


Bits error versus t



Bits error versus l



Bits error versus k
Results
if t < -5.965711922169134e+21 or 1.0053774240332047e-91 < t Initial program 22.8
rmApplied cube-mult22.8
Applied times-frac16.1
Applied associate-*l*14.2
rmApplied associate-/l*8.6
rmApplied associate-*l/7.4
Applied associate-*r/7.4
Applied associate-*l/4.5
Applied associate-*l/3.9
Applied associate-/r/3.7
rmApplied div-inv3.7
Applied associate-*l*3.6
if -5.965711922169134e+21 < t < 1.0053774240332047e-91Initial program 51.2
rmApplied cube-mult51.2
Applied times-frac43.7
Applied associate-*l*42.4
rmApplied associate-/l*36.7
rmApplied associate-*l/37.1
Applied associate-*r/37.1
Applied associate-*l/37.8
Applied associate-*l/34.7
Applied associate-/r/34.7
Taylor expanded around -inf 21.1
Simplified18.8
Final simplification8.7
herbie shell --seed 2019022
(FPCore (t l k)
:name "Toniolo and Linder, Equation (10+)"
(/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))
Time bar (total: 1.8m)Debug log
| 176.0ms | 63× | body | 1280 | valid |
| 73.0ms | 204× | body | 80 | valid |
| 69.0ms | 46× | body | 640 | valid |
| 66.0ms | 35× | body | 320 | valid |
| 8.0ms | 11× | body | 160 | valid |
| 149.0ms | (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))) |
1 alts after pruning (1 fresh and 0 done)
Merged error: 32.4b
Found 4 expressions with local error:
| 17.3b | (/ (pow t 3) (* l l)) |
| 12.2b | (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 4.1b | (* (/ (pow t 3) (* l l)) (sin k)) |
| 3.9b | (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) |
| 44× | add-exp-log |
| 34× | add-cbrt-cube |
| 22× | prod-exp |
| 20× | pow1 |
| 16× | cbrt-unprod |
| 16× | associate-*l/ |
| 11× | add-cube-cbrt |
| 11× | associate-*l* |
| 11× | *-un-lft-identity |
| 11× | add-sqr-sqrt |
| 10× | pow-prod-down |
| 10× | times-frac |
| 10× | frac-times |
| 9× | associate-*r* |
| 8× | tan-quot |
| 8× | div-exp |
| 6× | associate-*r/ |
| 5× | associate-/l* |
| 4× | add-log-exp |
| 4× | flip-+ |
| 4× | flip3-+ |
| 4× | cbrt-undiv |
| 4× | pow-to-exp |
| 3× | *-commutative |
| 3× | unpow3 |
| 3× | cube-mult |
| 2× | div-inv |
| 1× | distribute-lft-in |
| 1× | frac-2neg |
| 1× | associate-/r* |
| 1× | clear-num |
| 1× | distribute-rgt-in |
| 53.0ms | (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 23.0ms | (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) |
| 14.0ms | (* (/ (pow t 3) (* l l)) (sin k)) |
| 3.0ms | (/ (pow t 3) (* l l)) |
| 394.0ms | (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 143.0ms | (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) |
| 67.0ms | (* (/ (pow t 3) (* l l)) (sin k)) |
| 27.0ms | (/ (pow t 3) (* l l)) |
| 943.0ms | (* (* (* (/ (* (* (pow t 3) (pow t 3)) (pow t 3)) (* (* (* l l) (* l l)) (* l l))) (* (* (sin k) (sin k)) (sin k))) (* (* (tan k) (tan k)) (tan k))) (* (* (+ (+ 1 (pow (/ k t) 2)) 1) (+ (+ 1 (pow (/ k t) 2)) 1)) (+ (+ 1 (pow (/ k t) 2)) 1))) |
| 702.0ms | (* (* l l) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1)))) |
| 664.0ms | (* (* (* (pow t 3) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 637.0ms | (* (* l l) (- (+ 1 (pow (/ k t) 2)) 1)) |
| 611.0ms | (- (+ 1 (pow (/ k t) 2)) 1) |
13 alts after pruning (13 fresh and 0 done)
Merged error: 14.8b
Found 4 expressions with local error:
| 12.2b | (* (* (* (/ t l) (* (/ (* t t) l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 7.2b | (/ (* t t) l) |
| 6.8b | (* (/ (* t t) l) (sin k)) |
| 3.9b | (* (* (/ t l) (* (/ (* t t) l) (sin k))) (tan k)) |
| 44× | associate-*l/ |
| 34× | frac-times |
| 29× | add-exp-log |
| 29× | add-cbrt-cube |
| 29× | pow1 |
| 17× | cbrt-unprod |
| 17× | prod-exp |
| 17× | pow-prod-down |
| 16× | tan-quot |
| 14× | associate-*r/ |
| 9× | associate-*r* |
| 8× | flip-+ |
| 8× | add-cube-cbrt |
| 8× | flip3-+ |
| 8× | *-un-lft-identity |
| 8× | add-sqr-sqrt |
| 6× | associate-*l* |
| 4× | add-log-exp |
| 3× | *-commutative |
| 2× | div-inv |
| 1× | associate-/l* |
| 1× | distribute-lft-in |
| 1× | frac-2neg |
| 1× | clear-num |
| 1× | distribute-rgt-in |
| 79.0ms | (* (* (* (/ t l) (* (/ (* t t) l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 41.0ms | (* (* (/ t l) (* (/ (* t t) l) (sin k))) (tan k)) |
| 8.0ms | (* (/ (* t t) l) (sin k)) |
| 4.0ms | (/ (* t t) l) |
| 333.0ms | (* (* (* (/ t l) (* (/ (* t t) l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 175.0ms | (* (* (/ t l) (* (/ (* t t) l) (sin k))) (tan k)) |
| 76.0ms | (* (/ (* t t) l) (sin k)) |
| 16.0ms | (/ (* t t) l) |
| 967.0ms | (* (* l (cos k)) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1)))) |
| 659.0ms | (- (+ 1 (pow (/ k t) 2)) 1) |
| 649.0ms | (* (* l l) (- (+ 1 (pow (/ k t) 2)) 1)) |
| 622.0ms | (* (* l l) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1)))) |
| 593.0ms | (* (* (* (/ t l) (* (* t t) (sin k))) (tan k)) (- (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (* 1 1))) |
12 alts after pruning (12 fresh and 0 done)
Merged error: 8.5b
Found 4 expressions with local error:
| 12.2b | (* (* (* (/ t l) (* (/ t (/ l t)) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 6.8b | (* (/ t (/ l t)) (sin k)) |
| 3.9b | (* (* (/ t l) (* (/ t (/ l t)) (sin k))) (tan k)) |
| 0.2b | (/ 2 (* (* (* (/ t l) (* (/ t (/ l t)) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))) |
| 78× | associate-*l/ |
| 63× | frac-times |
| 29× | add-exp-log |
| 29× | add-cbrt-cube |
| 29× | pow1 |
| 28× | tan-quot |
| 25× | associate-*r/ |
| 24× | associate-/r/ |
| 17× | cbrt-unprod |
| 17× | prod-exp |
| 17× | pow-prod-down |
| 16× | flip-+ |
| 16× | flip3-+ |
| 9× | associate-*r* |
| 8× | add-cube-cbrt |
| 8× | *-un-lft-identity |
| 8× | add-sqr-sqrt |
| 7× | associate-*l* |
| 4× | add-log-exp |
| 3× | *-commutative |
| 2× | div-inv |
| 1× | distribute-lft-in |
| 1× | frac-2neg |
| 1× | associate-/r* |
| 1× | clear-num |
| 1× | distribute-rgt-in |
| 94.0ms | (* (* (* (/ t l) (* (/ t (/ l t)) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 36.0ms | (* (* (/ t l) (* (/ t (/ l t)) (sin k))) (tan k)) |
| 14.0ms | (/ 2 (* (* (* (/ t l) (* (/ t (/ l t)) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))) |
| 7.0ms | (* (/ t (/ l t)) (sin k)) |
| 378.0ms | (* (* (* (/ t l) (* (/ t (/ l t)) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 205.0ms | (/ 2 (* (* (* (/ t l) (* (/ t (/ l t)) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))) |
| 174.0ms | (* (* (/ t l) (* (/ t (/ l t)) (sin k))) (tan k)) |
| 85.0ms | (* (/ t (/ l t)) (sin k)) |
| 792.0ms | (* (* (/ l t) (cos k)) (- (+ 1 (pow (/ k t) 2)) 1)) |
| 773.0ms | (/ (* (pow t 3) (pow (sin k) 2)) (* (pow l 2) (cos k))) |
| 691.0ms | (* (/ l t) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1)))) |
| 678.0ms | (* (/ l t) (- (+ 1 (pow (/ k t) 2)) 1)) |
| 670.0ms | (- (+ 1 (pow (/ k t) 2)) 1) |
14 alts after pruning (14 fresh and 0 done)
Merged error: 5.7b
Found 4 expressions with local error:
| 11.0b | (* (* (* (/ t l) (* t (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 4.6b | (* (/ 2 (* (* (* (/ t l) (* t (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))) (/ l t)) |
| 2.3b | (* (* (/ t l) (* t (sin k))) (tan k)) |
| 2.3b | (* (/ t l) (* t (sin k))) |
| 29× | associate-*l/ |
| 22× | add-exp-log |
| 22× | add-cbrt-cube |
| 22× | pow1 |
| 21× | associate-*l* |
| 20× | frac-times |
| 14× | tan-quot |
| 12× | associate-*r/ |
| 11× | cbrt-unprod |
| 11× | associate-/r/ |
| 11× | prod-exp |
| 11× | pow-prod-down |
| 11× | associate-*r* |
| 9× | add-cube-cbrt |
| 9× | *-un-lft-identity |
| 9× | add-sqr-sqrt |
| 8× | flip-+ |
| 8× | flip3-+ |
| 4× | add-log-exp |
| 4× | *-commutative |
| 3× | div-inv |
| 1× | distribute-lft-in |
| 1× | distribute-rgt-in |
| 66.0ms | (* (* (* (/ t l) (* t (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 25.0ms | (* (/ 2 (* (* (* (/ t l) (* t (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))) (/ l t)) |
| 15.0ms | (* (* (/ t l) (* t (sin k))) (tan k)) |
| 10.0ms | (* (/ t l) (* t (sin k))) |
| 366.0ms | (* (* (* (/ t l) (* t (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 237.0ms | (* (/ 2 (* (* (* (/ t l) (* t (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))) (/ l t)) |
| 175.0ms | (* (* (/ t l) (* t (sin k))) (tan k)) |
| 93.0ms | (* (/ t l) (* t (sin k))) |
| 662.0ms | (* (cos k) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1)))) |
| 601.0ms | (- (+ 1 (pow (/ k t) 2)) 1) |
| 594.0ms | (* (* (* t (* t (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
| 593.0ms | (* (* l (cos k)) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1)))) |
| 583.0ms | (* (* (* t (* t (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)) |
16 alts after pruning (16 fresh and 0 done)
Merged error: 4.1b
68% (2.5b remaining)
Error of 8.7b against oracle of 6.2b and baseline of 14.0b
| 6.3s | 2022× | body | 1280 | valid |
| 3.6s | 1755× | body | 640 | valid |
| 3.0s | 5954× | body | 80 | valid |
| 1.8s | 852× | body | 320 | valid |
| 358.0ms | 394× | body | 160 | valid |