Average Error: 37.0 → 14.0
Time: 58.7s
Precision: 64
Internal Precision: 128
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.7531779447900738 \cdot 10^{-40}:\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan x + \tan \varepsilon\right) + \left(\sin x \cdot (\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + -1)_*\right))_*}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \le 2.649457653506729 \cdot 10^{-60}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left((\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(\sin x\right) + -1)_*\right) \cdot \left(\sin x\right) + \left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x\right))_*}{(\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(-\sin x\right) + 1)_* \cdot \cos x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
Target14.7
Herbie14.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.7531779447900738e-40

    1. Initial program 29.6

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

    3. Applied tan-sum3.6

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

    5. Applied tan-quot3.6

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\frac{\sin x}{\cos x}}\]

    6. Applied frac-sub3.6

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

    7. Simplified3.6

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

    if -1.7531779447900738e-40 < eps < 2.649457653506729e-60

    1. Initial program 46.7

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

    3. Applied tan-sum46.7

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

    5. Applied tan-quot46.7

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\cos x}} \cdot \tan \varepsilon} - \tan x\]

    6. Applied associate-*l/46.7

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

    8. Applied add-cube-cbrt47.6

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \color{blue}{\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}}\]

    9. Applied *-un-lft-identity47.6

      \[\leadsto \color{blue}{1 \cdot \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}} - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}\]

    10. Applied prod-diff47.7

      \[\leadsto \color{blue}{(1 \cdot \left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}\right) + \left(-\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right))_* + (\left(-\sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) + \left(\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right))_*}\]

    11. Simplified47.6

      \[\leadsto \color{blue}{\left(\frac{\tan \varepsilon + \tan x}{(\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(-\sin x\right) + 1)_*} - \tan x\right)} + (\left(-\sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) + \left(\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right))_*\]

    12. Simplified46.7

      \[\leadsto \left(\frac{\tan \varepsilon + \tan x}{(\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(-\sin x\right) + 1)_*} - \tan x\right) + \color{blue}{0}\]

    13. Taylor expanded around 0 27.1

      \[\leadsto \color{blue}{\left(x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)\right)} + 0\]

    14. Simplified27.1

      \[\leadsto \color{blue}{(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*} + 0\]

    if 2.649457653506729e-60 < eps

    1. Initial program 30.2

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

    3. Applied tan-sum4.9

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

    5. Applied tan-quot4.9

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\cos x}} \cdot \tan \varepsilon} - \tan x\]

    6. Applied associate-*l/4.9

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

    8. Applied add-cube-cbrt5.2

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \color{blue}{\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}}\]

    9. Applied *-un-lft-identity5.2

      \[\leadsto \color{blue}{1 \cdot \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}} - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}\]

    10. Applied prod-diff5.2

      \[\leadsto \color{blue}{(1 \cdot \left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}\right) + \left(-\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right))_* + (\left(-\sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) + \left(\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right))_*}\]

    11. Simplified4.9

      \[\leadsto \color{blue}{\left(\frac{\tan \varepsilon + \tan x}{(\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(-\sin x\right) + 1)_*} - \tan x\right)} + (\left(-\sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) + \left(\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right))_*\]

    12. Simplified4.9

      \[\leadsto \left(\frac{\tan \varepsilon + \tan x}{(\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(-\sin x\right) + 1)_*} - \tan x\right) + \color{blue}{0}\]
    13. Using strategy rm

    14. Applied tan-quot5.0

      \[\leadsto \left(\frac{\tan \varepsilon + \tan x}{(\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(-\sin x\right) + 1)_*} - \color{blue}{\frac{\sin x}{\cos x}}\right) + 0\]

    15. Applied frac-sub5.0

      \[\leadsto \color{blue}{\frac{\left(\tan \varepsilon + \tan x\right) \cdot \cos x - (\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(-\sin x\right) + 1)_* \cdot \sin x}{(\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(-\sin x\right) + 1)_* \cdot \cos x}} + 0\]

    16. Simplified5.0

      \[\leadsto \frac{\color{blue}{(\left((\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(\sin x\right) + -1)_*\right) \cdot \left(\sin x\right) + \left(\left(\tan \varepsilon + \tan x\right) \cdot \cos x\right))_*}}{(\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(-\sin x\right) + 1)_* \cdot \cos x} + 0\]
  3. Recombined 3 regimes into one program.

  4. Final simplification14.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.7531779447900738 \cdot 10^{-40}:\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan x + \tan \varepsilon\right) + \left(\sin x \cdot (\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + -1)_*\right))_*}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \le 2.649457653506729 \cdot 10^{-60}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left((\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(\sin x\right) + -1)_*\right) \cdot \left(\sin x\right) + \left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x\right))_*}{(\left(\frac{\tan \varepsilon}{\cos x}\right) \cdot \left(-\sin x\right) + 1)_* \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 +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: 57.3s)Debug log

sample367.0ms

Algorithm
intervals

simplify4.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
4.0ms
(- (tan (+ x eps)) (tan x))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 35.1b

localize68.0ms

Local error

Found 2 expressions with local error:

2.9b
(tan (+ x eps))
1.8b
(- (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
12.0ms
(- (tan (+ x eps)) (tan x))
3.0ms
(tan (+ x eps))

series206.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
132.0ms
(- (tan (+ x eps)) (tan x))
74.0ms
(tan (+ x eps))

simplify1.0s

Counts
35 → 47
Calls
35 calls:
Slowest
229.0ms
(- (* (+ (tan x) (tan eps)) (cos x)) (* (- 1 (* (tan x) (tan eps))) (sin x)))
212.0ms
(+ (* x (pow eps 2)) (+ eps (* (pow x 2) eps)))
143.0ms
(* (- 1 (* (tan x) (tan eps))) (cos x))
49.0ms
(+ x (+ (* 1/3 (pow x 3)) eps))
41.0ms
(fma (sqrt (tan (+ x eps))) (sqrt (tan (+ x eps))) (- (* (sqrt (tan x)) (sqrt (tan x)))))

prune511.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 16.6b

localize33.0ms

Local error

Found 4 expressions with local error:

4.1b
(- (/ (+ (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
(- 1 (* (tan x) (tan eps)))

rewrite78.0ms

Algorithm
rewrite-expression-head
Rules
85×*-un-lft-identity
62×add-sqr-sqrt
57×add-cube-cbrt
54×prod-diff
49×times-frac
18×fma-neg
13×distribute-lft-out
8×add-log-exp
8×associate-/r/
7×tan-quot
6×flip--
6×add-exp-log
6×add-cbrt-cube
6×flip3--
6×pow1
4×div-inv
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-*l*
3×associate-*r*
3×fma-def
2×difference-of-squares
2×sub-neg
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×sum-log
1×+-commutative
1×frac-times
Counts
4 → 143
Calls
4 calls:
Slowest
59.0ms
(- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x))
7.0ms
(+ (tan x) (tan eps))
5.0ms
(* (tan x) (tan eps))
2.0ms
(- 1 (* (tan x) (tan eps)))

series741.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
390.0ms
(- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x))
130.0ms
(- 1 (* (tan x) (tan eps)))
114.0ms
(+ (tan x) (tan eps))
107.0ms
(* (tan x) (tan eps))

simplify13.9s

Counts
163 → 155
Calls
163 calls:
Slowest
538.0ms
(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)))))
526.0ms
(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)))))
479.0ms
(fma (/ (+ (tan x) (tan eps)) (- (pow 1 3) (pow (* (tan x) (tan eps)) 3))) (+ (* 1 1) (+ (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* 1 (* (tan x) (tan eps))))) (- (* (cbrt (tan x)) (* (cbrt (tan x)) (cbrt (tan x))))))
469.0ms
(fma (/ 1 (* (cbrt (- 1 (* (tan x) (tan eps)))) (cbrt (- 1 (* (tan x) (tan eps)))))) (/ (+ (tan x) (tan eps)) (cbrt (- 1 (* (tan x) (tan eps))))) (- (* (cbrt (tan x)) (* (cbrt (tan x)) (cbrt (tan x))))))
466.0ms
(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)))

prune2.0s

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 16.4b

localize34.0ms

Local error

Found 4 expressions with local error:

4.1b
(- (/ (+ (tan x) (tan eps)) (- 1 (/ (* (sin x) (tan eps)) (cos x)))) (tan x))
0.2b
(* (sin x) (tan eps))
0.1b
(/ (* (sin x) (tan eps)) (cos x))
0.1b
(+ (tan x) (tan eps))

rewrite105.0ms

Algorithm
rewrite-expression-head
Rules
87×*-un-lft-identity
64×add-sqr-sqrt
59×add-cube-cbrt
54×prod-diff
52×times-frac
18×fma-neg
13×distribute-lft-out
11×add-exp-log
11×add-cbrt-cube
8×add-log-exp
8×associate-/r/
6×pow1
5×flip--
5×div-inv
5×tan-quot
5×flip3--
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-*l*
3×associate-*r*
3×fma-def
3×associate-/r*
2×difference-of-squares
2×cbrt-unprod
2×associate-*r/
2×prod-exp
2×div-exp
2×cbrt-undiv
1×associate-/l/
1×distribute-lft-out--
1×associate-/l*
1×flip-+
1×frac-sub
1×*-commutative
1×frac-add
1×pow-prod-down
1×diff-log
1×flip3-+
1×frac-2neg
1×sub-neg
1×sum-log
1×clear-num
1×+-commutative
Counts
4 → 152
Calls
4 calls:
Slowest
70.0ms
(- (/ (+ (tan x) (tan eps)) (- 1 (/ (* (sin x) (tan eps)) (cos x)))) (tan x))
14.0ms
(/ (* (sin x) (tan eps)) (cos x))
8.0ms
(+ (tan x) (tan eps))
8.0ms
(* (sin x) (tan eps))

series748.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
421.0ms
(- (/ (+ (tan x) (tan eps)) (- 1 (/ (* (sin x) (tan eps)) (cos x)))) (tan x))
145.0ms
(/ (* (sin x) (tan eps)) (cos x))
95.0ms
(+ (tan x) (tan eps))
87.0ms
(* (sin x) (tan eps))

simplify14.1s

Counts
175 → 164
Calls
175 calls:
Slowest
515.0ms
(fma (/ 1 (sqrt (- 1 (/ (* (sin x) (tan eps)) (cos x))))) (/ (+ (tan x) (tan eps)) (sqrt (- 1 (/ (* (sin x) (tan eps)) (cos x))))) (- (* (tan x) 1)))
500.0ms
(fma (/ 1 (* (cbrt (- 1 (/ (* (sin x) (tan eps)) (cos x)))) (cbrt (- 1 (/ (* (sin x) (tan eps)) (cos x)))))) (/ (+ (tan x) (tan eps)) (cbrt (- 1 (/ (* (sin x) (tan eps)) (cos x))))) (- (* (sqrt (tan x)) (sqrt (tan x)))))
455.0ms
(fma (/ 1 (sqrt (- 1 (/ (* (sin x) (tan eps)) (cos x))))) (/ (+ (tan x) (tan eps)) (sqrt (- 1 (/ (* (sin x) (tan eps)) (cos x))))) (- (* (cbrt (tan x)) (* (cbrt (tan x)) (cbrt (tan x))))))
448.0ms
(fma (/ 1 (* (cbrt (- 1 (/ (* (sin x) (tan eps)) (cos x)))) (cbrt (- 1 (/ (* (sin x) (tan eps)) (cos x)))))) (/ (+ (tan x) (tan eps)) (cbrt (- 1 (/ (* (sin x) (tan eps)) (cos x))))) (- (* (tan x) 1)))
427.0ms
(fma (/ 1 (* (cbrt (- 1 (/ (* (sin x) (tan eps)) (cos x)))) (cbrt (- 1 (/ (* (sin x) (tan eps)) (cos x)))))) (/ (+ (tan x) (tan eps)) (cbrt (- 1 (/ (* (sin x) (tan eps)) (cos x))))) (- (* (tan x) 1)))

prune1.5s

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 16.4b

localize18.0ms

Local error

Found 4 expressions with local error:

4.1b
(- (/ (+ (tan eps) (tan x)) (fma (/ (tan eps) (cos x)) (- (sin x)) 1)) (tan x))
0.1b
(/ (tan eps) (cos x))
0.1b
(+ (tan eps) (tan x))
0.1b
(/ (+ (tan eps) (tan x)) (fma (/ (tan eps) (cos x)) (- (sin x)) 1))

rewrite51.0ms

Algorithm
rewrite-expression-head
Rules
106×*-un-lft-identity
75×add-sqr-sqrt
70×add-cube-cbrt
70×times-frac
48×prod-diff
17×distribute-lft-out
16×fma-neg
8×add-log-exp
8×add-exp-log
8×add-cbrt-cube
7×associate-/l*
6×div-inv
6×tan-quot
6×associate-/r*
4×associate-/l/
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×fma-def
2×difference-of-squares
2×flip-+
2×frac-add
2×div-exp
2×flip3-+
2×frac-2neg
2×clear-num
2×cbrt-undiv
1×distribute-lft-out--
1×flip--
1×frac-sub
1×diff-log
1×flip3--
1×sub-neg
1×sum-log
1×+-commutative
Counts
4 → 166
Calls
4 calls:
Slowest
28.0ms
(- (/ (+ (tan eps) (tan x)) (fma (/ (tan eps) (cos x)) (- (sin x)) 1)) (tan x))
11.0ms
(/ (+ (tan eps) (tan x)) (fma (/ (tan eps) (cos x)) (- (sin x)) 1))
6.0ms
(/ (tan eps) (cos x))
3.0ms
(+ (tan eps) (tan x))

series644.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
326.0ms
(- (/ (+ (tan eps) (tan x)) (fma (/ (tan eps) (cos x)) (- (sin x)) 1)) (tan x))
179.0ms
(/ (+ (tan eps) (tan x)) (fma (/ (tan eps) (cos x)) (- (sin x)) 1))
84.0ms
(+ (tan eps) (tan x))
55.0ms
(/ (tan eps) (cos x))

simplify8.7s

Counts
197 → 178
Calls
197 calls:
Slowest
323.0ms
(fma (/ 1 (* (cbrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1)) (cbrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1)))) (/ (+ (tan eps) (tan x)) (cbrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1))) (- (* (sqrt (tan x)) (sqrt (tan x)))))
319.0ms
(fma (/ 1 (* (cbrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1)) (cbrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1)))) (/ (+ (tan eps) (tan x)) (cbrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1))) (- (* (sqrt (tan x)) (sqrt (tan x)))))
304.0ms
(fma (/ 1 (* (cbrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1)) (cbrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1)))) (/ (+ (tan eps) (tan x)) (cbrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1))) (- (* (tan x) 1)))
303.0ms
(fma (/ 1 (* (cbrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1)) (cbrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1)))) (/ (+ (tan eps) (tan x)) (cbrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1))) (- (* (tan x) 1)))
302.0ms
(fma (/ 1 (sqrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1))) (/ (+ (tan eps) (tan x)) (sqrt (fma (/ (tan eps) (cos x)) (- (sin x)) 1))) (- (* (cbrt (tan x)) (* (cbrt (tan x)) (cbrt (tan x))))))

prune1.6s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 15.0b

regimes198.0ms

Accuracy

93.7% (0.6b remaining)

Error of 14.0b against oracle of 13.4b and baseline of 22.3b

bsearch508.0ms

end0.0ms

sample10.2s

Algorithm
intervals