Average Error: 52.7 → 1.4
Time: 8.1s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\left(\left(-0.5 \cdot \frac{c}{b} - 0.375 \cdot \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right) - 1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}\right) - 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\left(\left(-0.5 \cdot \frac{c}{b} - 0.375 \cdot \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right) - 1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}\right) - 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 52.7

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

  2. Simplified52.7

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

  3. Taylor expanded around inf 1.4

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

  4. Simplified1.4

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

  5. Final simplification1.4

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

Reproduce

herbie shell --seed 2021206 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))