Average Error: 52.2 → 0.1
Time: 8.9s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[-\frac{c}{b + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
-\frac{c}{b + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}
(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 (sqrt (- (* b b) (* c (* 3.0 a))))))))
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 + sqrt((b * b) - (c * (3.0 * a)))));
}

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 52.2

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

  2. Simplified52.2

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

  4. Applied flip--_binary6452.2

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

  5. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{\left(b \cdot b\right) \cdot 0 - \left(3 \cdot a\right) \cdot c}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\]

  6. Simplified0.4

    \[\leadsto \frac{\frac{\left(b \cdot b\right) \cdot 0 - \left(3 \cdot a\right) \cdot c}{\color{blue}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]
  7. Using strategy rm

  8. Applied div-sub_binary640.4

    \[\leadsto \frac{\color{blue}{\frac{\left(b \cdot b\right) \cdot 0}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} - \frac{\left(3 \cdot a\right) \cdot c}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  9. Applied div-sub_binary640.4

    \[\leadsto \color{blue}{\frac{\frac{\left(b \cdot b\right) \cdot 0}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} - \frac{\frac{\left(3 \cdot a\right) \cdot c}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}}\]

  10. Simplified0.4

    \[\leadsto \color{blue}{0} - \frac{\frac{\left(3 \cdot a\right) \cdot c}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

  11. Simplified0.4

    \[\leadsto 0 - \color{blue}{1 \cdot \frac{a \cdot c}{a \cdot \left(b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}\]
  12. Using strategy rm

  13. Applied associate-/r*_binary640.2

    \[\leadsto 0 - 1 \cdot \color{blue}{\frac{\frac{a \cdot c}{a}}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}\]

  14. Simplified0.1

    \[\leadsto 0 - 1 \cdot \frac{\color{blue}{c \cdot 1}}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\]

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2020277 
(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)))