;
}

↓

double f(double a, double b_2, double c) {
        double r8682 = b_2;
        double r8683 = -2.763265898390889e-72;
        bool r8684 = r8682 <= r8683;
        double r8685 = -0.5;
        double r8686 = c;
        double r8687 = r8686 / r8682;
        double r8688 = r8685 * r8687;
        double r8689 = 6.568668442325333e+48;
        bool r8690 = r8682 <= r8689;
        double r8691 = -r8682;
        double r8692 = r8682 * r8682;
        double r8693 = a;
        double r8694 = r8693 * r8686;
        double r8695 = r8692 - r8694;
        double r8696 = sqrt(r8695);
        double r8697 = r8691 - r8696;
        double r8698 = 1.0;
        double r8699 = r8698 / r8693;
        double r8700 = r8697 * r8699;
        double r8701 = 0.5;
        double r8702 = r8701 * r8687;
        double r8703 = 2.0;
        double r8704 = r8682 / r8693;
        double r8705 = r8703 * r8704;
        double r8706 = r8702 - r8705;
        double r8707 = r8690 ? r8700 : r8706;
        double r8708 = r8684 ? r8688 : r8707;
        return r8708;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.763265898390889e-72

   1. Initial program 52.9

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around -inf 9.2

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

   if -2.763265898390889e-72 < b_2 < 6.568668442325333e+48

   1. Initial program 14.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied div-inv 14.1

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]

   if 6.568668442325333e+48 < b_2

   1. Initial program 39.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around inf 5.3

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.763265898390888870746272508253147533908 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))