Average Error: 52.5 → 1.3
Time: 12.2s
Precision: binary64
Cost: 34688
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (-
  (-
   (* -0.5 (/ c b))
   (+
    (* 0.375 (/ (* a (* c c)) (pow b 3.0)))
    (* 0.5625 (/ (* (* c c) (* c (* a a))) (pow b 5.0)))))
  (* 1.0546875 (/ (* (pow c 4.0) (pow a 3.0)) (pow b 7.0)))))
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 ((-0.5 * (c / b)) - ((0.375 * ((a * (c * c)) / pow(b, 3.0))) + (0.5625 * (((c * c) * (c * (a * a))) / pow(b, 5.0))))) - (1.0546875 * ((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 7.0)));
}

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
Cost20992
\[-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\]
Alternative 2
Error2.9
Cost7424
\[-0.5 \cdot \frac{c}{b} - 0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\]
Alternative 3
Error6.2
Cost320
\[-0.5 \cdot \frac{c}{b}\]
Alternative 4
Error57.6
Cost64
\[-1\]
Alternative 5
Error61.9
Cost64
\[0\]
Alternative 6
Error63.0
Cost64
\[1\]

Error

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. Taylor expanded around inf 1.3

    \[\leadsto \color{blue}{-\left(1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} + \left(0.5 \cdot \frac{c}{b} + \left(0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + 0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}}\right)\right)\right)}\]

  4. Simplified1.3

    \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}}\]
  5. Using strategy rm

  6. Applied unpow3_binary64_11671.3

    \[\leadsto \left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\color{blue}{\left(\left(c \cdot c\right) \cdot c\right)} \cdot \left(a \cdot a\right)}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}\]

  7. Applied associate-*l*_binary64_10421.3

    \[\leadsto \left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\color{blue}{\left(c \cdot c\right) \cdot \left(c \cdot \left(a \cdot a\right)\right)}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}\]

  8. Simplified1.3

    \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}}\]

  9. Final simplification1.3

    \[\leadsto \left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}\]

Reproduce

herbie shell --seed 2021044 
(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)))