Average Error: 47.7 → 8.0

Time: 33.8s

Precision: binary64

\[\]

↓

\[\]

Error

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

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

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

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

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

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

Derivation

Split input into 2 regimes
if k < -1.1653754517657999e-138 or 2.2473460020406468e-14 < k
1. Initial program 45.6
  \[\]
2. Simplified35.7
  \[\leadsto \]
3. Taylor expanded around inf 52.9
  \[\leadsto \]
4. Simplified14.1
  \[\leadsto \]
5. Using strategy rm
6. Applied sqr-pow14.1
  \[\leadsto \]
7. Applied unpow-prod-down14.1
  \[\leadsto \]
8. Applied associate-*l*10.5
  \[\leadsto \]
9. Simplified10.5
  \[\leadsto \]
10. Using strategy rm
11. Applied unpow-prod-down10.5
  \[\leadsto \]
12. Applied associate-*l*5.8
  \[\leadsto \]
13. Simplified6.5
  \[\leadsto \]
14. Using strategy rm
15. Applied associate-*r*5.8
  \[\leadsto \]
if -1.1653754517657999e-138 < k < 2.2473460020406468e-14
1. Initial program 61.9
  \[\]
2. Simplified57.7
  \[\leadsto \]
3. Taylor expanded around inf 47.8
  \[\leadsto \]
4. Simplified27.8
  \[\leadsto \]
5. Using strategy rm
6. Applied sqr-pow27.9
  \[\leadsto \]
7. Applied unpow-prod-down27.9
  \[\leadsto \]
8. Applied associate-*l*27.9
  \[\leadsto \]
9. Simplified27.9
  \[\leadsto \]
10. Using strategy rm
11. Applied unpow227.9
  \[\leadsto \]
12. Applied *-un-lft-identity27.9
  \[\leadsto \]
13. Applied times-frac28.0
  \[\leadsto \]
14. Applied associate-*l*23.1
  \[\leadsto \]
15. Simplified23.1
  \[\leadsto \]
Recombined 2 regimes into one program.
Final simplification8.0
\[\leadsto \]

Reproduce

herbie shell --seed 2020181 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))