Average Error: 43.7 → 0.4
Time: 10.9s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(\left(-b\right) - \left|\frac{\sqrt{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}}\right|\right) \cdot \left(2 \cdot a\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(\left(-b\right) - \left|\frac{\sqrt{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}}\right|\right) \cdot \left(2 \cdot a\right)}
double f(double a, double b, double c) {
        double r37610 = b;
        double r37611 = -r37610;
        double r37612 = r37610 * r37610;
        double r37613 = 4.0;
        double r37614 = a;
        double r37615 = r37613 * r37614;
        double r37616 = c;
        double r37617 = r37615 * r37616;
        double r37618 = r37612 - r37617;
        double r37619 = sqrt(r37618);
        double r37620 = r37611 + r37619;
        double r37621 = 2.0;
        double r37622 = r37621 * r37614;
        double r37623 = r37620 / r37622;
        return r37623;
}
↓
double f(double a, double b, double c) {
        double r37624 = b;
        double r37625 = r37624 - r37624;
        double r37626 = r37624 * r37625;
        double r37627 = 4.0;
        double r37628 = a;
        double r37629 = c;
        double r37630 = r37628 * r37629;
        double r37631 = r37627 * r37630;
        double r37632 = r37626 + r37631;
        double r37633 = -r37624;
        double r37634 = 4.0;
        double r37635 = pow(r37624, r37634);
        double r37636 = r37631 * r37631;
        double r37637 = r37635 - r37636;
        double r37638 = sqrt(r37637);
        double r37639 = r37624 * r37624;
        double r37640 = r37627 * r37628;
        double r37641 = r37640 * r37629;
        double r37642 = r37639 + r37641;
        double r37643 = sqrt(r37642);
        double r37644 = r37638 / r37643;
        double r37645 = fabs(r37644);
        double r37646 = r37633 - r37645;
        double r37647 = 2.0;
        double r37648 = r37647 * r37628;
        double r37649 = r37646 * r37648;
        double r37650 = r37632 / r37649;
        return r37650;
}

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 43.7

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

  3. Applied flip-+43.7

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

  4. Simplified0.4

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

  6. Applied flip--0.4

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

  7. Simplified0.4

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

  9. Applied add-sqr-sqrt0.4

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied times-frac0.4

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

  13. Applied div-inv0.5

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

  14. Applied associate-/l*0.5

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))