Average Error: 52.6 → 0.4
Time: 6.4s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\frac{\frac{\left(a \cdot c\right) \cdot 4}{b + \sqrt{\left(a \cdot c\right) \cdot -4 + b \cdot b}}}{a \cdot -2} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{\left(a \cdot c\right) \cdot 4}{b + \sqrt{\left(a \cdot c\right) \cdot -4 + b \cdot b}}}{a \cdot -2}

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (/ (/ (* (* a c) 4.0) (+ b (sqrt (+ (* (* a c) -4.0) (* b b))))) (* 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 (((a * c) * 4.0) / (b + sqrt(((a * c) * -4.0) + (b * b)))) / (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

Derivation

  1. Initial program 52.6

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

  2. Simplified52.6

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

  4. Applied flip--_binary6452.6

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

  5. Simplified52.3

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

  6. Simplified52.3

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

  8. Applied frac-2neg_binary6452.3

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2021190 
(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)))