Average Error: 52.5 → 0.5
Time: 11.5s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{-3 \cdot \left(a \cdot c\right)}{b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{a \cdot 3}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{-3 \cdot \left(a \cdot c\right)}{b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{a \cdot 3}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (/ (/ (* -3.0 (* a c)) (+ b (sqrt (- (* b b) (* (* a c) 3.0))))) (* a 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 ((-3.0 * (a * c)) / (b + sqrt((b * b) - ((a * c) * 3.0)))) / (a * 3.0);
}

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 52.5

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

  2. Simplified52.5

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

  4. Applied flip--_binary64_141752.5

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

  5. Simplified52.2

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

  6. Simplified52.2

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

  7. Taylor expanded around 0 0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2021019 
(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)))