Average Error: 34.1 → 6.8

Time: 5.2s

Precision: binary64

\[\]

↓

\[\]

Error

The X axis uses an exponential scale — Bits error versus a

The X axis uses an exponential scale — Bits error versus a

The X axis uses an exponential scale — Bits error versus b_2

The X axis uses an exponential scale — Bits error versus b_2

The X axis uses an exponential scale — Bits error versus c

The X axis uses an exponential scale — Bits error versus c

Derivation

Split input into 4 regimes
if b_2 < -2.4345943263922276e105
1. Initial program 59.9
  \[\]
2. Using strategy rm
3. Applied clear-num59.9
  \[\leadsto \]
4. Using strategy rm
5. Applied flip--59.9
  \[\leadsto \]
6. Applied associate-/r/59.9
  \[\leadsto \]
7. Applied associate-/r*59.9
  \[\leadsto \]
8. Simplified30.5
  \[\leadsto \]
9. Taylor expanded around 0 29.6
  \[\leadsto \]
10. Taylor expanded around -inf 6.4
  \[\leadsto \]
11. Simplified2.5
  \[\leadsto \]
if -2.4345943263922276e105 < b_2 < -7.7601710165977257e-270
1. Initial program 33.8
  \[\]
2. Using strategy rm
3. Applied clear-num33.9
  \[\leadsto \]
4. Using strategy rm
5. Applied flip--33.9
  \[\leadsto \]
6. Applied associate-/r/33.9
  \[\leadsto \]
7. Applied associate-/r*33.9
  \[\leadsto \]
8. Simplified15.6
  \[\leadsto \]
9. Taylor expanded around 0 8.6
  \[\leadsto \]
if -7.7601710165977257e-270 < b_2 < 2.8975813688974251e60
1. Initial program 9.6
  \[\]
2. Using strategy rm
3. Applied div-sub9.6
  \[\leadsto \]
if 2.8975813688974251e60 < b_2
1. Initial program 39.9
  \[\]
2. Using strategy rm
3. Applied clear-num40.0
  \[\leadsto \]
4. Taylor expanded around 0 4.9
  \[\leadsto \]
Recombined 4 regimes into one program.
Final simplification6.8
\[\leadsto \]

Reproduce

herbie shell --seed 2020180 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (neg b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))