Average Error: 52.4 → 1.4
Time: 9.9s
Precision: binary64
Cost: 34432
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\left(\left(-2 \cdot \frac{c \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\left(\left(-2 \cdot \frac{c \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5
(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 (* (* a a) (* c c))) (pow b 5.0)))
    (/ (* a (* c c)) (pow b 3.0)))
   (/ c b))
  (* (/ (* (pow c 4.0) (pow a 3.0)) (pow b 7.0)) -5.0)))
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 * ((a * a) * (c * c))) / pow(b, 5.0))) - ((a * (c * c)) / pow(b, 3.0))) - (c / b)) + (((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 7.0)) * -5.0);
}

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

Alternatives

Alternative 1
Error1.9
Cost20736
\[\left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\]
Alternative 2
Error3.0
Cost7232
\[\left(-\frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\]
Alternative 3
Error6.2
Cost256
\[-\frac{c}{b}\]
Alternative 4
Error57.5
Cost64
\[-1\]
Alternative 5
Error61.9
Cost64
\[0\]
Alternative 6
Error63.0
Cost64
\[1\]

Error

Derivation

  1. Initial program 52.4

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

  2. Simplified52.4

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

  3. Taylor expanded around inf 1.4

    \[\leadsto \color{blue}{-\left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} + \left(\frac{c}{b} + \left(\frac{a \cdot {c}^{2}}{{b}^{3}} + 2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}}\right)\right)\right)}\]

  4. Simplified1.4

    \[\leadsto \color{blue}{\left(\left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} \cdot -5}\]
  5. Using strategy rm

  6. Applied unpow3_binary64_1441.4

    \[\leadsto \left(\left(-2 \cdot \frac{\left(a \cdot a\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot c\right)}}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} \cdot -5\]

  7. Applied associate-*r*_binary64_181.4

    \[\leadsto \left(\left(-2 \cdot \frac{\color{blue}{\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot c}}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} \cdot -5\]

  8. Simplified1.4

    \[\leadsto \color{blue}{\left(\left(-2 \cdot \frac{c \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5}\]

  9. Final simplification1.4

    \[\leadsto \left(\left(-2 \cdot \frac{c \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5\]

Reproduce

herbie shell --seed 2021044 
(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)))