Average Error: 28.3 → 0.4
Time: 5.1s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\left(-4\right) \cdot \frac{\frac{c}{b + \sqrt[3]{{\left(\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right)}^{3}}}}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\left(-4\right) \cdot \frac{\frac{c}{b + \sqrt[3]{{\left(\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right)}^{3}}}}{2}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (*
  (- 4.0)
  (/ (/ c (+ b (cbrt (pow (sqrt (- (* b b) (* c (* 4.0 a)))) 3.0)))) 2.0)))
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))) / ((double) (2.0 * a)));
}
↓
double code(double a, double b, double c) {
	return ((double) (((double) -(4.0)) * ((c / ((double) (b + ((double) cbrt(((double) pow(((double) sqrt(((double) (((double) (b * b)) - ((double) (c * ((double) (4.0 * a)))))))), 3.0))))))) / 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 Error: 28.3 bits

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

  2. SimplifiedError: 28.3 bits

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

  4. Applied flip--Error: 28.3 bits

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

  5. SimplifiedError: 0.5 bits

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

  6. SimplifiedError: 0.5 bits

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

  8. Applied *-un-lft-identityError: 0.5 bits

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

  9. Applied distribute-lft-neg-inError: 0.5 bits

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

  10. Applied times-fracError: 0.3 bits

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

  11. Applied times-fracError: 0.3 bits

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

  12. SimplifiedError: 0.3 bits

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

  14. Applied add-cbrt-cubeError: 0.4 bits

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

  15. SimplifiedError: 0.4 bits

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

  16. Final simplificationError: 0.4 bits

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

Reproduce

herbie shell --seed 2020205 
(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)))