Average Error: 28.2 → 0.4
Time: 17.9s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{\left(a \cdot 4\right) \cdot c}{a}}{-\mathsf{fma}\left(\sqrt{b}, \sqrt{b}, \sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)}\right)} \cdot \frac{1}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{\left(a \cdot 4\right) \cdot c}{a}}{-\mathsf{fma}\left(\sqrt{b}, \sqrt{b}, \sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)}\right)} \cdot \frac{1}{2}
double f(double a, double b, double c) {
        double r41146 = b;
        double r41147 = -r41146;
        double r41148 = r41146 * r41146;
        double r41149 = 4.0;
        double r41150 = a;
        double r41151 = r41149 * r41150;
        double r41152 = c;
        double r41153 = r41151 * r41152;
        double r41154 = r41148 - r41153;
        double r41155 = sqrt(r41154);
        double r41156 = r41147 + r41155;
        double r41157 = 2.0;
        double r41158 = r41157 * r41150;
        double r41159 = r41156 / r41158;
        return r41159;
}
↓
double f(double a, double b, double c) {
        double r41160 = a;
        double r41161 = 4.0;
        double r41162 = r41160 * r41161;
        double r41163 = c;
        double r41164 = r41162 * r41163;
        double r41165 = r41164 / r41160;
        double r41166 = b;
        double r41167 = sqrt(r41166);
        double r41168 = -r41163;
        double r41169 = r41166 * r41166;
        double r41170 = fma(r41168, r41162, r41169);
        double r41171 = sqrt(r41170);
        double r41172 = fma(r41167, r41167, r41171);
        double r41173 = -r41172;
        double r41174 = r41165 / r41173;
        double r41175 = 1.0;
        double r41176 = 2.0;
        double r41177 = r41175 / r41176;
        double r41178 = r41174 * r41177;
        return r41178;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.2

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

  3. Applied flip-+28.2

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

  4. Simplified0.5

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

  5. Simplified0.5

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

  7. Applied *-un-lft-identity0.5

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

  8. Applied *-un-lft-identity0.5

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

  9. Applied times-frac0.5

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

  10. Applied times-frac0.5

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

  11. Simplified0.5

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

  12. Simplified0.3

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\frac{\left(a \cdot 4\right) \cdot c}{a}}{-\left(b + \sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)}\right)}}\]
  13. Using strategy rm

  14. Applied add-sqr-sqrt0.5

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

  15. Applied fma-def0.4

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

  16. Final simplification0.4

    \[\leadsto \frac{\frac{\left(a \cdot 4\right) \cdot c}{a}}{-\mathsf{fma}\left(\sqrt{b}, \sqrt{b}, \sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)}\right)} \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2019194 +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) (* (* 4.0 a) c)))) (* 2.0 a)))