Average Error: 52.2 → 0.1
Time: 5.3s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[-2 \cdot \frac{c}{b + \frac{\sqrt{{b}^{4} - a \cdot \left(c \cdot \left(\left(c \cdot a\right) \cdot 16\right)\right)}}{\sqrt{b \cdot b + c \cdot \left(4 \cdot a\right)}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
-2 \cdot \frac{c}{b + \frac{\sqrt{{b}^{4} - a \cdot \left(c \cdot \left(\left(c \cdot a\right) \cdot 16\right)\right)}}{\sqrt{b \cdot b + c \cdot \left(4 \cdot a\right)}}}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (*
  -2.0
  (/
   c
   (+
    b
    (/
     (sqrt (- (pow b 4.0) (* a (* c (* (* c a) 16.0)))))
     (sqrt (+ (* b b) (* c (* 4.0 a)))))))))
double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((4.0 * a) * c))) / (2.0 * a);
}
↓
double code(double a, double b, double c) {
	return -2.0 * (c / (b + (sqrt(pow(b, 4.0) - (a * (c * ((c * a) * 16.0)))) / sqrt((b * b) + (c * (4.0 * 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

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

  2. Simplified52.2

    \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}}\]
  3. Using strategy rm

  4. Applied flip--_binary64_5252.2

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b \cdot b}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}}{a \cdot 2}\]

  5. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{a \cdot \left(c \cdot -4\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{a \cdot 2}\]

  6. Simplified0.4

    \[\leadsto \frac{\frac{a \cdot \left(c \cdot -4\right)}{\color{blue}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity_binary64_770.4

    \[\leadsto \frac{\frac{a \cdot \left(c \cdot -4\right)}{\color{blue}{1 \cdot \left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{a \cdot 2}\]

  9. Applied times-frac_binary64_830.2

    \[\leadsto \frac{\color{blue}{\frac{a}{1} \cdot \frac{c \cdot -4}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2}\]

  10. Applied times-frac_binary64_830.1

    \[\leadsto \color{blue}{\frac{\frac{a}{1}}{a} \cdot \frac{\frac{c \cdot -4}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2}}\]

  11. Simplified0.1

    \[\leadsto \color{blue}{1} \cdot \frac{\frac{c \cdot -4}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2}\]

  12. Simplified0.1

    \[\leadsto 1 \cdot \color{blue}{\left(-2 \cdot \frac{c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\right)}\]
  13. Using strategy rm

  14. Applied flip--_binary64_520.1

    \[\leadsto 1 \cdot \left(-2 \cdot \frac{c}{b + \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(c \cdot \left(a \cdot 4\right)\right) \cdot \left(c \cdot \left(a \cdot 4\right)\right)}{b \cdot b + c \cdot \left(a \cdot 4\right)}}}}\right)\]

  15. Applied sqrt-div_binary64_930.1

    \[\leadsto 1 \cdot \left(-2 \cdot \frac{c}{b + \color{blue}{\frac{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(c \cdot \left(a \cdot 4\right)\right) \cdot \left(c \cdot \left(a \cdot 4\right)\right)}}{\sqrt{b \cdot b + c \cdot \left(a \cdot 4\right)}}}}\right)\]

  16. Simplified0.1

    \[\leadsto 1 \cdot \left(-2 \cdot \frac{c}{b + \frac{\color{blue}{\sqrt{{b}^{4} - a \cdot \left(c \cdot \left(\left(c \cdot a\right) \cdot 16\right)\right)}}}{\sqrt{b \cdot b + c \cdot \left(a \cdot 4\right)}}}\right)\]

  17. Final simplification0.1

    \[\leadsto -2 \cdot \frac{c}{b + \frac{\sqrt{{b}^{4} - a \cdot \left(c \cdot \left(\left(c \cdot a\right) \cdot 16\right)\right)}}{\sqrt{b \cdot b + c \cdot \left(4 \cdot a\right)}}}\]

Reproduce

herbie shell --seed 2020277 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))