Average Error: 34.2 → 34.2
Time: 32.6s
Precision: 64
Internal Precision: 128
\[\left(\left(\cosh c\right) \bmod \left(\log_* (1 + a)\right)\right)\]
↓
\[\left(\sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{e^{c}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)} \cdot \sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{e^{c}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)}\right) \cdot \sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{\sqrt{e^{c}}} \cdot \sqrt[3]{\sqrt{e^{c}}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)}\]

Error

Bits error versus a

Bits error versus c

Derivation

  1. Initial program 34.2

    \[\left(\left(\cosh c\right) \bmod \left(\log_* (1 + a)\right)\right)\]

  2. Taylor expanded around -inf 34.2

    \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \left(e^{-1 \cdot c} + e^{c}\right)\right)} \bmod \left(\log_* (1 + a)\right)\right)\]

  3. Simplified34.2

    \[\leadsto \left(\color{blue}{\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{1}{2}}{e^{c}}\right))_*\right)} \bmod \left(\log_* (1 + a)\right)\right)\]
  4. Using strategy rm

  5. Applied add-cube-cbrt34.2

    \[\leadsto \left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{1}{2}}{\color{blue}{\left(\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}\right) \cdot \sqrt[3]{e^{c}}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)\]

  6. Applied associate-/r*34.2

    \[\leadsto \left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \color{blue}{\left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{e^{c}}}\right)})_*\right) \bmod \left(\log_* (1 + a)\right)\right)\]
  7. Using strategy rm

  8. Applied add-cube-cbrt34.2

    \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{e^{c}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)} \cdot \sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{e^{c}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)}\right) \cdot \sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{e^{c}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)}}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt34.2

    \[\leadsto \left(\sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{e^{c}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)} \cdot \sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{e^{c}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)}\right) \cdot \sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{\color{blue}{\sqrt{e^{c}} \cdot \sqrt{e^{c}}}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)}\]

  11. Applied cbrt-prod34.2

    \[\leadsto \left(\sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{e^{c}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)} \cdot \sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{e^{c}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)}\right) \cdot \sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\color{blue}{\sqrt[3]{\sqrt{e^{c}}} \cdot \sqrt[3]{\sqrt{e^{c}}}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)}\]

  12. Final simplification34.2

    \[\leadsto \left(\sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{e^{c}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)} \cdot \sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{e^{c}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)}\right) \cdot \sqrt[3]{\left(\left((\frac{1}{2} \cdot \left(e^{c}\right) + \left(\frac{\frac{\frac{1}{2}}{\sqrt[3]{e^{c}} \cdot \sqrt[3]{e^{c}}}}{\sqrt[3]{\sqrt{e^{c}}} \cdot \sqrt[3]{\sqrt{e^{c}}}}\right))_*\right) \bmod \left(\log_* (1 + a)\right)\right)}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (a c)
  :name "Random Jason Timeout Test 004"
  (fmod (cosh c) (log1p a)))

Details

Time bar (total: 29.5s)Debug log

sample774.0ms

Algorithm
intervals

simplify14.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

13.0ms
(fmod (cosh c) (log1p a))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 36.1b

localize38.0ms

Local error

Found 2 expressions with local error:

40.7b
(fmod (cosh c) (log1p a))
0.0b
(cosh c)

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
2 → 19
Calls

2 calls. Slowest were:

1.0ms
(fmod (cosh c) (log1p a))
1.0ms
(cosh c)

series80.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

64.0ms
(cosh c)
15.0ms
(fmod (cosh c) (log1p a))

simplify306.0ms

Counts
6 → 25
Calls

6 calls. Slowest were:

213.0ms
(* 1/2 (+ (exp c) (exp (- c))))
40.0ms
(* 1/2 (+ (exp (* -1 c)) (exp c)))
36.0ms
(+ (* 1/2 (pow c 2)) (+ (* 1/24 (pow c 4)) 1))

prune286.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 35.9b

localize18.0ms

Local error

Found 3 expressions with local error:

40.7b
(fmod (fma 1/2 (exp c) (/ 1/2 (exp c))) (log1p a))
0.0b
(/ 1/2 (exp c))
0.0b
(fma 1/2 (exp c) (/ 1/2 (exp c)))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
3 → 34
Calls

3 calls. Slowest were:

1.0ms
(/ 1/2 (exp c))
0.0ms
(fma 1/2 (exp c) (/ 1/2 (exp c)))
0.0ms
(fmod (fma 1/2 (exp c) (/ 1/2 (exp c))) (log1p a))

series71.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

55.0ms
(fma 1/2 (exp c) (/ 1/2 (exp c)))
12.0ms
(/ 1/2 (exp c))
3.0ms
(fmod (fma 1/2 (exp c) (/ 1/2 (exp c))) (log1p a))

simplify229.0ms

Counts
12 → 43
Calls

12 calls. Slowest were:

107.0ms
(- (+ (* 1/4 (pow c 2)) 1/2) (* 1/2 c))
35.0ms
(+ (* 1/2 (exp c)) (* 1/2 (/ 1 (exp c))))
23.0ms
(+ (* 1/2 (exp c)) (* 1/2 (/ 1 (exp c))))

prune467.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 35.8b

localize27.0ms

Local error

Found 4 expressions with local error:

40.8b
(fmod (fma 1/2 (exp c) (/ (/ 1/2 (* (cbrt (exp c)) (cbrt (exp c)))) (cbrt (exp c)))) (log1p a))
0.5b
(cbrt (exp c))
0.5b
(cbrt (exp c))
0.5b
(cbrt (exp c))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

1.0ms
(cbrt (exp c))
1.0ms
(cbrt (exp c))
1.0ms
(cbrt (exp c))

series373.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

136.0ms
(cbrt (exp c))
117.0ms
(cbrt (exp c))
116.0ms
(cbrt (exp c))
4.0ms
(fmod (fma 1/2 (exp c) (/ (/ 1/2 (* (cbrt (exp c)) (cbrt (exp c)))) (cbrt (exp c)))) (log1p a))

simplify249.0ms

Counts
21 → 60
Calls

21 calls. Slowest were:

66.0ms
(+ (* 1/18 (pow c 2)) (+ (* 1/3 c) 1))
65.0ms
(+ (* 1/18 (pow c 2)) (+ (* 1/3 c) 1))
61.0ms
(+ (* 1/18 (pow c 2)) (+ (* 1/3 c) 1))

prune847.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 35.8b

localize28.0ms

Local error

Found 4 expressions with local error:

40.8b
(fmod (fma 1/2 (exp c) (/ (/ 1/2 (* (cbrt (exp c)) (cbrt (exp c)))) (cbrt (exp c)))) (log1p a))
40.8b
(fmod (fma 1/2 (exp c) (/ (/ 1/2 (* (cbrt (exp c)) (cbrt (exp c)))) (cbrt (exp c)))) (log1p a))
40.8b
(fmod (fma 1/2 (exp c) (/ (/ 1/2 (* (cbrt (exp c)) (cbrt (exp c)))) (cbrt (exp c)))) (log1p a))
0.5b
(cbrt (exp c))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 40
Calls

4 calls. Slowest were:

1.0ms
(cbrt (exp c))
0.0ms
(fmod (fma 1/2 (exp c) (/ (/ 1/2 (* (cbrt (exp c)) (cbrt (exp c)))) (cbrt (exp c)))) (log1p a))
0.0ms
(fmod (fma 1/2 (exp c) (/ (/ 1/2 (* (cbrt (exp c)) (cbrt (exp c)))) (cbrt (exp c)))) (log1p a))

series122.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

105.0ms
(cbrt (exp c))
7.0ms
(fmod (fma 1/2 (exp c) (/ (/ 1/2 (* (cbrt (exp c)) (cbrt (exp c)))) (cbrt (exp c)))) (log1p a))
6.0ms
(fmod (fma 1/2 (exp c) (/ (/ 1/2 (* (cbrt (exp c)) (cbrt (exp c)))) (cbrt (exp c)))) (log1p a))
4.0ms
(fmod (fma 1/2 (exp c) (/ (/ 1/2 (* (cbrt (exp c)) (cbrt (exp c)))) (cbrt (exp c)))) (log1p a))

simplify102.0ms

Counts
15 → 52
Calls

15 calls. Slowest were:

32.0ms
(+ (* 1/18 (pow c 2)) (+ (* 1/3 c) 1))
8.0ms
(fmod (fma 1/2 (exp c) (/ 1/2 (exp c))) (log1p a))
7.0ms
(fmod (fma 1/2 (exp c) (/ 1/2 (exp c))) (log1p a))

prune874.0ms

Pruning

18 alts after pruning (18 fresh and 0 done)

Merged error: 35.7b

regimes825.0ms

Accuracy

0% (0.1b remaining)

Error of 34.2b against oracle of 34.1b and baseline of 34.2b

bsearch2.0ms

end0.0ms

sample23.8s

Algorithm
intervals