Average Error: 28.7 → 0.3
Time: 5.1s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{\mathsf{fma}\left(4 \cdot c, a, 0\right)}{a}}{2 \cdot \frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}{1}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{\mathsf{fma}\left(4 \cdot c, a, 0\right)}{a}}{2 \cdot \frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}{1}}
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) / (2.0 * ((-b - sqrt(fma(b, b, -((a * c) * 4.0)))) / 1.0)));
}

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 28.7

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

  3. Applied flip-+28.8

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

  4. Simplified0.4

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

  6. Applied *-commutative0.4

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

  7. Applied div-inv0.5

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

  8. Applied times-frac0.4

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

  10. Applied fma-neg0.4

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

  11. Simplified0.4

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

  13. Applied clear-num0.4

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

  14. Applied associate-/l/0.4

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

  15. Applied un-div-inv0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2020071 +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)))