Average Error: 37.0 → 0.6
Time: 42.8s
Precision: 64
Internal Precision: 128
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[(\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)}\right) + \left(\frac{(\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x}\right) + 1)_* \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)}\right))_* + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)} - \frac{\sin x}{\cos x}\right)\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
Target15.0
Herbie0.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.0

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

  3. Applied tan-sum21.9

    \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
  4. Using strategy rm

  5. Applied div-inv21.9

    \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]

  6. Applied fma-neg21.9

    \[\leadsto \color{blue}{(\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right))_*}\]
  7. Using strategy rm

  8. Applied flip--21.9

    \[\leadsto (\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}}}\right) + \left(-\tan x\right))_*\]

  9. Applied associate-/r/21.9

    \[\leadsto (\left(\tan x + \tan \varepsilon\right) \cdot \color{blue}{\left(\frac{1}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right)} + \left(-\tan x\right))_*\]

  10. Simplified21.9

    \[\leadsto (\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \color{blue}{(\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*}\right) + \left(-\tan x\right))_*\]

  11. Taylor expanded around -inf 22.1

    \[\leadsto \color{blue}{\left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)} + \left(\frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\cos x \cdot \left({\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)\right)} + \left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)\right)} + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)}\right)\right)\right) - \frac{\sin x}{\cos x}}\]

  12. Simplified0.6

    \[\leadsto \color{blue}{(\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)}\right) + \left(\frac{(\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x}\right) + 1)_* \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)}\right))_* + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)} - \frac{\sin x}{\cos x}\right)}\]

  13. Final simplification0.6

    \[\leadsto (\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)}\right) + \left(\frac{(\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x}\right) + 1)_* \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)}\right))_* + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)} - \frac{\sin x}{\cos x}\right)\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))

Details

Time bar (total: 41.9s)Debug log

sample450.0ms

Algorithm
intervals
Results
256×(pre true 80)
115×(body real 1280)
60×(body real 640)
27×(body real 2560)
25×(body real 80)
17×(body real 320)
12×(body real 160)

simplify8.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
8.0ms
(- (tan (+ x eps)) (tan x))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 38.3b

localize66.0ms

Local error

Found 2 expressions with local error:

3.9b
(tan (+ x eps))
1.6b
(- (tan (+ x eps)) (tan x))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
11×*-un-lft-identity
11×add-sqr-sqrt
9×add-cube-cbrt
9×prod-diff
4×add-log-exp
4×tan-quot
3×fma-neg
2×log1p-expm1-u
2×frac-sub
2×add-exp-log
2×add-cbrt-cube
2×tan-sum
2×pow1
2×expm1-log1p-u
1×difference-of-squares
1×distribute-lft-out--
1×flip--
1×diff-log
1×flip3--
1×sub-neg
Counts
2 → 41
Calls
2 calls:
Slowest
11.0ms
(- (tan (+ x eps)) (tan x))
3.0ms
(tan (+ x eps))

series201.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
128.0ms
(- (tan (+ x eps)) (tan x))
72.0ms
(tan (+ x eps))

simplify1.3s

Counts
35 → 47
Calls
35 calls:
Slowest
296.0ms
(- (* (+ (tan x) (tan eps)) (cos x)) (* (- 1 (* (tan x) (tan eps))) (sin x)))
268.0ms
(+ (* x (pow eps 2)) (+ eps (* (pow x 2) eps)))
155.0ms
(* (- 1 (* (tan x) (tan eps))) (cos x))
47.0ms
(+ x (+ (* 1/3 (pow x 3)) eps))
42.0ms
(fma (sqrt (tan (+ x eps))) (sqrt (tan (+ x eps))) (- (* (sqrt (tan x)) (sqrt (tan x)))))

prune466.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 17.2b

localize36.0ms

Local error

Found 4 expressions with local error:

2.9b
(- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x))
0.2b
(* (tan x) (tan eps))
0.1b
(+ (tan x) (tan eps))
0.1b
(/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps))))

rewrite103.0ms

Algorithm
rewrite-expression-head
Rules
102×*-un-lft-identity
71×add-sqr-sqrt
66×add-cube-cbrt
61×times-frac
54×prod-diff
18×fma-neg
17×distribute-lft-out
10×associate-/r/
9×tan-quot
8×add-log-exp
8×add-exp-log
8×add-cbrt-cube
6×flip--
6×flip3--
6×pow1
5×div-inv
4×associate-/l*
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-/l/
3×associate-*l*
3×associate-*r*
3×fma-def
3×associate-/r*
2×difference-of-squares
2×flip-+
2×frac-add
2×flip3-+
1×distribute-lft-out--
1×cbrt-unprod
1×frac-sub
1×*-commutative
1×associate-*r/
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×div-exp
1×diff-log
1×frac-2neg
1×sub-neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
1×frac-times
Counts
4 → 168
Calls
4 calls:
Slowest
63.0ms
(- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x))
20.0ms
(/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps))))
8.0ms
(* (tan x) (tan eps))
8.0ms
(+ (tan x) (tan eps))

series913.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
397.0ms
(- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x))
271.0ms
(/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps))))
127.0ms
(* (tan x) (tan eps))
118.0ms
(+ (tan x) (tan eps))

simplify15.2s

Counts
199 → 180
Calls
199 calls:
Slowest
510.0ms
(fma (/ (+ (tan x) (tan eps)) (- (* 1 1) (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))) (+ 1 (* (tan x) (tan eps))) (- (* (cbrt (tan x)) (* (cbrt (tan x)) (cbrt (tan x))))))
490.0ms
(fma (/ 1 (* (cbrt (- 1 (* (tan x) (tan eps)))) (cbrt (- 1 (* (tan x) (tan eps)))))) (/ (+ (tan x) (tan eps)) (cbrt (- 1 (* (tan x) (tan eps))))) (- (* (sqrt (tan x)) (sqrt (tan x)))))
485.0ms
(fma (/ 1 (* (cbrt (- 1 (* (tan x) (tan eps)))) (cbrt (- 1 (* (tan x) (tan eps)))))) (/ (+ (tan x) (tan eps)) (cbrt (- 1 (* (tan x) (tan eps))))) (- (* (tan x) 1)))
472.0ms
(fma (/ 1 (* (cbrt (- 1 (* (tan x) (tan eps)))) (cbrt (- 1 (* (tan x) (tan eps)))))) (/ (+ (tan x) (tan eps)) (cbrt (- 1 (* (tan x) (tan eps))))) (- (* (sqrt (tan x)) (sqrt (tan x)))))
465.0ms
(fma (/ 1 (* (cbrt (- 1 (* (tan x) (tan eps)))) (cbrt (- 1 (* (tan x) (tan eps)))))) (/ (+ (tan x) (tan eps)) (cbrt (- 1 (* (tan x) (tan eps))))) (- (* (tan x) 1)))

prune2.2s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 17.0b

localize10.0ms

Local error

Found 4 expressions with local error:

2.9b
(fma (+ (tan x) (tan eps)) (/ 1 (- 1 (* (tan x) (tan eps)))) (- (tan x)))
0.2b
(* (tan x) (tan eps))
0.1b
(+ (tan x) (tan eps))
0.1b
(/ 1 (- 1 (* (tan x) (tan eps))))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
10×*-un-lft-identity
8×add-cube-cbrt
8×add-sqr-sqrt
7×add-exp-log
7×pow1
6×add-log-exp
6×tan-quot
6×add-cbrt-cube
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-*l*
3×associate-*r*
3×fma-def
3×associate-/r*
2×associate-/r/
1×inv-pow
1×distribute-lft-out
1×flip--
1×pow-flip
1×div-inv
1×fma-udef
1×flip-+
1×cbrt-unprod
1×*-commutative
1×associate-*r/
1×prod-exp
1×frac-add
1×associate-*l/
1×pow-prod-down
1×flip3--
1×flip3-+
1×frac-2neg
1×sum-log
1×clear-num
1×rec-exp
1×+-commutative
1×frac-times
Counts
4 → 70
Calls
4 calls:
Slowest
4.0ms
(* (tan x) (tan eps))
4.0ms
(+ (tan x) (tan eps))
1.0ms
(/ 1 (- 1 (* (tan x) (tan eps))))
0.0ms
(fma (+ (tan x) (tan eps)) (/ 1 (- 1 (* (tan x) (tan eps)))) (- (tan x)))

series800.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
363.0ms
(fma (+ (tan x) (tan eps)) (/ 1 (- 1 (* (tan x) (tan eps)))) (- (tan x)))
172.0ms
(/ 1 (- 1 (* (tan x) (tan eps))))
133.0ms
(* (tan x) (tan eps))
131.0ms
(+ (tan x) (tan eps))

simplify1.8s

Counts
36 → 82
Calls
36 calls:
Slowest
283.0ms
(- (+ (/ (sin eps) (* (cos eps) (- 1 (/ (* (sin x) (sin eps)) (* (cos x) (cos eps)))))) (/ (sin x) (* (cos x) (- 1 (/ (* (sin x) (sin eps)) (* (cos x) (cos eps))))))) (/ (sin x) (cos x)))
227.0ms
(- (+ (/ (sin eps) (* (cos eps) (- 1 (/ (* (sin x) (sin eps)) (* (cos x) (cos eps)))))) (/ (sin x) (* (cos x) (- 1 (/ (* (sin x) (sin eps)) (* (cos x) (cos eps))))))) (/ (sin x) (cos x)))
221.0ms
(+ (* x (pow eps 2)) (+ eps (* (pow x 2) eps)))
209.0ms
(+ (* 1/3 (* (pow x 3) eps)) (+ (* 1/3 (* x (pow eps 3))) (* x eps)))
194.0ms
(+ (* 1 1) (+ (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* 1 (* (tan x) (tan eps)))))

prune685.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 17.0b

localize19.0ms

Local error

Found 4 expressions with local error:

2.9b
(fma (+ (tan x) (tan eps)) (* (/ 1 (- (* 1 1) (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))) (fma (tan eps) (tan x) 1)) (- (tan x)))
0.3b
(* (* (tan x) (tan eps)) (* (tan x) (tan eps)))
0.2b
(* (tan x) (tan eps))
0.2b
(* (tan x) (tan eps))

rewrite29.0ms

Algorithm
rewrite-expression-head
Rules
40×tan-quot
35×pow1
20×add-exp-log
20×add-cbrt-cube
19×frac-times
15×pow-prod-down
13×associate-*r/
13×associate-*l/
10×cbrt-unprod
10×prod-exp
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×associate-*l*
7×associate-*r*
4×add-log-exp
4×log1p-expm1-u
4×pow-prod-up
4×expm1-log1p-u
3×*-commutative
2×pow-plus
1×fma-udef
1×pow2
Counts
4 → 100
Calls
4 calls:
Slowest
17.0ms
(* (* (tan x) (tan eps)) (* (tan x) (tan eps)))
4.0ms
(* (tan x) (tan eps))
4.0ms
(* (tan x) (tan eps))
0.0ms
(fma (+ (tan x) (tan eps)) (* (/ 1 (- (* 1 1) (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))) (fma (tan eps) (tan x) 1)) (- (tan x)))

series787.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
494.0ms
(fma (+ (tan x) (tan eps)) (* (/ 1 (- (* 1 1) (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))) (fma (tan eps) (tan x) 1)) (- (tan x)))
127.0ms
(* (* (tan x) (tan eps)) (* (tan x) (tan eps)))
86.0ms
(* (tan x) (tan eps))
81.0ms
(* (tan x) (tan eps))

simplify3.7s

Counts
84 → 112
Calls
84 calls:
Slowest
416.0ms
(* (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan eps) (tan eps)) (tan eps))) (* (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* (tan x) (tan eps))))
409.0ms
(* (* (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* (tan x) (tan eps))) (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan eps) (tan eps)) (tan eps))))
376.0ms
(* (* (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* (tan x) (tan eps))) (* (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* (tan x) (tan eps))))
258.0ms
(/ (* (pow (sin x) 2) (pow (sin eps) 2)) (* (pow (cos x) 2) (pow (cos eps) 2)))
258.0ms
(/ (* (pow (sin x) 2) (pow (sin eps) 2)) (* (pow (cos x) 2) (pow (cos eps) 2)))

prune1.3s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.2b

regimes106.0ms

Accuracy

0% (0.3b remaining)

Error of 0.6b against oracle of 0.3b and baseline of 0.6b

bsearch3.0ms

end0.0ms

sample11.6s

Algorithm
intervals
Results
8000×(pre true 80)
3314×(body real 1280)
1793×(body real 640)
1090×(body real 80)
769×(body real 320)
664×(body real 2560)
370×(body real 160)