jeff quadratic root 1

Average Error: 20.4 → 6.9

Time: 6.3s

Precision: binary64

Math

\[\]

↓

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Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -1.71075788809424871e87

   1. Initial program 30.1

      \[\]

   2. Taylor expanded around inf 30.1

      \[\leadsto \]

   3. Simplified 30.1

      \[\leadsto \]

   4. Taylor expanded around -inf 7.4

      \[\leadsto \]

   5. Simplified 3.4

      \[\leadsto \]

   6. Taylor expanded around 0 3.4

      \[\leadsto \]

   7. Simplified 3.4

      \[\leadsto \]
   8. Using strategy rm

   9. Applied add-exp-log 3.4

      \[\leadsto \]

   if -1.71075788809424871e87 < b < 2.3540792276628007e82

   1. Initial program 9.2

      \[\]

   2. Simplified 9.3

      \[\leadsto \]

   if 2.3540792276628007e82 < b

   1. Initial program 43.8

      \[\]

   2. Taylor expanded around inf 10.1

      \[\leadsto \]

   3. Simplified 4.3

      \[\leadsto \]

   4. Taylor expanded around -inf 4.3

      \[\leadsto \]

   5. Simplified 4.3

      \[\leadsto \]

   6. Taylor expanded around 0 4.1

      \[\leadsto \]

   7. Simplified 4.1

      \[\leadsto \]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 4.1

      \[\leadsto \]

   10. Applied times-frac 4.1

       \[\leadsto \]

   11. Simplified 4.1

       \[\leadsto \]

   12. Simplified 4.1

       \[\leadsto \]
3. Recombined 3 regimes into one program.

4. Final simplification 6.9

   \[\leadsto \]

Reproduce

herbie shell --seed 2020179 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))