Average Error: 28.4 → 0.4
Time: 4.1s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{a \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(a \cdot \left(c \cdot \sqrt[3]{3}\right)\right)}}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{a \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(a \cdot \left(c \cdot \sqrt[3]{3}\right)\right)}}}{a}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (/
  (*
   a
   (/
    c
    (-
     (- b)
     (sqrt (- (* b b) (* (* (cbrt 3.0) (cbrt 3.0)) (* a (* c (cbrt 3.0)))))))))
  a))
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (3.0 * a)) * c)))))))) / ((double) (3.0 * a)));
}
↓
double code(double a, double b, double c) {
	return (((double) (a * (c / ((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))) * ((double) (a * ((double) (c * ((double) cbrt(3.0))))))))))))))))) / 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

Derivation

  1. Initial program Error: 28.4 bits

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

  3. Applied flip-+Error: 28.5 bits

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

  4. SimplifiedError: 0.6 bits

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

  5. SimplifiedError: 0.6 bits

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

  7. Applied associate-/r*Error: 0.6 bits

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

  8. SimplifiedError: 0.4 bits

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

  10. Applied add-cube-cbrtError: 0.4 bits

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

  11. Applied associate-*l*Error: 0.4 bits

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

  12. SimplifiedError: 0.4 bits

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

  13. Final simplificationError: 0.4 bits

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

Reproduce

herbie shell --seed 2020204 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))