Average Error: 28.6 → 5.9
Time: 14.9s
Precision: binary64
Cost: 40640
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\left(\left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\left(\left(-2 \cdot \frac{\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{{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 (/ (* (* a a) (pow c 3.0)) (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 * (((a * a) * pow(c, 3.0)) / 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
Error7.9
Cost53952
\[\frac{\frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}} \cdot -10 + \left(-2 \cdot \left(\frac{a \cdot c}{b} + \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{3}}\right) + -4 \cdot \log \left(e^{\frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}}\right)\right)}{a \cdot 2}\]
Alternative 2
Error11.9
Cost53440
\[\frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5 + \left(\left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}} - \log \left(e^{\frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}}\right)\right) - \frac{c}{b}\right)\]
Alternative 3
Error7.9
Cost48000
\[\frac{\left(-2 \cdot \left(\frac{a \cdot c}{b} + \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{3}}\right) + -4 \cdot \log \left(e^{\frac{\left(a \cdot c\right) \cdot \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}{{b}^{5}}}\right)\right) + \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}} \cdot -10}{a \cdot 2}\]
Alternative 4
Error28.6
Cost41408
\[\sqrt[3]{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}} \cdot \left(\sqrt[3]{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}} \cdot \sqrt[3]{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}}\right)\]
Alternative 5
Error6.0
Cost41152
\[\frac{\frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}} \cdot -10 + \left(-2 \cdot \left(\frac{a \cdot c}{b} + \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{3}}\right) + -4 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right)}{a \cdot 2}\]
Alternative 6
Error28.6
Cost40896
\[\frac{\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{a} \cdot \frac{\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2}\]
Alternative 7
Error28.6
Cost40896
\[\frac{\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b} \cdot \left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\right)}{a \cdot 2}\]
Alternative 8
Error30.7
Cost40640
\[\frac{\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) - b}{a \cdot 2}\]
Alternative 9
Error29.8
Cost40128
\[\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} + \sqrt{b}}{\frac{a}{\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - \sqrt{b}}{2}}}\]
Alternative 10
Error29.8
Cost40128
\[\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - \sqrt{b}}{2} \cdot \frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} + \sqrt{b}}{a}\]
Alternative 11
Error29.8
Cost40128
\[\frac{\left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} + \sqrt{b}\right) \cdot \left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - \sqrt{b}\right)}{a \cdot 2}\]
Alternative 12
Error6.0
Cost35200
\[\frac{\frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}} \cdot -10 + \left(-2 \cdot \left(\frac{a \cdot c}{b} + \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{3}}\right) + -4 \cdot \frac{\left(a \cdot c\right) \cdot \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}{{b}^{5}}\right)}{a \cdot 2}\]
Alternative 13
Error6.0
Cost34752
\[\frac{\left(-2 \cdot \left(\frac{a \cdot c}{b} + \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{3}}\right) + -4 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) + \frac{{\left(a \cdot c\right)}^{4}}{-{b}^{7}} \cdot 10}{a \cdot 2}\]
Alternative 14
Error6.0
Cost34688
\[\frac{\left(-2 \cdot \left(\frac{a \cdot c}{b} + \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{3}}\right) + -4 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}\right) + -10 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}}}{a \cdot 2}\]
Alternative 15
Error5.9
Cost34432
\[\left(\left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + -5 \cdot \frac{\left(a \cdot a\right) \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}\]
Alternative 16
Error29.0
Cost34176
\[\frac{\frac{\sqrt{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + 4 \cdot \left(\left(a \cdot c\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right)\right)}} - b}{a \cdot 2}\]
Alternative 17
Error28.7
Cost28096
\[\frac{{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}^{3} - {b}^{3}}{\left(a \cdot 2\right) \cdot \left(b \cdot \left(b + \left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\right) + a \cdot \left(c \cdot -4\right)\right)}\]
Alternative 18
Error28.7
Cost28096
\[\frac{\frac{{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}^{3} - {b}^{3}}{b \cdot \left(b + \left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\right) + a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\]
Alternative 19
Error29.9
Cost27840
\[\frac{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2}\]
Alternative 20
Error28.8
Cost27776
\[\frac{\sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{{b}^{4} + 4 \cdot \left(\left(a \cdot c\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right)\right)}} - b}{a \cdot 2}\]
Alternative 21
Error29.3
Cost27200
\[\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2}\]
Alternative 22
Error29.9
Cost27200
\[\frac{\left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2}\]
Alternative 23
Error32.3
Cost26432
\[\log \left(\sqrt{e^{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}}}\right)\]
Alternative 24
Error28.7
Cost21696
\[\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b \cdot b}{\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(a \cdot 2\right)}\]
Alternative 25
Error7.9
Cost21248
\[\frac{-2 \cdot \left(\frac{a \cdot c}{b} + \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{3}}\right) + -4 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}}{a \cdot 2}\]
Alternative 26
Error29.0
Cost20992
\[\frac{\frac{\sqrt{{b}^{4} - \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 16}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2}\]
Alternative 27
Error7.8
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 28
Error29.0
Cost20224
\[\frac{\sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}} - b}{a \cdot 2}\]
Alternative 29
Error28.6
Cost20224
\[\sqrt[3]{{\left(\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\right)}^{3}}\]
Alternative 30
Error28.6
Cost20224
\[\frac{\sqrt[3]{{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)}^{3}}}{a \cdot 2}\]
Alternative 31
Error29.4
Cost20224
\[\frac{\sqrt[3]{{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}^{3}} - b}{a \cdot 2}\]
Alternative 32
Error31.5
Cost20160
\[\frac{\log \left(e^{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\right)}{a \cdot 2}\]
Alternative 33
Error30.5
Cost20160
\[\frac{\sqrt{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2}\]
Alternative 34
Error30.5
Cost20160
\[\frac{e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2}\]
Alternative 35
Error28.8
Cost14592
\[\frac{\sqrt{\frac{{b}^{4} - \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 16}{b \cdot b + \left(4 \cdot a\right) \cdot c}} - b}{a \cdot 2}\]
Alternative 36
Error27.7
Cost8256
\[\frac{\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2}\]
Alternative 37
Error11.8
Cost7808
\[\frac{-2 \cdot \left(\frac{a \cdot c}{b} + \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{3}}\right)}{a \cdot 2}\]
Alternative 38
Error29.1
Cost7616
\[\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} - \frac{b}{a \cdot 2}\]
Alternative 39
Error28.6
Cost7360
\[\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\]
Alternative 40
Error11.7
Cost7232
\[\left(-\frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\]
Alternative 41
Error22.8
Cost704
\[\frac{-2 \cdot \frac{a \cdot c}{b}}{a \cdot 2}\]
Alternative 42
Error22.8
Cost256
\[-\frac{c}{b}\]
Alternative 43
Error63.0
Cost64
\[1\]
Alternative 44
Error62.0
Cost64
\[0\]
Alternative 45
Error55.6
Cost64
\[-1\]

Error

Derivation

  1. Initial program 28.6

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

  2. Simplified28.6

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

  3. Taylor expanded around inf 5.9

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

  4. Simplified5.9

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

  5. Simplified5.9

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

  6. Final simplification5.9

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

Reproduce

herbie shell --seed 2021022 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))