Average Error: 33.8 → 6.9

Time: 6.0s

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 < -446010382391396610
1. Initial program 56.1
  \[\]
2. Using strategy rm
3. Applied flip--56.1
  \[\leadsto \]
4. Simplified26.7
  \[\leadsto \]
5. Simplified26.7
  \[\leadsto \]
6. Using strategy rm
7. Applied *-un-lft-identity26.7
  \[\leadsto \]
8. Applied *-un-lft-identity26.7
  \[\leadsto \]
9. Applied times-frac26.7
  \[\leadsto \]
10. Simplified26.7
  \[\leadsto \]
11. Simplified23.0
  \[\leadsto \]
12. Taylor expanded around -inf 7.7
  \[\leadsto \]
13. Simplified5.1
  \[\leadsto \]
if -446010382391396610 < b_2 < 3.11018247161006561e-199
1. Initial program 24.7
  \[\]
2. Using strategy rm
3. Applied flip--24.8
  \[\leadsto \]
4. Simplified16.6
  \[\leadsto \]
5. Simplified16.6
  \[\leadsto \]
6. Using strategy rm
7. Applied *-un-lft-identity16.6
  \[\leadsto \]
8. Applied *-un-lft-identity16.6
  \[\leadsto \]
9. Applied times-frac16.6
  \[\leadsto \]
10. Simplified16.6
  \[\leadsto \]
11. Simplified10.9
  \[\leadsto \]
if 3.11018247161006561e-199 < b_2 < 1.1464830517773706e127
1. Initial program 6.2
  \[\]
2. Using strategy rm
3. Applied clear-num6.4
  \[\leadsto \]
if 1.1464830517773706e127 < b_2
1. Initial program 53.1
  \[\]
2. Using strategy rm
3. Applied flip--63.7
  \[\leadsto \]
4. Simplified62.7
  \[\leadsto \]
5. Simplified62.7
  \[\leadsto \]
6. Taylor expanded around 0 2.8
  \[\leadsto \]
7. Simplified2.8
  \[\leadsto \]
Recombined 4 regimes into one program.
Final simplification6.9
\[\leadsto \]

Reproduce

herbie shell --seed 2020181 
(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))