Average Error: 52.4 → 0.2
Time: 3.6min
Precision: binary64
Cost: 7680
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a} \cdot \left(a \cdot 2\right)\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a} \cdot \left(a \cdot 2\right)
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (* (/ (/ c (- (- b) (sqrt (- (* b b) (* c (* 4.0 a)))))) a) (* a 2.0)))
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 ((c / (-b - sqrt((b * b) - (c * (4.0 * a))))) / a) * (a * 2.0);
}

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
Error3.1
Cost7232
\[\left(-\frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\]
Alternative 2
Error3.2
Cost1216
\[\frac{\frac{c \cdot \left(4 \cdot a\right)}{2 \cdot \left(\frac{c \cdot a}{b} - b\right)}}{a \cdot 2}\]
Alternative 3
Error6.3
Cost256
\[-\frac{c}{b}\]
Alternative 4
Error57.5
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. Using strategy rm

  3. Applied flip-+_binary64_39352.4

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

  4. Simplified0.4

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

  6. Applied *-un-lft-identity_binary64_4190.4

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

  7. Applied times-frac_binary64_4250.2

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

  8. Applied times-frac_binary64_4250.2

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

  9. Simplified0.2

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

  10. Simplified0.2

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

  12. Applied *-commutative_binary64_3500.2

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2021014 
(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)))