quad2m (problem 3.2.1, negative)

Average Error: 34.5 → 6.8

Time: 11.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.02458994604675516 \cdot 10^{154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.9151734797750612 \cdot 10^{-228}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 9.3479963141541371 \cdot 10^{42}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{{b_2}^{2} - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r17172 = b_2;
        double r17173 = -r17172;
        double r17174 = r17172 * r17172;
        double r17175 = a;
        double r17176 = c;
        double r17177 = r17175 * r17176;
        double r17178 = r17174 - r17177;
        double r17179 = sqrt(r17178);
        double r17180 = r17173 - r17179;
        double r17181 = r17180 / r17175;
        return r17181;
}

↓

double f(double a, double b_2, double c) {
        double r17182 = b_2;
        double r17183 = -1.0245899460467552e+154;
        bool r17184 = r17182 <= r17183;
        double r17185 = -0.5;
        double r17186 = c;
        double r17187 = r17186 / r17182;
        double r17188 = r17185 * r17187;
        double r17189 = -1.9151734797750612e-228;
        bool r17190 = r17182 <= r17189;
        double r17191 = r17182 * r17182;
        double r17192 = a;
        double r17193 = r17192 * r17186;
        double r17194 = r17191 - r17193;
        double r17195 = sqrt(r17194);
        double r17196 = r17195 - r17182;
        double r17197 = r17186 / r17196;
        double r17198 = 9.347996314154137e+42;
        bool r17199 = r17182 <= r17198;
        double r17200 = -r17182;
        double r17201 = 2.0;
        double r17202 = pow(r17182, r17201);
        double r17203 = r17202 - r17193;
        double r17204 = sqrt(r17203);
        double r17205 = r17200 - r17204;
        double r17206 = r17205 / r17192;
        double r17207 = -2.0;
        double r17208 = r17182 / r17192;
        double r17209 = r17207 * r17208;
        double r17210 = r17199 ? r17206 : r17209;
        double r17211 = r17190 ? r17197 : r17210;
        double r17212 = r17184 ? r17188 : r17211;
        return r17212;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.0245899460467552e+154

   1. Initial program 64.0

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

   2. Taylor expanded around -inf 1.3

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

   if -1.0245899460467552e+154 < b_2 < -1.9151734797750612e-228

   1. Initial program 37.3

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

   3. Applied flip-- 37.3

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

   4. Simplified 15.7

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Simplified 15.7

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
   6. Using strategy rm

   7. Applied div-inv 15.7

      \[\leadsto \color{blue}{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 15.7

      \[\leadsto \color{blue}{\left(1 \cdot \frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\right)} \cdot \frac{1}{a}\]

   10. Applied associate-*l* 15.7

       \[\leadsto \color{blue}{1 \cdot \left(\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}\right)}\]

   11. Simplified 13.6

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

   12. Taylor expanded around 0 6.9

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

   if -1.9151734797750612e-228 < b_2 < 9.347996314154137e+42

   1. Initial program 10.7

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

   2. Taylor expanded around 0 10.7

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

   if 9.347996314154137e+42 < b_2

   1. Initial program 37.1

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

   3. Applied flip-- 61.4

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

   4. Simplified 60.8

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Simplified 60.8

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

   6. Taylor expanded around 0 5.9

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.02458994604675516 \cdot 10^{154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.9151734797750612 \cdot 10^{-228}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 9.3479963141541371 \cdot 10^{42}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{{b_2}^{2} - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))