Average Error: 52.2 → 0.1
Time: 6.8s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\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 (* 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 c / (-b - sqrt((b * b) - (c * (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 52.2

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

  3. Applied flip-+_binary64_108552.2

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

  4. Simplified0.4

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

  6. Applied *-un-lft-identity_binary64_11110.4

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

  7. Applied times-frac_binary64_11170.2

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

  8. Applied times-frac_binary64_11170.3

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

  9. Simplified0.2

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

  10. Simplified0.2

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

  12. Applied pow1_binary64_11720.2

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

  13. Applied pow1_binary64_11720.2

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

  14. Applied pow-prod-down_binary64_11820.2

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

  15. Simplified0.1

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

  16. Final simplification0.1

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

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)))