Quadratic roots, wide range

Average Error: 52.5 → 0.4

Time: 6.2s

Precision: 64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}\right)}\]

C

double f(double a, double b, double c) {
        double r31541 = b;
        double r31542 = -r31541;
        double r31543 = r31541 * r31541;
        double r31544 = 4.0;
        double r31545 = a;
        double r31546 = r31544 * r31545;
        double r31547 = c;
        double r31548 = r31546 * r31547;
        double r31549 = r31543 - r31548;
        double r31550 = sqrt(r31549);
        double r31551 = r31542 + r31550;
        double r31552 = 2.0;
        double r31553 = r31552 * r31545;
        double r31554 = r31551 / r31553;
        return r31554;
}

↓

double f(double a, double b, double c) {
        double r31555 = 0.0;
        double r31556 = 4.0;
        double r31557 = a;
        double r31558 = c;
        double r31559 = r31557 * r31558;
        double r31560 = r31556 * r31559;
        double r31561 = r31555 + r31560;
        double r31562 = 2.0;
        double r31563 = r31562 * r31557;
        double r31564 = b;
        double r31565 = -r31564;
        double r31566 = 6.0;
        double r31567 = pow(r31564, r31566);
        double r31568 = r31556 * r31557;
        double r31569 = r31568 * r31558;
        double r31570 = 3.0;
        double r31571 = pow(r31569, r31570);
        double r31572 = r31567 - r31571;
        double r31573 = 2.0;
        double r31574 = pow(r31564, r31573);
        double r31575 = r31569 + r31574;
        double r31576 = r31569 * r31575;
        double r31577 = r31564 * r31564;
        double r31578 = r31577 * r31577;
        double r31579 = r31576 + r31578;
        double r31580 = r31572 / r31579;
        double r31581 = sqrt(r31580);
        double r31582 = r31565 - r31581;
        double r31583 = r31563 * r31582;
        double r31584 = r31561 / r31583;
        return r31584;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.5

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Using strategy rm

3. Applied flip-+ 52.5

   \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

4. Simplified 0.4

   \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
5. Using strategy rm

6. Applied div-inv 0.5

   \[\leadsto \frac{\color{blue}{\left(0 + 4 \cdot \left(a \cdot c\right)\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

7. Applied associate-/l* 0.5

   \[\leadsto \color{blue}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\frac{2 \cdot a}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

8. Simplified 0.4

   \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]
9. Using strategy rm

10. Applied flip3-- 0.4

    \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}\right)}\]

11. Simplified 0.4

    \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{\color{blue}{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}\right)}\]

12. Simplified 0.4

    \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\color{blue}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}}\right)}\]

13. Final simplification 0.4

    \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}\right)}\]

Reproduce

herbie shell --seed 2020003 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))