Average Error: 29.0 → 0.3
Time: 6.4s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\frac{c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \frac{-a}{a} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \frac{-a}{a}

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (* (/ c (+ b (sqrt (fma a (* c -3.0) (* b b))))) (/ (- a) a)))
double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((3.0 * a) * c))) / (3.0 * a);
}
↓
double code(double a, double b, double c) {
	return (c / (b + sqrt(fma(a, (c * -3.0), (b * b))))) * (-a / a);
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 29.0

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

  2. Simplified29.0

    \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

  3. Applied flip--_binary6429.0

    \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b}} \cdot \frac{0.3333333333333333}{a} \]

  4. Applied associate-*l/_binary6429.0

    \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b}} \]

  5. Simplified0.6

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, c \cdot -3, 0\right) \cdot \frac{0.3333333333333333}{a}}}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b} \]

  6. Applied associate-*r/_binary640.6

    \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(a, c \cdot -3, 0\right) \cdot 0.3333333333333333}{a}}}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b} \]

  7. Applied associate-/l/_binary640.6

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c \cdot -3, 0\right) \cdot 0.3333333333333333}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b\right) \cdot a}} \]

  8. Taylor expanded in a around 0 0.4

    \[\leadsto \frac{\color{blue}{-1 \cdot \left(c \cdot a\right)}}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b\right) \cdot a} \]

  9. Simplified0.4

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

  10. Applied distribute-rgt-neg-in_binary640.4

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

  11. Applied times-frac_binary640.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2021215 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))