Average Error: 28.2 → 0.5
Time: 7.0s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}\right)}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}\right)}}{2 \cdot a}
double f(double a, double b, double c) {
        double r36567 = b;
        double r36568 = -r36567;
        double r36569 = r36567 * r36567;
        double r36570 = 4.0;
        double r36571 = a;
        double r36572 = r36570 * r36571;
        double r36573 = c;
        double r36574 = r36572 * r36573;
        double r36575 = r36569 - r36574;
        double r36576 = sqrt(r36575);
        double r36577 = r36568 + r36576;
        double r36578 = 2.0;
        double r36579 = r36578 * r36571;
        double r36580 = r36577 / r36579;
        return r36580;
}
↓
double f(double a, double b, double c) {
        double r36581 = 0.0;
        double r36582 = 4.0;
        double r36583 = a;
        double r36584 = c;
        double r36585 = r36583 * r36584;
        double r36586 = r36582 * r36585;
        double r36587 = r36581 + r36586;
        double r36588 = b;
        double r36589 = sqrt(r36588);
        double r36590 = -r36589;
        double r36591 = r36588 * r36588;
        double r36592 = r36582 * r36583;
        double r36593 = r36592 * r36584;
        double r36594 = r36591 - r36593;
        double r36595 = 3.0;
        double r36596 = pow(r36594, r36595);
        double r36597 = cbrt(r36596);
        double r36598 = sqrt(r36597);
        double r36599 = -r36598;
        double r36600 = fma(r36589, r36590, r36599);
        double r36601 = r36587 / r36600;
        double r36602 = 2.0;
        double r36603 = r36602 * r36583;
        double r36604 = r36601 / r36603;
        return r36604;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.2

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

  3. Applied flip-+28.2

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

  4. Simplified0.5

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

  6. Applied add-sqr-sqrt0.5

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

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

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

  8. Applied fma-neg0.5

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

  10. Applied add-cbrt-cube0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}}\right)}}{2 \cdot a}\]

  11. Simplified0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{\sqrt[3]{\color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}\right)}}{2 \cdot a}\]

  12. Final simplification0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}\right)}}{2 \cdot a}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(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)))