Average Error: 52.7 → 0.2
Time: 6.7s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{2} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{-\left(b + \sqrt{\mathsf{fma}\left(b, b, 0 - 4 \cdot \left(a \cdot c\right)\right)}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1}{2} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{-\left(b + \sqrt{\mathsf{fma}\left(b, b, 0 - 4 \cdot \left(a \cdot c\right)\right)}\right)}
double f(double a, double b, double c) {
        double r39775 = b;
        double r39776 = -r39775;
        double r39777 = r39775 * r39775;
        double r39778 = 4.0;
        double r39779 = a;
        double r39780 = r39778 * r39779;
        double r39781 = c;
        double r39782 = r39780 * r39781;
        double r39783 = r39777 - r39782;
        double r39784 = sqrt(r39783);
        double r39785 = r39776 + r39784;
        double r39786 = 2.0;
        double r39787 = r39786 * r39779;
        double r39788 = r39785 / r39787;
        return r39788;
}
↓
double f(double a, double b, double c) {
        double r39789 = 1.0;
        double r39790 = 2.0;
        double r39791 = r39789 / r39790;
        double r39792 = 4.0;
        double r39793 = a;
        double r39794 = c;
        double r39795 = r39793 * r39794;
        double r39796 = r39792 * r39795;
        double r39797 = r39796 / r39793;
        double r39798 = b;
        double r39799 = 0.0;
        double r39800 = r39799 - r39796;
        double r39801 = fma(r39798, r39798, r39800);
        double r39802 = sqrt(r39801);
        double r39803 = r39798 + r39802;
        double r39804 = -r39803;
        double r39805 = r39797 / r39804;
        double r39806 = r39791 * r39805;
        return r39806;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.7

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

  3. Applied flip-+52.7

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

  4. Simplified0.4

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

  6. Applied fma-neg0.4

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

  7. Simplified0.4

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

  9. Applied add-sqr-sqrt0.5

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

  10. Applied distribute-rgt-neg-in0.5

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

  11. Applied fma-neg0.4

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

  13. Applied *-un-lft-identity0.4

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

  14. Applied *-un-lft-identity0.4

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

  15. Applied times-frac0.4

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

  16. Applied times-frac0.4

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

  17. Simplified0.4

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

  18. Simplified0.2

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

  19. Final simplification0.2

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))