Average Error: 28.9 → 5.8
Time: 6.6s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[-\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \frac{1}{{b}^{7}} \cdot \left({c}^{4} \cdot {a}^{3}\right), \mathsf{fma}\left(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\right)\right) \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
-\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \frac{1}{{b}^{7}} \cdot \left({c}^{4} \cdot {a}^{3}\right), \mathsf{fma}\left(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\right)\right)

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (-
  (fma
   0.5625
   (/ (* (* a a) (pow c 3.0)) (pow b 5.0))
   (fma
    1.0546875
    (* (/ 1.0 (pow b 7.0)) (* (pow c 4.0) (pow a 3.0)))
    (fma 0.5 (/ c b) (* 0.375 (/ (* c (* a c)) (pow b 3.0))))))))
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 -fma(0.5625, (((a * a) * pow(c, 3.0)) / pow(b, 5.0)), fma(1.0546875, ((1.0 / pow(b, 7.0)) * (pow(c, 4.0) * pow(a, 3.0))), fma(0.5, (c / b), (0.375 * ((c * (a * c)) / pow(b, 3.0))))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.9

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

  2. Taylor expanded in b around inf 5.8

    \[\leadsto \color{blue}{-\left(0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 0.5 \cdot \frac{c}{b}\right)\right)\right)} \]

  3. Simplified5.8

    \[\leadsto \color{blue}{-\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \mathsf{fma}\left(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right)\right)} \]

  4. Applied add-log-exp_binary646.4

    \[\leadsto -\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \color{blue}{\log \left(e^{\frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}}\right)}, \mathsf{fma}\left(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right)\right) \]

  5. Applied div-inv_binary646.4

    \[\leadsto -\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \log \left(e^{\color{blue}{\left({a}^{3} \cdot {c}^{4}\right) \cdot \frac{1}{{b}^{7}}}}\right), \mathsf{fma}\left(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right)\right) \]

  6. Applied exp-prod_binary6422.3

    \[\leadsto -\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \log \color{blue}{\left({\left(e^{{a}^{3} \cdot {c}^{4}}\right)}^{\left(\frac{1}{{b}^{7}}\right)}\right)}, \mathsf{fma}\left(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right)\right) \]

  7. Applied log-pow_binary6422.1

    \[\leadsto -\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \color{blue}{\frac{1}{{b}^{7}} \cdot \log \left(e^{{a}^{3} \cdot {c}^{4}}\right)}, \mathsf{fma}\left(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right)\right) \]

  8. Simplified5.8

    \[\leadsto -\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \frac{1}{{b}^{7}} \cdot \color{blue}{\left({c}^{4} \cdot {a}^{3}\right)}, \mathsf{fma}\left(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right)\right) \]

  9. Final simplification5.8

    \[\leadsto -\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \frac{1}{{b}^{7}} \cdot \left({c}^{4} \cdot {a}^{3}\right), \mathsf{fma}\left(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\right)\right) \]

Reproduce

herbie shell --seed 2022104 
(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)))