Average Error: 52.8 → 2.9
Time: 6.6s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{b} \cdot -0.5 + \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} \cdot -0.375\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{b} \cdot -0.5 + \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} \cdot -0.375
(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) -0.5) (* (/ (* a (* c c)) (pow b 3.0)) -0.375)))
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) * -0.5) + (((a * (c * c)) / pow(b, 3.0)) * -0.375);
}

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 52.8

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

  2. Simplified52.8

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

  3. Taylor expanded around inf 2.9

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

  4. Simplified2.9

    \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5 + \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} \cdot -0.375}\]

  5. Final simplification2.9

    \[\leadsto \frac{c}{b} \cdot -0.5 + \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} \cdot -0.375\]

Reproduce

herbie shell --seed 2020339 
(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)))