Average Error: 28.4 → 0.4
Time: 55.1s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[c \cdot \frac{-2}{b + \sqrt{(b \cdot b + \left(\left(a \cdot -4\right) \cdot c\right))_*}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
c \cdot \frac{-2}{b + \sqrt{(b \cdot b + \left(\left(a \cdot -4\right) \cdot c\right))_*}}
double f(double a, double b, double c) {
        double r5550170 = b;
        double r5550171 = -r5550170;
        double r5550172 = r5550170 * r5550170;
        double r5550173 = 4.0;
        double r5550174 = a;
        double r5550175 = r5550173 * r5550174;
        double r5550176 = c;
        double r5550177 = r5550175 * r5550176;
        double r5550178 = r5550172 - r5550177;
        double r5550179 = sqrt(r5550178);
        double r5550180 = r5550171 + r5550179;
        double r5550181 = 2.0;
        double r5550182 = r5550181 * r5550174;
        double r5550183 = r5550180 / r5550182;
        return r5550183;
}
↓
double f(double a, double b, double c) {
        double r5550184 = c;
        double r5550185 = -2.0;
        double r5550186 = b;
        double r5550187 = a;
        double r5550188 = -4.0;
        double r5550189 = r5550187 * r5550188;
        double r5550190 = r5550189 * r5550184;
        double r5550191 = fma(r5550186, r5550186, r5550190);
        double r5550192 = sqrt(r5550191);
        double r5550193 = r5550186 + r5550192;
        double r5550194 = r5550185 / r5550193;
        double r5550195 = r5550184 * r5550194;
        return r5550195;
}

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. Simplified28.4

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

  4. Applied *-un-lft-identity28.4

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

  5. Applied div-inv28.4

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

  6. Applied times-frac28.4

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

  7. Simplified28.4

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

  8. Simplified28.4

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

  10. Applied flip--28.5

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

  11. Applied frac-times28.5

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

  12. Simplified0.4

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

  14. Applied times-frac0.4

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019104 +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)))