Average Error: 28.2 → 0.3
Time: 7.8s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\frac{\frac{c \cdot 2}{-1}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{c \cdot 2}{-1}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (/ (/ (* c 2.0) -1.0) (+ b (sqrt (fma a (* c -4.0) (* b b))))))
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 ((c * 2.0) / -1.0) / (b + sqrt(fma(a, (c * -4.0), (b * b))));
}

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. Simplified28.2

    \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]
  3. Using strategy rm

  4. Applied flip--_binary6428.2

    \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + b}} \cdot \frac{0.5}{a} \]

  5. Simplified0.5

    \[\leadsto \frac{\color{blue}{c \cdot \left(a \cdot -4\right) + 0}}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + b} \cdot \frac{0.5}{a} \]

  6. Simplified0.5

    \[\leadsto \frac{c \cdot \left(a \cdot -4\right) + 0}{\color{blue}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}} \cdot \frac{0.5}{a} \]
  7. Using strategy rm

  8. Applied associate-*l/_binary640.4

    \[\leadsto \color{blue}{\frac{\left(c \cdot \left(a \cdot -4\right) + 0\right) \cdot \frac{0.5}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}} \]

  9. Simplified0.3

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

  11. Applied frac-2neg_binary640.3

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

  12. Applied distribute-frac-neg_binary640.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2021211 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-8 a 94906265.62425156) (< 1.0536712127723509e-8 b 94906265.62425156) (< 1.0536712127723509e-8 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))