Average Error: 52.5 → 0.1
Time: 46.7s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{\frac{{\left(b \cdot b\right)}^{3} + {\left(c \cdot \left(-3 \cdot a\right)\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(c \cdot \left(-3 \cdot a\right)\right) \cdot \left(c \cdot \left(-3 \cdot a\right)\right) - \left(b \cdot b\right) \cdot \left(c \cdot \left(-3 \cdot a\right)\right)\right)}}}\]

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 52.5

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

  3. Applied flip-+52.5

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

  4. Applied associate-/l/52.5

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

  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 associate-*l*0.5

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

  9. Applied times-frac0.4

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

  10. Simplified0.4

    \[\leadsto \color{blue}{1} \cdot \frac{c \cdot a}{a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}\]

  11. Simplified0.1

    \[\leadsto 1 \cdot \color{blue}{\frac{c}{\left(-b\right) - \sqrt{b \cdot b + \left(-3 \cdot a\right) \cdot c}}}\]
  12. Using strategy rm

  13. Applied flip3-+0.1

    \[\leadsto 1 \cdot \frac{c}{\left(-b\right) - \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} + {\left(\left(-3 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(-3 \cdot a\right) \cdot c\right) \cdot \left(\left(-3 \cdot a\right) \cdot c\right) - \left(b \cdot b\right) \cdot \left(\left(-3 \cdot a\right) \cdot c\right)\right)}}}}\]

  14. Final simplification0.1

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

Runtime

Time bar (total: 46.7s)Debug logProfile

herbie shell --seed 2018278 
(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)))