Average Error: 28.7 → 0.3
Time: 7.5s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\\ \frac{-c}{\mathsf{fma}\left(t_0, t_0, b\right)} \end{array} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
t_0 := \sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\\
\frac{-c}{\mathsf{fma}\left(t_0, t_0, b\right)}
\end{array}

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt (fma a (* c -3.0) (* b b))))))
   (/ (- c) (fma t_0 t_0 b))))
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) {
	double t_0 = sqrt(sqrt(fma(a, (c * -3.0), (b * b))));
	return -c / fma(t_0, t_0, b);
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.7

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

  2. Simplified28.7

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

  3. Applied flip--_binary6428.7

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

  4. Applied associate-*l/_binary6428.7

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

  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. Taylor expanded in a around 0 0.3

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

  7. Simplified0.3

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

  8. Applied add-sqr-sqrt_binary640.4

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

  9. Applied fma-def_binary640.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2021280 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))