Quadratic roots, narrow range

Average Error: 28.9 → 0.3

Time: 2.0m

Precision: 64

\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r9477111 = b;
        double r9477112 = -r9477111;
        double r9477113 = r9477111 * r9477111;
        double r9477114 = 4.0;
        double r9477115 = a;
        double r9477116 = r9477114 * r9477115;
        double r9477117 = c;
        double r9477118 = r9477116 * r9477117;
        double r9477119 = r9477113 - r9477118;
        double r9477120 = sqrt(r9477119);
        double r9477121 = r9477112 + r9477120;
        double r9477122 = 2.0;
        double r9477123 = r9477122 * r9477115;
        double r9477124 = r9477121 / r9477123;
        return r9477124;
}

↓

double f(double a, double b, double c) {
        double r9477125 = -2.0;
        double r9477126 = c;
        double r9477127 = b;
        double r9477128 = sqrt(r9477127);
        double r9477129 = -4.0;
        double r9477130 = a;
        double r9477131 = r9477126 * r9477130;
        double r9477132 = r9477129 * r9477131;
        double r9477133 = fma(r9477127, r9477127, r9477132);
        double r9477134 = sqrt(r9477133);
        double r9477135 = fma(r9477128, r9477128, r9477134);
        double r9477136 = r9477126 / r9477135;
        double r9477137 = r9477125 * r9477136;
        return r9477137;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.9

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

2. Simplified 28.9

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

4. Applied flip-- 29.0

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

5. Taylor expanded around 0 0.4

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

7. Applied *-un-lft-identity 0.4

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

8. Applied *-un-lft-identity 0.4

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

9. Applied *-un-lft-identity 0.4

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

10. Applied times-frac 0.4

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

11. Applied times-frac 0.4

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

12. Simplified 0.4

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

13. Simplified 0.3

    \[\leadsto 1 \cdot \color{blue}{\left(-2 \cdot \frac{c}{b + \sqrt{(b \cdot b + \left(\left(c \cdot a\right) \cdot -4\right))_*}}\right)}\]
14. Using strategy rm

15. Applied add-sqr-sqrt 0.4

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

16. Applied fma-def 0.3

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

17. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :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)))