Average Error: 43.8 → 0.5
Time: 17.6s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{-\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{\left(a \cdot 2\right) \cdot \left(\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(-4 \cdot \left(a \cdot c\right)\right)\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{-\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{\left(a \cdot 2\right) \cdot \left(\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(-4 \cdot \left(a \cdot c\right)\right)\right)}
double f(double a, double b, double c) {
        double r36581 = b;
        double r36582 = -r36581;
        double r36583 = r36581 * r36581;
        double r36584 = 4.0;
        double r36585 = a;
        double r36586 = r36584 * r36585;
        double r36587 = c;
        double r36588 = r36586 * r36587;
        double r36589 = r36583 - r36588;
        double r36590 = sqrt(r36589);
        double r36591 = r36582 + r36590;
        double r36592 = 2.0;
        double r36593 = r36592 * r36585;
        double r36594 = r36591 / r36593;
        return r36594;
}
↓
double f(double a, double b, double c) {
        double r36595 = 4.0;
        double r36596 = a;
        double r36597 = r36595 * r36596;
        double r36598 = c;
        double r36599 = r36597 * r36598;
        double r36600 = r36599 * r36599;
        double r36601 = -r36600;
        double r36602 = 2.0;
        double r36603 = r36596 * r36602;
        double r36604 = b;
        double r36605 = -r36604;
        double r36606 = r36604 * r36604;
        double r36607 = r36606 - r36599;
        double r36608 = sqrt(r36607);
        double r36609 = r36605 - r36608;
        double r36610 = r36596 * r36598;
        double r36611 = r36595 * r36610;
        double r36612 = -r36611;
        double r36613 = r36609 * r36612;
        double r36614 = r36603 * r36613;
        double r36615 = r36601 / r36614;
        return r36615;
}

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 43.8

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

  3. Applied flip-+43.8

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

  4. Simplified0.4

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

  6. Applied div-inv0.5

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

  7. Applied associate-/l*0.5

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

  8. Simplified0.4

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

  10. Applied flip-+0.5

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

  11. Applied associate-/l/0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019212 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))