Average Error: 28.7 → 0.4
Time: 6.3s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4}}}{2} \cdot \frac{a \cdot c}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4}}}{2} \cdot \frac{a \cdot c}{a}
double f(double a, double b, double c) {
        double r36167 = b;
        double r36168 = -r36167;
        double r36169 = r36167 * r36167;
        double r36170 = 4.0;
        double r36171 = a;
        double r36172 = r36170 * r36171;
        double r36173 = c;
        double r36174 = r36172 * r36173;
        double r36175 = r36169 - r36174;
        double r36176 = sqrt(r36175);
        double r36177 = r36168 + r36176;
        double r36178 = 2.0;
        double r36179 = r36178 * r36171;
        double r36180 = r36177 / r36179;
        return r36180;
}
↓
double f(double a, double b, double c) {
        double r36181 = 1.0;
        double r36182 = b;
        double r36183 = -r36182;
        double r36184 = r36182 * r36182;
        double r36185 = 4.0;
        double r36186 = a;
        double r36187 = r36185 * r36186;
        double r36188 = c;
        double r36189 = r36187 * r36188;
        double r36190 = r36184 - r36189;
        double r36191 = sqrt(r36190);
        double r36192 = r36183 - r36191;
        double r36193 = r36192 / r36185;
        double r36194 = r36181 / r36193;
        double r36195 = 2.0;
        double r36196 = r36194 / r36195;
        double r36197 = r36186 * r36188;
        double r36198 = r36197 / r36186;
        double r36199 = r36196 * r36198;
        return r36199;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied flip-+28.7

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

  4. Simplified0.5

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

  6. Applied clear-num0.5

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

  7. Simplified0.5

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

  9. Applied div-inv0.5

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

  10. Applied add-sqr-sqrt0.5

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

  11. Applied times-frac0.6

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

  12. Applied times-frac0.5

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

  13. Simplified0.5

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019353 
(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)))