Average Error: 44.0 → 0.4
Time: 2.6m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot \left(-a\right)\right)}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 44.0

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

  3. Applied flip-+44.0

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

  4. Applied associate-/l/44.0

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

  5. Simplified0.4

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

  7. Applied sub-neg0.4

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

  8. Applied distribute-rgt-in0.4

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

  9. Final simplification0.4

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))