quad2m (problem 3.2.1, negative)

Average Error: 34.5 → 6.9

Time: 9.8s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.52829345465496796 \cdot 10^{148}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -7.5932715112131794 \cdot 10^{-252}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{c}}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 9.9656763960867421 \cdot 10^{45}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r17105 = b_2;
        double r17106 = -r17105;
        double r17107 = r17105 * r17105;
        double r17108 = a;
        double r17109 = c;
        double r17110 = r17108 * r17109;
        double r17111 = r17107 - r17110;
        double r17112 = sqrt(r17111);
        double r17113 = r17106 - r17112;
        double r17114 = r17113 / r17108;
        return r17114;
}

↓

double f(double a, double b_2, double c) {
        double r17115 = b_2;
        double r17116 = -1.528293454654968e+148;
        bool r17117 = r17115 <= r17116;
        double r17118 = -0.5;
        double r17119 = c;
        double r17120 = r17119 / r17115;
        double r17121 = r17118 * r17120;
        double r17122 = -7.593271511213179e-252;
        bool r17123 = r17115 <= r17122;
        double r17124 = 1.0;
        double r17125 = r17124 / r17119;
        double r17126 = r17124 / r17125;
        double r17127 = r17115 * r17115;
        double r17128 = a;
        double r17129 = r17128 * r17119;
        double r17130 = r17127 - r17129;
        double r17131 = sqrt(r17130);
        double r17132 = r17131 - r17115;
        double r17133 = r17126 / r17132;
        double r17134 = 9.965676396086742e+45;
        bool r17135 = r17115 <= r17134;
        double r17136 = -r17115;
        double r17137 = r17136 / r17128;
        double r17138 = r17131 / r17128;
        double r17139 = r17137 - r17138;
        double r17140 = 0.5;
        double r17141 = r17140 * r17120;
        double r17142 = 2.0;
        double r17143 = r17115 / r17128;
        double r17144 = r17142 * r17143;
        double r17145 = r17141 - r17144;
        double r17146 = r17135 ? r17139 : r17145;
        double r17147 = r17123 ? r17133 : r17146;
        double r17148 = r17117 ? r17121 : r17147;
        return r17148;
}

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 < -1.528293454654968e+148

   1. Initial program 63.4

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

   2. Taylor expanded around -inf 1.6

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

   if -1.528293454654968e+148 < b_2 < -7.593271511213179e-252

   1. Initial program 36.5

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

   3. Applied flip-- 36.6

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

   4. Simplified 16.1

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

   5. Simplified 16.1

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

   7. Applied div-inv 16.1

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

   9. Applied associate-*l/ 14.3

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

   10. Simplified 14.2

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

   12. Applied clear-num 14.3

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

   13. Simplified 7.8

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

   if -7.593271511213179e-252 < b_2 < 9.965676396086742e+45

   1. Initial program 10.6

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

   3. Applied div-sub 10.6

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

   if 9.965676396086742e+45 < b_2

   1. Initial program 36.8

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

   2. Taylor expanded around inf 5.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.52829345465496796 \cdot 10^{148}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -7.5932715112131794 \cdot 10^{-252}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{c}}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 9.9656763960867421 \cdot 10^{45}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))