Average Error: 0.0 → 0.0
Time: 26.1s
Precision: 64
Internal Precision: 128
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \left(\left(2 - \frac{2}{1 + t}\right) \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) + 2}\]

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{1 - \frac{1}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.0

    \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right)} \cdot \left(2 - \frac{2}{1 + t}\right)}\]

  5. Applied associate-*l*0.0

    \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \left(2 - \frac{2}{1 + t}\right)\right)}}\]

  6. Final simplification0.0

    \[\leadsto 1 - \frac{1}{\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \left(\left(2 - \frac{2}{1 + t}\right) \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) + 2}\]

Reproduce

herbie shell --seed 2019010 
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))

Details

Time bar (total: 25.6s)Debug log

sample60.0ms

Algorithm
intervals

simplify418.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
418.0ms
(- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))))))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

2.3b
(- 2 (/ 2 (+ 1 t)))
2.3b
(- 2 (/ 2 (+ 1 t)))
0.0b
(* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))
0.0b
(/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
10×pow1
8×flip--
8×flip3--
7×add-cube-cbrt
7×add-exp-log
7×*-un-lft-identity
7×add-sqr-sqrt
6×add-cbrt-cube
4×add-log-exp
4×sub-neg
4×frac-times
3×associate-*l*
3×associate-*r*
3×associate-/r*
2×associate-/r/
2×associate-*r/
2×associate-*l/
2×distribute-lft-in
2×distribute-rgt-in
1×inv-pow
1×pow-flip
1×div-inv
1×flip-+
1×cbrt-unprod
1×pow-prod-up
1×*-commutative
1×prod-exp
1×pow-prod-down
1×pow-plus
1×flip3-+
1×frac-2neg
1×clear-num
1×rec-exp
1×pow2
Counts
4 → 72
Calls
4 calls:
Slowest
4.0ms
(* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))
1.0ms
(/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))))
1.0ms
(- 2 (/ 2 (+ 1 t)))
0.0ms
(- 2 (/ 2 (+ 1 t)))

series61.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
31.0ms
(/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))))
12.0ms
(* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))
10.0ms
(- 2 (/ 2 (+ 1 t)))
9.0ms
(- 2 (/ 2 (+ 1 t)))

simplify7.7s

Counts
45 → 84
Calls
45 calls:
Slowest
687.0ms
(sqrt (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))))
593.0ms
(* (- (pow 2 3) (pow (/ 2 (+ 1 t)) 3)) (- (pow 2 3) (pow (/ 2 (+ 1 t)) 3)))
581.0ms
(* (- (pow 2 3) (pow (/ 2 (+ 1 t)) 3)) (- 2 (/ 2 (+ 1 t))))
545.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
530.0ms
(* (+ (* 2 2) (+ (* (/ 2 (+ 1 t)) (/ 2 (+ 1 t))) (* 2 (/ 2 (+ 1 t))))) (+ (* 2 2) (+ (* (/ 2 (+ 1 t)) (/ 2 (+ 1 t))) (* 2 (/ 2 (+ 1 t))))))

prune751.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize37.0ms

Local error

Found 4 expressions with local error:

2.3b
(- 2 (/ 2 (+ 1 t)))
2.3b
(- 2 (/ 2 (+ 1 t)))
2.3b
(- 2 (/ 2 (+ 1 t)))
2.3b
(- 2 (/ 2 (+ 1 t)))

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×flip--
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×flip3--
4×*-un-lft-identity
4×sub-neg
4×pow1
4×add-sqr-sqrt
Counts
4 → 44
Calls
4 calls:
Slowest
1.0ms
(- 2 (/ 2 (+ 1 t)))
1.0ms
(- 2 (/ 2 (+ 1 t)))
1.0ms
(- 2 (/ 2 (+ 1 t)))
1.0ms
(- 2 (/ 2 (+ 1 t)))

series61.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
16.0ms
(- 2 (/ 2 (+ 1 t)))
16.0ms
(- 2 (/ 2 (+ 1 t)))
15.0ms
(- 2 (/ 2 (+ 1 t)))
14.0ms
(- 2 (/ 2 (+ 1 t)))

simplify2.7s

Counts
12 → 56
Calls
12 calls:
Slowest
481.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
456.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
453.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
393.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
147.0ms
(- (+ (* 2 (/ 1 (pow t 2))) 2) (* 2 (/ 1 t)))

prune657.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize63.0ms

Local error

Found 4 expressions with local error:

2.3b
(+ (/ 4 (+ t 1)) -4)
2.3b
(- 2 (/ 2 (+ 1 t)))
2.3b
(- 2 (/ 2 (+ 1 t)))
2.3b
(- 2 (/ 2 (+ t 1)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
3×flip--
3×flip3--
3×sub-neg
1×flip-+
1×flip3-+
1×+-commutative
Counts
4 → 43
Calls
4 calls:
Slowest
5.0ms
(+ (/ 4 (+ t 1)) -4)
0.0ms
(- 2 (/ 2 (+ 1 t)))
0.0ms
(- 2 (/ 2 (+ 1 t)))
0.0ms
(- 2 (/ 2 (+ t 1)))

series46.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
15.0ms
(- 2 (/ 2 (+ t 1)))
11.0ms
(- 2 (/ 2 (+ 1 t)))
10.0ms
(+ (/ 4 (+ t 1)) -4)
9.0ms
(- 2 (/ 2 (+ 1 t)))

simplify2.9s

Counts
12 → 55
Calls
12 calls:
Slowest
526.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
476.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
373.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
337.0ms
(- (* 4 (pow t 2)) (+ (* 4 t) (* 4 (pow t 3))))
214.0ms
(- (* 4 (/ 1 t)) (+ (* 4 (/ 1 (pow t 2))) 4))

prune1.0s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

2.3b
(- 2 (/ 2 (+ 1 t)))
2.3b
(- 2 (/ 2 (+ 1 t)))
1.0b
(log (exp (/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))))))
0.5b
(exp (/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t)))))))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
12×exp-prod
7×add-cube-cbrt
7×log-pow
7×*-un-lft-identity
7×add-sqr-sqrt
5×add-log-exp
5×pow1
4×associate-/r/
4×add-exp-log
4×add-cbrt-cube
3×log-prod
2×flip--
2×div-inv
2×flip-+
2×flip3--
2×flip3-+
2×sub-neg
1×rem-exp-log
1×rem-log-exp
Counts
4 → 54
Calls
4 calls:
Slowest
2.0ms
(log (exp (/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))))))
2.0ms
(exp (/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t)))))))
1.0ms
(- 2 (/ 2 (+ 1 t)))
0.0ms
(- 2 (/ 2 (+ 1 t)))

series115.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
58.0ms
(exp (/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t)))))))
32.0ms
(log (exp (/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))))))
15.0ms
(- 2 (/ 2 (+ 1 t)))
9.0ms
(- 2 (/ 2 (+ 1 t)))

simplify6.3s

Counts
28 → 66
Calls
28 calls:
Slowest
506.0ms
(- (+ (exp 1/2) (* 2 (* (pow t 3) (exp 1/2)))) (* (pow t 2) (exp 1/2)))
491.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
430.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
399.0ms
(/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))))
379.0ms
(exp (* (cbrt (/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))))) (cbrt (/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t)))))))))

prune655.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes23.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch2.0ms

end0.0ms

sample1.9s

Algorithm
intervals