Average Error: 52.3 → 0.5
Time: 37.5s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left(3 \cdot c\right) \cdot a}{\left(\left(-b\right) - \sqrt{b \cdot b - \sqrt[3]{\left(\left(a \cdot 3\right) \cdot c\right) \cdot \left(\left(\left(a \cdot 3\right) \cdot c\right) \cdot \left(\left(a \cdot 3\right) \cdot c\right)\right)}}\right) \cdot \left(a \cdot 3\right)}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 52.3

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

  3. Applied flip-+52.3

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

  4. Applied associate-/l/52.3

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

  5. Simplified0.5

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

  7. Applied add-cbrt-cube0.5

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

  9. Applied associate-*r*0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019030 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, wide range"
  :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) (* (* 3 a) c)))) (* 3 a)))