Quadratic roots, narrow range

Average Error: 28.4 → 0.4

Time: 19.3s

Precision: 64

\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r37630 = b;
        double r37631 = -r37630;
        double r37632 = r37630 * r37630;
        double r37633 = 4.0;
        double r37634 = a;
        double r37635 = r37633 * r37634;
        double r37636 = c;
        double r37637 = r37635 * r37636;
        double r37638 = r37632 - r37637;
        double r37639 = sqrt(r37638);
        double r37640 = r37631 + r37639;
        double r37641 = 2.0;
        double r37642 = r37641 * r37634;
        double r37643 = r37640 / r37642;
        return r37643;
}

↓

double f(double a, double b, double c) {
        double r37644 = a;
        double r37645 = c;
        double r37646 = r37644 * r37645;
        double r37647 = 4.0;
        double r37648 = r37646 * r37647;
        double r37649 = b;
        double r37650 = r37649 * r37649;
        double r37651 = r37647 * r37644;
        double r37652 = r37651 * r37645;
        double r37653 = r37650 - r37652;
        double r37654 = sqrt(r37653);
        double r37655 = r37654 * r37644;
        double r37656 = r37649 * r37644;
        double r37657 = r37655 + r37656;
        double r37658 = -r37657;
        double r37659 = 2.0;
        double r37660 = r37658 * r37659;
        double r37661 = r37648 / r37660;
        return r37661;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.4

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

3. Applied flip-+ 28.4

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

4. Simplified 0.5

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

6. Applied div-inv 0.5

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

7. Applied associate-/l* 0.5

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

8. Simplified 0.5

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

10. Applied sub-neg 0.5

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

11. Applied distribute-lft-in 0.4

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

12. Simplified 0.4

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

13. Simplified 0.4

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

14. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))