Average Error: 43.4 → 0.4
Time: 6.0s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(-b\right) + \left(2 \cdot a\right) \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(-b\right) + \left(2 \cdot a\right) \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}
double f(double a, double b, double c) {
        double r33754 = b;
        double r33755 = -r33754;
        double r33756 = r33754 * r33754;
        double r33757 = 4.0;
        double r33758 = a;
        double r33759 = r33757 * r33758;
        double r33760 = c;
        double r33761 = r33759 * r33760;
        double r33762 = r33756 - r33761;
        double r33763 = sqrt(r33762);
        double r33764 = r33755 + r33763;
        double r33765 = 2.0;
        double r33766 = r33765 * r33758;
        double r33767 = r33764 / r33766;
        return r33767;
}
↓
double f(double a, double b, double c) {
        double r33768 = 0.0;
        double r33769 = 4.0;
        double r33770 = a;
        double r33771 = c;
        double r33772 = r33770 * r33771;
        double r33773 = r33769 * r33772;
        double r33774 = r33768 + r33773;
        double r33775 = 2.0;
        double r33776 = r33775 * r33770;
        double r33777 = b;
        double r33778 = -r33777;
        double r33779 = r33776 * r33778;
        double r33780 = r33777 * r33777;
        double r33781 = r33769 * r33770;
        double r33782 = r33781 * r33771;
        double r33783 = r33780 - r33782;
        double r33784 = sqrt(r33783);
        double r33785 = -r33784;
        double r33786 = r33776 * r33785;
        double r33787 = r33779 + r33786;
        double r33788 = r33774 / r33787;
        return r33788;
}

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 43.4

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

  3. Applied flip-+43.4

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

  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. Simplified0.4

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

  10. Applied sub-neg0.4

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

  11. Applied distribute-lft-in0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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)))