Average Error: 52.2 → 0.2
Time: 5.0s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{\frac{\mathsf{fma}\left(4 \cdot c, a, 0\right)}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{\frac{\mathsf{fma}\left(4 \cdot c, a, 0\right)}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}}{2}
double code(double a, double b, double c) {
	return ((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a));
}
↓
double code(double a, double b, double c) {
	return (((fma((4.0 * c), a, 0.0) / a) / (-b - sqrt(fma(b, b, -((a * c) * 4.0))))) / 2.0);
}

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 52.2

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

  3. Applied flip-+52.2

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

  4. Simplified0.4

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

  6. Applied fma-neg0.4

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

  7. Simplified0.4

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

  9. Applied *-commutative0.4

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

  10. Applied div-inv0.5

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

  11. Applied times-frac0.4

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

  13. Applied associate-*r/0.4

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

  14. Simplified0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2020078 +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)))