Average Error: 52.7 → 0.4
Time: 2.9m
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c \cdot \left(4 \cdot a\right) + \left(b \cdot b - b \cdot b\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(c \cdot \left(4 \cdot a\right)\right)}^{3}}{\left(\left(b \cdot b\right) \cdot \left(c \cdot \left(4 \cdot a\right)\right) + \left(c \cdot \left(4 \cdot a\right)\right) \cdot \left(c \cdot \left(4 \cdot a\right)\right)\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}\right)}\]

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 52.7

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

  3. Applied flip-+52.7

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

  4. Applied associate-/l/52.7

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

  5. Simplified0.4

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

  7. Applied flip3--0.4

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

  8. Final simplification0.4

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

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018214 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))