Quadratic roots, narrow range

Average Error: 28.7 → 27.5

Time: 52.6s

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

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

↓

\[\frac{\frac{\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right), \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}\right), \left(\left(b \cdot b\right) \cdot \left(-b\right)\right)\right)}{\sqrt{\mathsf{fma}\left(b, \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} + b\right), \left(\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right)\right)\right)\right)\right)\right)}}}{\sqrt{\mathsf{fma}\left(b, \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} + b\right), \left(\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right)\right)\right)\right)\right)\right)}}}{2}}{a}\]

C

double f(double a, double b, double c) {
        double r1578187 = b;
        double r1578188 = -r1578187;
        double r1578189 = r1578187 * r1578187;
        double r1578190 = 4.0;
        double r1578191 = a;
        double r1578192 = r1578190 * r1578191;
        double r1578193 = c;
        double r1578194 = r1578192 * r1578193;
        double r1578195 = r1578189 - r1578194;
        double r1578196 = sqrt(r1578195);
        double r1578197 = r1578188 + r1578196;
        double r1578198 = 2.0;
        double r1578199 = r1578198 * r1578191;
        double r1578200 = r1578197 / r1578199;
        return r1578200;
}

↓

double f(double a, double b, double c) {
        double r1578201 = a;
        double r1578202 = -4.0;
        double r1578203 = r1578201 * r1578202;
        double r1578204 = c;
        double r1578205 = b;
        double r1578206 = r1578205 * r1578205;
        double r1578207 = fma(r1578203, r1578204, r1578206);
        double r1578208 = sqrt(r1578207);
        double r1578209 = -r1578205;
        double r1578210 = r1578206 * r1578209;
        double r1578211 = fma(r1578207, r1578208, r1578210);
        double r1578212 = r1578208 + r1578205;
        double r1578213 = log1p(r1578207);
        double r1578214 = expm1(r1578213);
        double r1578215 = fma(r1578205, r1578212, r1578214);
        double r1578216 = sqrt(r1578215);
        double r1578217 = r1578211 / r1578216;
        double r1578218 = r1578217 / r1578216;
        double r1578219 = 2.0;
        double r1578220 = r1578218 / r1578219;
        double r1578221 = r1578220 / r1578201;
        return r1578221;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.7

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

2. Simplified 28.7

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

4. Applied flip3-- 28.8

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

5. Simplified 28.1

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

6. Simplified 28.1

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

8. Applied fma-neg 27.5

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

10. Applied expm1-log1p-u 27.5

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right), \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}\right), \left(-b \cdot \left(b \cdot b\right)\right)\right)}{\mathsf{fma}\left(b, \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} + b\right), \color{blue}{\left(\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right)\right)\right)\right)\right)}\right)}}{2}}{a}\]
11. Using strategy rm

12. Applied add-sqr-sqrt 27.5

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right), \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}\right), \left(-b \cdot \left(b \cdot b\right)\right)\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(b, \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} + b\right), \left(\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right)\right)\right)\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(b, \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} + b\right), \left(\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right)\right)\right)\right)\right)\right)}}}}{2}}{a}\]

13. Applied associate-/r* 27.5

    \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right), \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}\right), \left(-b \cdot \left(b \cdot b\right)\right)\right)}{\sqrt{\mathsf{fma}\left(b, \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} + b\right), \left(\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right)\right)\right)\right)\right)\right)}}}{\sqrt{\mathsf{fma}\left(b, \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} + b\right), \left(\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right)\right)\right)\right)\right)\right)}}}}{2}}{a}\]

14. Final simplification 27.5

    \[\leadsto \frac{\frac{\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right), \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}\right), \left(\left(b \cdot b\right) \cdot \left(-b\right)\right)\right)}{\sqrt{\mathsf{fma}\left(b, \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} + b\right), \left(\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right)\right)\right)\right)\right)\right)}}}{\sqrt{\mathsf{fma}\left(b, \left(\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} + b\right), \left(\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)\right)\right)\right)\right)\right)\right)}}}{2}}{a}\]

Reproduce

herbie shell --seed 2019132 +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)))