Average Error: 52.6 → 0.1
Time: 8.7s
Precision: binary64
\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\frac{\frac{c}{-0.5}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{c}{-0.5}}{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 -0.5) (+ 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 / -0.5) / (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 52.6

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

  2. Simplified52.6

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

  3. Applied flip--_binary6452.7

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

  4. Applied associate-*l/_binary6452.7

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

  5. Simplified0.3

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

  6. Applied associate-/l*_binary640.5

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

  7. Applied associate-/r/_binary640.4

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

  8. Applied *-un-lft-identity_binary640.4

    \[\leadsto \frac{\color{blue}{1 \cdot \left(c \cdot \left(a \cdot -4\right) + 0\right)}}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + b}{0.5} \cdot a} \]

  9. Applied times-frac_binary640.4

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + b}{0.5}} \cdot \frac{c \cdot \left(a \cdot -4\right) + 0}{a}} \]

  10. Simplified0.4

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

  11. Simplified0.3

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

  12. Applied associate-*l/_binary640.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2022055 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))