Average Error: 34.9 → 7.1

Time: 5.4s

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

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

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 < -9.8297983381069687e64
1. Initial program 41.4
  \[\]
2. Taylor expanded around -inf 5.1
  \[\leadsto \]
3. Simplified5.1
  \[\leadsto \]
if -9.8297983381069687e64 < b < -1.0616583834882925e-277
1. Initial program 9.9
  \[\]
2. Using strategy rm
3. Applied div-inv9.9
  \[\leadsto \]
if -1.0616583834882925e-277 < b < 4.36830952149196385e74
1. Initial program 31.0
  \[\]
2. Using strategy rm
3. Applied flip-+31.0
  \[\leadsto \]
4. Simplified16.7
  \[\leadsto \]
5. Simplified16.6
  \[\leadsto \]
6. Using strategy rm
7. Applied *-un-lft-identity16.6
  \[\leadsto \]
8. Applied times-frac16.6
  \[\leadsto \]
9. Applied times-frac16.5
  \[\leadsto \]
10. Simplified16.5
  \[\leadsto \]
11. Simplified9.4
  \[\leadsto \]
if 4.36830952149196385e74 < b
1. Initial program 58.1
  \[\]
2. Using strategy rm
3. Applied flip-+58.1
  \[\leadsto \]
4. Simplified30.7
  \[\leadsto \]
5. Simplified30.7
  \[\leadsto \]
6. Using strategy rm
7. Applied *-un-lft-identity30.7
  \[\leadsto \]
8. Applied times-frac30.7
  \[\leadsto \]
9. Applied times-frac30.6
  \[\leadsto \]
10. Simplified30.6
  \[\leadsto \]
11. Simplified28.0
  \[\leadsto \]
12. Taylor expanded around inf 6.5
  \[\leadsto \]
13. Simplified3.4
  \[\leadsto \]
Recombined 4 regimes into one program.
Final simplification7.1
\[\leadsto \]

Reproduce

herbie shell --seed 2020180 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))