Average Error: 28.1 → 0.3
Time: 6.4s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\frac{-2 \cdot c}{b + \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)}} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{-2 \cdot c}{b + \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)}}

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.1

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

  2. Simplified28.1

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

  3. Applied flip--_binary6428.1

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

  4. Applied associate-*l/_binary6428.1

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

  5. Simplified0.5

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

  6. Taylor expanded in c around 0 0.3

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

  7. Applied add-cbrt-cube_binary640.3

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

  8. Simplified0.3

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

  9. Taylor expanded in a around 0 0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2022125 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))