quad2m (problem 3.2.1, negative)

Average Error: 34.3 → 9.8

Time: 10.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.271875817960049862049445045905922921165 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.583649041028097672662453906912522703022 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r16627 = b_2;
        double r16628 = -r16627;
        double r16629 = r16627 * r16627;
        double r16630 = a;
        double r16631 = c;
        double r16632 = r16630 * r16631;
        double r16633 = r16629 - r16632;
        double r16634 = sqrt(r16633);
        double r16635 = r16628 - r16634;
        double r16636 = r16635 / r16630;
        return r16636;
}

↓

double f(double a, double b_2, double c) {
        double r16637 = b_2;
        double r16638 = -2.27187581796005e-81;
        bool r16639 = r16637 <= r16638;
        double r16640 = -0.5;
        double r16641 = c;
        double r16642 = r16641 / r16637;
        double r16643 = r16640 * r16642;
        double r16644 = 3.5836490410280977e+84;
        bool r16645 = r16637 <= r16644;
        double r16646 = -r16637;
        double r16647 = -r16641;
        double r16648 = a;
        double r16649 = r16637 * r16637;
        double r16650 = fma(r16647, r16648, r16649);
        double r16651 = sqrt(r16650);
        double r16652 = r16646 - r16651;
        double r16653 = r16652 / r16648;
        double r16654 = r16646 - r16637;
        double r16655 = r16654 / r16648;
        double r16656 = r16645 ? r16653 : r16655;
        double r16657 = r16639 ? r16643 : r16656;
        return r16657;
}

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 < -2.27187581796005e-81

   1. Initial program 53.4

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

   2. Simplified 53.4

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

   3. Taylor expanded around -inf 8.8

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

   if -2.27187581796005e-81 < b_2 < 3.5836490410280977e+84

   1. Initial program 13.1

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

   2. Simplified 13.1

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

   3. Taylor expanded around 0 13.1

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

   4. Simplified 13.1

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

   if 3.5836490410280977e+84 < b_2

   1. Initial program 43.9

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

   2. Simplified 43.9

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

   3. Taylor expanded around 0 3.7

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{b_2}}{a}\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.271875817960049862049445045905922921165 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.583649041028097672662453906912522703022 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))