Average Error: 52.8 → 1.3
Time: 23.6s
Precision: binary64
Cost: 40896
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}\]
\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(a \cdot a\right) \cdot {c}^{3}}{{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 (/ (* (* a a) (pow c 3.0)) (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 * (((a * a) * pow(c, 3.0)) / 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
Error3.6
Cost251648
\[\frac{\left(-1.5 \cdot \left(a \cdot c\right)\right) \cdot \left({b}^{7} \cdot \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right) \cdot \left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right) + \left(\left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) \cdot \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) - \left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right) \cdot \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right)\right)\right)\right) - b \cdot \left({b}^{7} \cdot \left({\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right)}^{3} + {\left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right)}^{3}\right) + \left(3.1640625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right) \cdot \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right) \cdot \left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right) + \left(\left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) \cdot \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) - \left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right) \cdot \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right)\right)\right)\right)}{\left(a \cdot 3\right) \cdot \left(b \cdot \left({b}^{7} \cdot \left(1.265625 \cdot \frac{{a}^{4}}{\frac{{b}^{6}}{{c}^{4}}} + \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} \cdot \left(\frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} \cdot 2.84765625 - \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} \cdot 1.8984375\right)\right)\right)\right)}\]
Alternative 2
Error1.9
Cost169344
\[\frac{\left(-1.5 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) - 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right) - b \cdot \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) \cdot \left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) - \left(3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right) \cdot \left(3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)}{\left(a \cdot 3\right) \cdot \left(b \cdot \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} - 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)\right)}\]
Alternative 3
Error2.9
Cost168576
\[\frac{\left(-1.5 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) - 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right) - \left(b \cdot \left(\left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right) \cdot \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right)\right) - b \cdot \frac{\frac{\left({a}^{8} \cdot {c}^{8}\right) \cdot 10.01129150390625}{{b}^{7}}}{{b}^{7}}\right)}{\left(a \cdot 3\right) \cdot \left(b \cdot \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} - 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)\right)}\]
Alternative 4
Error2.9
Cost168448
\[\frac{\left(-1.5 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) - 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right) - b \cdot \left(\left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right) \cdot \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right) - \frac{\frac{\left({a}^{8} \cdot {c}^{8}\right) \cdot 10.01129150390625}{{b}^{7}}}{{b}^{7}}\right)}{\left(a \cdot 3\right) \cdot \left(b \cdot \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} - 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)\right)}\]
Alternative 5
Error1.9
Cost163392
\[\frac{\left(-1.5 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) - 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right) - b \cdot \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) \cdot \left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) - \left(3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right) \cdot \left(3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)}{\left(a \cdot 3\right) \cdot \left(b \cdot \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{\left(a \cdot c\right) \cdot \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}{{b}^{5}} - 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)\right)}\]
Alternative 6
Error2.6
Cost161792
\[\sqrt[3]{\frac{-1.5 \cdot \frac{a \cdot c}{b} - \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3}} \cdot \left(\sqrt[3]{\frac{-1.5 \cdot \frac{a \cdot c}{b} - \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3}} \cdot \sqrt[3]{\frac{-1.5 \cdot \frac{a \cdot c}{b} - \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3}}\right)\]
Alternative 7
Error2.6
Cost161280
\[\frac{\sqrt[3]{-1.5 \cdot \frac{a \cdot c}{b} - \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)} \cdot \sqrt[3]{-1.5 \cdot \frac{a \cdot c}{b} - \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)}}{3} \cdot \frac{\sqrt[3]{-1.5 \cdot \frac{a \cdot c}{b} - \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)}}{a}\]
Alternative 8
Error2.6
Cost148864
\[\frac{\sqrt[3]{-1.5 \cdot \frac{a \cdot c}{b} - \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)} \cdot \sqrt[3]{-1.5 \cdot \frac{a \cdot c}{b} - \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)}}{\frac{3}{\frac{\sqrt[3]{-1.5 \cdot \frac{a \cdot c}{b} - \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)}}{a}}}\]
Alternative 9
Error1.9
Cost143488
\[\frac{\left(-1.5 \cdot \left(a \cdot c\right)\right) \cdot \left({b}^{7} \cdot \left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} - 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right)\right) - b \cdot \left({b}^{7} \cdot \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right) \cdot \left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}}\right) - \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) \cdot \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right)\right) + \left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} - 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) \cdot \left(3.1640625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)\right)}{\left(a \cdot 3\right) \cdot \left(b \cdot \left({b}^{7} \cdot \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} - 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right)\right)\right)}\]
Alternative 10
Error2.0
Cost141312
\[\frac{\left(-1.5 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) - 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right) - \left(3.796875 \cdot \frac{{a}^{5} \cdot {c}^{5}}{{b}^{7}} + \left(1.265625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{5}} + 2.84765625 \cdot \frac{{\left(a \cdot c\right)}^{6}}{{b}^{9}}\right)\right)}{\left(a \cdot 3\right) \cdot \left(b \cdot \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} - 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)\right)}\]
Alternative 11
Error3.5
Cost107968
\[\frac{\left(-1.5 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) - 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right) - 1.265625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{5}}}{\left(a \cdot 3\right) \cdot \left(b \cdot \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} - 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)\right)}\]
Alternative 12
Error27.0
Cost94144
\[\frac{\left(-1.5 \cdot \left(a \cdot c\right)\right) \cdot \left({b}^{7} \cdot \left({b}^{3} \cdot {b}^{5}\right)\right) - b \cdot \left({b}^{7} \cdot \left({b}^{5} \cdot \left(1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) + {b}^{3} \cdot \left(1.6875 \cdot {\left(a \cdot c\right)}^{3}\right)\right) + \left(3.1640625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right) \cdot \left({b}^{3} \cdot {b}^{5}\right)\right)}{\left(a \cdot 3\right) \cdot \left(b \cdot \left({b}^{7} \cdot \left({b}^{3} \cdot {b}^{5}\right)\right)\right)}\]
Alternative 13
Error1.9
Cost66816
\[\sqrt[3]{\frac{{\left(-1.5 \cdot \frac{a \cdot c}{b} - \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)\right)}^{3}}{{a}^{3} \cdot 27}}\]
Alternative 14
Error2.0
Cost60352
\[\sqrt[3]{{\left(\frac{-1.5 \cdot \frac{a \cdot c}{b} - \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3}\right)}^{3}}\]
Alternative 15
Error1.8
Cost47616
\[\frac{1}{\frac{3}{\frac{-1.5 \cdot \frac{a \cdot c}{b} - \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)}{a}}}\]
Alternative 16
Error1.6
Cost47616
\[-0.5 \cdot \frac{\frac{a \cdot c}{b}}{a} - \frac{1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)}{a \cdot 3}\]
Alternative 17
Error1.8
Cost47488
\[0.3333333333333333 \cdot \frac{-1.5 \cdot \frac{a \cdot c}{b} - \left(1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + \left(1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right)}{a}\]
Alternative 18
Error52.8
Cost41408
\[\sqrt[3]{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}} \cdot \left(\sqrt[3]{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}} \cdot \sqrt[3]{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}}\right)\]
Alternative 19
Error1.8
Cost41280
\[\left(-1.5 \cdot \frac{a \cdot c}{b} - \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)\right) \cdot \frac{0.3333333333333333}{a}\]
Alternative 20
Error1.7
Cost41280
\[\frac{-1.5 \cdot \frac{a \cdot c}{b} - \left(\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) + 3.1640625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}\right)}{a \cdot 3}\]
Alternative 21
Error52.8
Cost40896
\[\frac{\sqrt[3]{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b} \cdot \left(\sqrt[3]{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b} \cdot \sqrt[3]{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}\right)}{a \cdot 3}\]
Alternative 22
Error53.2
Cost40128
\[\frac{\left(\sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}} + \sqrt{b}\right) \cdot \left(\sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}} - \sqrt{b}\right)}{a \cdot 3}\]
Alternative 23
Error1.3
Cost34688
\[\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}\]
Alternative 24
Error52.8
Cost28096
\[\frac{{\left(\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)}^{3} - {b}^{3}}{\left(a \cdot 3\right) \cdot \left(b \cdot \left(b + \left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)\right) + a \cdot \left(c \cdot -3\right)\right)}\]
Alternative 25
Error52.8
Cost28096
\[\frac{\frac{{\left(\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)}^{3} - {b}^{3}}{b \cdot \left(b + \left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)\right) + a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\]
Alternative 26
Error52.8
Cost27840
\[\frac{\sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 3\right)} \cdot \left(\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 3\right)} \cdot \sqrt[3]{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)} - b}{a \cdot 3}\]
Alternative 27
Error52.8
Cost27776
\[\frac{\sqrt{\frac{{b}^{6} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{3}}{{b}^{4} + 3 \cdot \left(\left(a \cdot c\right) \cdot \left(b \cdot b + c \cdot \left(a \cdot 3\right)\right)\right)}} - b}{a \cdot 3}\]
Alternative 28
Error58.4
Cost26432
\[\log \left(\sqrt[3]{e^{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a}}}\right)\]
Alternative 29
Error52.8
Cost21696
\[\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b \cdot b}{\left(a \cdot 3\right) \cdot \left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)}\]
Alternative 30
Error2.2
Cost21376
\[\frac{-1.5 \cdot \frac{a \cdot c}{b} - \left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}} + 1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right)}{a \cdot 3}\]
Alternative 31
Error1.8
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 32
Error52.8
Cost20224
\[\frac{\sqrt{\sqrt[3]{{\left(b \cdot b - c \cdot \left(a \cdot 3\right)\right)}^{3}}} - b}{a \cdot 3}\]
Alternative 33
Error52.8
Cost20224
\[\frac{\sqrt[3]{{\left(\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b\right)}^{3}}}{a \cdot 3}\]
Alternative 34
Error58.2
Cost20160
\[\frac{\log \left(e^{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}\right)}{a \cdot 3}\]
Alternative 35
Error52.9
Cost20160
\[\frac{\sqrt{e^{\log \left(b \cdot b - c \cdot \left(a \cdot 3\right)\right)}} - b}{a \cdot 3}\]
Alternative 36
Error62.1
Cost20160
\[\frac{\sqrt{\log \left(e^{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)} - b}{a \cdot 3}\]
Alternative 37
Error52.8
Cost14592
\[\frac{\sqrt{\frac{{b}^{4} - \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot 9}{b \cdot b + c \cdot \left(a \cdot 3\right)}} - b}{a \cdot 3}\]
Alternative 38
Error52.5
Cost8256
\[\frac{\frac{\left(b \cdot b - c \cdot \left(a \cdot 3\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3}\]
Alternative 39
Error3.2
Cost7936
\[\frac{-1.5 \cdot \frac{a \cdot c}{b} - 1.125 \cdot \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}}}{a \cdot 3}\]
Alternative 40
Error53.0
Cost7616
\[\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}{a \cdot 3} - \frac{b}{a \cdot 3}\]
Alternative 41
Error2.8
Cost7424
\[-0.5 \cdot \frac{c}{b} - 0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\]
Alternative 42
Error52.8
Cost7360
\[\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\]
Alternative 43
Error52.8
Cost7360
\[0.3333333333333333 \cdot \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a}\]
Alternative 44
Error6.3
Cost704
\[\frac{-1.5 \cdot \frac{a \cdot c}{b}}{a \cdot 3}\]
Alternative 45
Error6.0
Cost320
\[-0.5 \cdot \frac{c}{b}\]
Alternative 46
Error63.0
Cost64
\[1\]
Alternative 47
Error61.9
Cost64
\[0\]
Alternative 48
Error57.5
Cost64
\[-1\]

Error

Derivation

  1. Initial program 52.8

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

  2. Simplified52.8

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

  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. Simplified1.3

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

  6. Final simplification1.3

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

Reproduce

herbie shell --seed 2021022 
(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)))