Average Error: 28.4 → 0.3
Time: 5.9s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{\frac{2}{4}} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1}{\frac{2}{4}} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}
double f(double a, double b, double c) {
        double r38442 = b;
        double r38443 = -r38442;
        double r38444 = r38442 * r38442;
        double r38445 = 4.0;
        double r38446 = a;
        double r38447 = r38445 * r38446;
        double r38448 = c;
        double r38449 = r38447 * r38448;
        double r38450 = r38444 - r38449;
        double r38451 = sqrt(r38450);
        double r38452 = r38443 + r38451;
        double r38453 = 2.0;
        double r38454 = r38453 * r38446;
        double r38455 = r38452 / r38454;
        return r38455;
}
↓
double f(double a, double b, double c) {
        double r38456 = 1.0;
        double r38457 = 2.0;
        double r38458 = 4.0;
        double r38459 = r38457 / r38458;
        double r38460 = r38456 / r38459;
        double r38461 = c;
        double r38462 = b;
        double r38463 = -r38462;
        double r38464 = r38462 * r38462;
        double r38465 = a;
        double r38466 = r38458 * r38465;
        double r38467 = r38466 * r38461;
        double r38468 = r38464 - r38467;
        double r38469 = sqrt(r38468);
        double r38470 = r38463 - r38469;
        double r38471 = r38461 / r38470;
        double r38472 = r38460 * r38471;
        return r38472;
}

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 28.4

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

  3. Applied flip-+28.5

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

  4. Simplified0.4

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

  6. Applied div-inv0.5

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

  7. Applied associate-/l*0.5

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

  8. Simplified0.5

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

  10. Applied clear-num0.5

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

  11. Simplified0.5

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

  13. Applied add-sqr-sqrt0.5

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

  14. Applied times-frac0.5

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

  15. Simplified0.5

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

  16. Simplified0.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2019346 
(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)))