Average Error: 28.6 → 0.3
Time: 2.6min
Precision: binary64
Cost: 7680
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\left(a \cdot 2\right) \cdot \frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\left(a \cdot 2\right) \cdot \frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (* (* a 2.0) (/ (/ c (- (- b) (sqrt (- (* b b) (* c (* a 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 (a * 2.0) * ((c / (-b - sqrt((b * b) - (c * (a * 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

Alternatives

Alternative 1
Error9.2
Cost15105
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.4379107732521437:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 2\right) \cdot \frac{\frac{c}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}}{a}\\ \end{array}\]
Alternative 2
Error9.2
Cost15105
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.4379107732521437:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 2\right) \cdot \frac{\frac{c}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}}{a}\\ \end{array}\]
Alternative 3
Error9.2
Cost14977
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.4379107732521437:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 2\right) \cdot \frac{\frac{c}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}}{a}\\ \end{array}\]
Alternative 4
Error11.5
Cost1088
\[\left(a \cdot 2\right) \cdot \frac{\frac{c}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}}{a}\]
Alternative 5
Error22.8
Cost256
\[\frac{-c}{b}\]
Alternative 6
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. Using strategy rm

  3. Applied flip-+_binary64_73428.6

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

  4. Simplified0.5

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

  6. Applied *-un-lft-identity_binary64_7600.5

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

  7. Applied times-frac_binary64_7660.4

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

  8. Applied times-frac_binary64_7660.3

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

  9. Simplified0.3

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

  10. Simplified0.3

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

  12. Applied pow1_binary64_8210.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2021014 
(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)))