Average Error: 13.3 → 0.3
Time: 31.5s
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\log \left(e^{x} \cdot e^{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a}\right)\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.3

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  2. Using strategy rm

  3. Applied tan-sum0.2

    \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right)\]
  4. Using strategy rm

  5. Applied add-log-exp0.3

    \[\leadsto \color{blue}{\log \left(e^{x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)}\right)}\]
  6. Using strategy rm

  7. Applied exp-sum0.3

    \[\leadsto \log \color{blue}{\left(e^{x} \cdot e^{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a}\right)}\]

  8. Final simplification0.3

    \[\leadsto \log \left(e^{x} \cdot e^{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a}\right)\]

Reproduce

herbie shell --seed 2019010 
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :pre (and (or (== x 0) (<= 0.5884142 x 505.5909)) (or (<= -1.796658e+308 y -9.425585e-310) (<= 1.284938e-309 y 1.751224e+308)) (or (<= -1.776707e+308 z -8.599796e-310) (<= 3.293145e-311 z 1.725154e+308)) (or (<= -1.796658e+308 a -9.425585e-310) (<= 1.284938e-309 a 1.751224e+308)))
  (+ x (- (tan (+ y z)) (tan a))))

Details

Time bar (total: 30.7s)Debug log

sample379.0ms

Algorithm
intervals

simplify36.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
36.0ms
(+ x (- (tan (+ y z)) (tan a)))

prune36.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 10.7b

localize27.0ms

Local error

Found 3 expressions with local error:

3.0b
(tan (+ y z))
0.2b
(- (tan (+ y z)) (tan a))
0.1b
(+ x (- (tan (+ y z)) (tan a)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
5×add-log-exp
5×*-un-lft-identity
5×add-sqr-sqrt
4×tan-quot
3×add-cube-cbrt
3×add-exp-log
3×add-cbrt-cube
3×pow1
2×frac-sub
2×sub-neg
2×associate-+r+
2×tan-sum
1×difference-of-squares
1×distribute-lft-out--
1×flip--
1×flip-+
1×diff-log
1×flip3--
1×flip3-+
1×associate-+r-
1×+-commutative
Counts
3 → 38
Calls
3 calls:
Slowest
5.0ms
(- (tan (+ y z)) (tan a))
3.0ms
(+ x (- (tan (+ y z)) (tan a)))
1.0ms
(tan (+ y z))

series289.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
126.0ms
(+ x (- (tan (+ y z)) (tan a)))
121.0ms
(- (tan (+ y z)) (tan a))
42.0ms
(tan (+ y z))

simplify469.0ms

Counts
21 → 47
Calls
21 calls:
Slowest
198.0ms
(- (* (+ (tan y) (tan z)) (cos a)) (* (- 1 (* (tan y) (tan z))) (sin a)))
94.0ms
(* (- 1 (* (tan y) (tan z))) (cos a))
35.0ms
(- (+ (/ (sin (+ z y)) (cos (+ z y))) x) (/ (sin a) (cos a)))
27.0ms
(+ (* 1/3 (pow y 3)) (+ z y))
25.0ms
(/ (exp (tan (+ y z))) (exp (tan a)))

prune487.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.1b

localize29.0ms

Local error

Found 4 expressions with local error:

0.2b
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
0.2b
(* (tan y) (tan z))
0.1b
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
0.1b
(+ (tan y) (tan z))

rewrite106.0ms

Algorithm
rewrite-expression-head
Rules
11×add-sqr-sqrt
10×*-un-lft-identity
8×add-log-exp
7×tan-quot
6×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×pow1
3×associate-*l*
3×associate-*r*
2×difference-of-squares
2×flip-+
2×flip3-+
2×sub-neg
2×associate-+r+
2×+-commutative
1×distribute-lft-out--
1×distribute-lft-out
1×flip--
1×cbrt-unprod
1×frac-sub
1×*-commutative
1×associate-*r/
1×prod-exp
1×frac-add
1×associate-*l/
1×pow-prod-down
1×diff-log
1×flip3--
1×times-frac
1×sum-log
1×associate-+r-
1×frac-times
Counts
4 → 62
Calls
4 calls:
Slowest
51.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
36.0ms
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
8.0ms
(* (tan y) (tan z))
7.0ms
(+ (tan y) (tan z))

series1.0s

Counts
4 → 12
Calls
4 calls:
Slowest
406.0ms
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
399.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
124.0ms
(+ (tan y) (tan z))
108.0ms
(* (tan y) (tan z))

simplify2.2s

Counts
39 → 74
Calls
39 calls:
Slowest
370.0ms
(+ (* 1/3 (* (pow z 3) y)) (+ (* 1/3 (* z (pow y 3))) (* z y)))
307.0ms
(- (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (+ (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) x)) (/ (sin a) (cos a)))
290.0ms
(- (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z))))))) (/ (sin a) (cos a)))
256.0ms
(- (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z))))))) (/ (sin a) (cos a)))
255.0ms
(- (* (+ (tan y) (tan z)) (cos a)) (* (- 1 (* (tan y) (tan z))) (sin a)))

prune1.1s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.1b

localize17.0ms

Local error

Found 4 expressions with local error:

2.6b
(exp (+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))))
0.2b
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
0.2b
(* (tan y) (tan z))
0.1b
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))

rewrite144.0ms

Algorithm
rewrite-expression-head
Rules
12×add-sqr-sqrt
9×*-un-lft-identity
7×add-log-exp
7×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×pow1
5×tan-quot
3×associate-*l*
3×associate-*r*
3×exp-prod
2×difference-of-squares
2×sub-neg
2×associate-+r-
2×associate-+r+
1×rem-exp-log
1×distribute-lft-out--
1×flip--
1×flip-+
1×cbrt-unprod
1×frac-sub
1×*-commutative
1×associate-*r/
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×exp-sum
1×diff-log
1×exp-diff
1×flip3--
1×times-frac
1×flip3-+
1×+-commutative
1×frac-times
Counts
4 → 62
Calls
4 calls:
Slowest
77.0ms
(exp (+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))))
33.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
29.0ms
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
4.0ms
(* (tan y) (tan z))

series1.4s

Counts
4 → 12
Calls
4 calls:
Slowest
548.0ms
(exp (+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))))
395.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
386.0ms
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
102.0ms
(* (tan y) (tan z))

simplify2.5s

Counts
41 → 74
Calls
41 calls:
Slowest
355.0ms
(exp (- (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (+ (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) x)) (/ (sin a) (cos a))))
276.0ms
(+ (* 1/3 (* (pow z 3) y)) (+ (* 1/3 (* z (pow y 3))) (* z y)))
268.0ms
(exp (- (+ x (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))))) (/ (sin a) (cos a))))
245.0ms
(- (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (+ (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) x)) (/ (sin a) (cos a)))
242.0ms
(- (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (+ (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) x)) (/ (sin a) (cos a)))

prune1.1s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.5b
(exp (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
0.3b
(* (exp x) (exp (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))))
0.2b
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
0.2b
(* (tan y) (tan z))

rewrite262.0ms

Algorithm
rewrite-expression-head
Rules
19×add-sqr-sqrt
13×*-un-lft-identity
9×add-log-exp
9×add-cube-cbrt
8×associate-*r*
8×add-cbrt-cube
8×pow1
6×add-exp-log
6×associate-*l*
6×exp-prod
5×tan-quot
4×difference-of-squares
4×exp-sum
3×sub-neg
2×rem-exp-log
2×distribute-lft-out--
2×cbrt-unprod
2×*-commutative
2×associate-*r/
2×prod-exp
2×pow-prod-down
2×diff-log
2×exp-diff
2×times-frac
1×flip--
1×frac-sub
1×associate-*l/
1×flip3--
1×frac-times
Counts
4 → 74
Calls
4 calls:
Slowest
162.0ms
(* (exp x) (exp (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))))
53.0ms
(exp (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
42.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
4.0ms
(* (tan y) (tan z))

series1.5s

Counts
4 → 12
Calls
4 calls:
Slowest
567.0ms
(* (exp x) (exp (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))))
460.0ms
(exp (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
366.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
91.0ms
(* (tan y) (tan z))

simplify2.6s

Counts
54 → 86
Calls
54 calls:
Slowest
274.0ms
(* (exp x) (exp (- (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z))))))) (/ (sin a) (cos a)))))
235.0ms
(+ (* 1/3 (* (pow z 3) y)) (+ (* 1/3 (* z (pow y 3))) (* z y)))
228.0ms
(exp (- (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z))))))) (/ (sin a) (cos a))))
225.0ms
(- (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z))))))) (/ (sin a) (cos a)))
217.0ms
(exp (- (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z))))))) (/ (sin a) (cos a))))

prune1.3s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

regimes373.0ms

Accuracy

0% (0.2b remaining)

Error of 0.3b against oracle of 0.1b and baseline of 0.3b

bsearch2.0ms

end0.0ms

sample13.2s

Algorithm
intervals