Average Error: 60.9 → 0.0
Time: 26.7s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}} \cdot \sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)} \cdot \frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.9
Target0.3
Herbie0.0
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 60.9

    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]

  2. Initial simplification60.0

    \[\leadsto \frac{\log \left(1 - x\right)}{\log_* (1 + x)}\]
  3. Using strategy rm

  4. Applied sub-neg60.0

    \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-x\right)\right)}}{\log_* (1 + x)}\]

  5. Applied log1p-def0.0

    \[\leadsto \frac{\color{blue}{\log_* (1 + \left(-x\right))}}{\log_* (1 + x)}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt0.1

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}} \cdot \sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}}\right) \cdot \sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}}}\]
  8. Using strategy rm

  9. Applied cbrt-unprod0.0

    \[\leadsto \color{blue}{\sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)} \cdot \frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}}} \cdot \sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}}\]

  10. Final simplification0.0

    \[\leadsto \sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}} \cdot \sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)} \cdot \frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 5/12 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))

Details

Time bar (total: 15.1s)Debug log

start336.0ms

Algorithm
intervals

setup23.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 59.6b

localize19.0ms

Local error

Found 2 expressions with local error:

5.3b
(log (- 1 x))
0.0b
(/ (log (- 1 x)) (log1p x))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 53
Calls

2 calls. Slowest were:

5.0ms
(/ (log (- 1 x)) (log1p x))
1.0ms
(log (- 1 x))

series139.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

112.0ms
(/ (log (- 1 x)) (log1p x))
26.0ms
(log (- 1 x))

simplify2.5s

Counts
47 → 59
Calls

47 calls. Slowest were:

748.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
509.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
404.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune350.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 1 expressions with local error:

0.0b
(/ (log1p (- x)) (log1p x))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

9.0ms
(/ (log1p (- x)) (log1p x))

series104.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

104.0ms
(/ (log1p (- x)) (log1p x))

simplify755.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

208.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
196.0ms
(- (+ (/ (log (/ -1 x)) (* x (pow (- (log -1) (log (/ -1 x))) 2))) (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (+ (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x)))) (+ (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 3))) (+ (/ 1 (* x (- (log -1) (log (/ -1 x))))) (+ (* 1/2 (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (* 1/2 (/ 1 (* (pow x 2) (- (log -1) (log (/ -1 x)))))))))))
143.0ms
(- (+ (* 2 (/ 1 (* (pow (log (/ 1 x)) 2) (pow x 2)))) (+ (* 2 (/ 1 (* (log (/ 1 x)) x))) (+ (* 1/2 (/ (log -1) (* (pow (log (/ 1 x)) 2) (pow x 2)))) 1))) (+ (/ (log -1) (* (pow (log (/ 1 x)) 3) (pow x 2))) (+ (/ (log -1) (log (/ 1 x))) (/ (log -1) (* (pow (log (/ 1 x)) 2) x)))))

prune310.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

0.0b
(/ (log1p (- x)) (log1p x))
0.0b
(/ (log1p (- x)) (log1p x))
0.0b
(/ (log1p (- x)) (log1p x))
0.0b
(* (cbrt (/ (log1p (- x)) (log1p x))) (cbrt (/ (log1p (- x)) (log1p x))))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 140
Calls

4 calls. Slowest were:

6.0ms
(* (cbrt (/ (log1p (- x)) (log1p x))) (cbrt (/ (log1p (- x)) (log1p x))))
4.0ms
(/ (log1p (- x)) (log1p x))
3.0ms
(/ (log1p (- x)) (log1p x))

series676.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

379.0ms
(* (cbrt (/ (log1p (- x)) (log1p x))) (cbrt (/ (log1p (- x)) (log1p x))))
103.0ms
(/ (log1p (- x)) (log1p x))
101.0ms
(/ (log1p (- x)) (log1p x))
93.0ms
(/ (log1p (- x)) (log1p x))

simplify2.7s

Counts
131 → 152
Calls

131 calls. Slowest were:

219.0ms
(- (+ (* 4/3 (* (pow (/ (log (/ 1 x)) (pow (- (log -1) (log (/ 1 x))) 4)) 1/3) (/ 1 x))) (+ (* 16/3 (* (pow (/ (log (/ 1 x)) (pow (- (log -1) (log (/ 1 x))) 10)) 1/3) (/ (log -1) (pow x 2)))) (+ (* 8/3 (* (pow (/ 1 (* (pow (log (/ 1 x)) 5) (pow (- (log -1) (log (/ 1 x))) 10))) 1/3) (/ (pow (log -1) 3) (pow x 2)))) (+ (pow (/ (pow (- (log -1) (log (/ 1 x))) 2) (pow (log (/ 1 x)) 2)) 1/3) (+ (* 2/3 (* (pow (/ 1 (* (pow (log (/ 1 x)) 5) (pow (- (log -1) (log (/ 1 x))) 4))) 1/3) (/ (pow (log -1) 2) x))) (+ (* (pow (/ 1 (* (pow (log (/ 1 x)) 8) (pow (- (log -1) (log (/ 1 x))) 4))) 1/3) (/ (pow (log -1) 2) (pow x 2))) (+ (* 8/3 (* (pow (/ 1 (* (pow (log (/ 1 x)) 2) (pow (- (log -1) (log (/ 1 x))) 4))) 1/3) (/ 1 (pow x 2)))) (* 1/3 (* (pow (/ 1 (* (pow (log (/ 1 x)) 2) (pow (- (log -1) (log (/ 1 x))) 4))) 1/3) (/ (log -1) (pow x 2))))))))))) (+ (* 16/9 (* (pow (/ (pow (log (/ 1 x)) 4) (pow (- (log -1) (log (/ 1 x))) 10)) 1/3) (/ 1 (pow x 2)))) (+ (* 1/3 (* (pow (/ 1 (* (pow (log (/ 1 x)) 5) (pow (- (log -1) (log (/ 1 x))) 4))) 1/3) (/ (pow (log -1) 2) (pow x 2)))) (+ (* 2 (* (/ (log -1) x) (pow (/ 1 (* (pow (log (/ 1 x)) 2) (pow (- (log -1) (log (/ 1 x))) 4))) 1/3))) (+ (* 10/3 (* (pow (/ 1 (* (pow (log (/ 1 x)) 5) (pow (- (log -1) (log (/ 1 x))) 4))) 1/3) (/ (log -1) (pow x 2)))) (+ (* 52/9 (* (pow (/ 1 (* (pow (log (/ 1 x)) 2) (pow (- (log -1) (log (/ 1 x))) 10))) 1/3) (/ (pow (log -1) 2) (pow x 2)))) (* 4/9 (* (pow (/ 1 (* (pow (log (/ 1 x)) 8) (pow (- (log -1) (log (/ 1 x))) 10))) 1/3) (/ (pow (log -1) 4) (pow x 2))))))))))
209.0ms
(- (+ (/ (log (/ -1 x)) (* x (pow (- (log -1) (log (/ -1 x))) 2))) (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (+ (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x)))) (+ (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 3))) (+ (/ 1 (* x (- (log -1) (log (/ -1 x))))) (+ (* 1/2 (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (* 1/2 (/ 1 (* (pow x 2) (- (log -1) (log (/ -1 x)))))))))))
206.0ms
(- (+ (/ (log (/ -1 x)) (* x (pow (- (log -1) (log (/ -1 x))) 2))) (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (+ (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x)))) (+ (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 3))) (+ (/ 1 (* x (- (log -1) (log (/ -1 x))))) (+ (* 1/2 (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (* 1/2 (/ 1 (* (pow x 2) (- (log -1) (log (/ -1 x)))))))))))

prune2.0s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize8.0ms

Local error

Found 4 expressions with local error:

0.0b
(/ (log1p (- x)) (log1p x))
0.0b
(/ (log1p (- x)) (log1p x))
0.0b
(/ (log1p (- x)) (log1p x))
0.0b
(cbrt (* (/ (log1p (- x)) (log1p x)) (/ (log1p (- x)) (log1p x))))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 101
Calls

4 calls. Slowest were:

6.0ms
(/ (log1p (- x)) (log1p x))
4.0ms
(cbrt (* (/ (log1p (- x)) (log1p x)) (/ (log1p (- x)) (log1p x))))
3.0ms
(/ (log1p (- x)) (log1p x))

series630.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

378.0ms
(cbrt (* (/ (log1p (- x)) (log1p x)) (/ (log1p (- x)) (log1p x))))
90.0ms
(/ (log1p (- x)) (log1p x))
84.0ms
(/ (log1p (- x)) (log1p x))
78.0ms
(/ (log1p (- x)) (log1p x))

simplify2.8s

Counts
91 → 113
Calls

91 calls. Slowest were:

246.0ms
(- (+ (/ (log (/ -1 x)) (* x (pow (- (log -1) (log (/ -1 x))) 2))) (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (+ (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x)))) (+ (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 3))) (+ (/ 1 (* x (- (log -1) (log (/ -1 x))))) (+ (* 1/2 (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (* 1/2 (/ 1 (* (pow x 2) (- (log -1) (log (/ -1 x)))))))))))
194.0ms
(- (+ (/ (log (/ -1 x)) (* x (pow (- (log -1) (log (/ -1 x))) 2))) (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (+ (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x)))) (+ (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 3))) (+ (/ 1 (* x (- (log -1) (log (/ -1 x))))) (+ (* 1/2 (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (* 1/2 (/ 1 (* (pow x 2) (- (log -1) (log (/ -1 x)))))))))))
189.0ms
(- (+ (/ (log (/ -1 x)) (* x (pow (- (log -1) (log (/ -1 x))) 2))) (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (+ (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x)))) (+ (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 3))) (+ (/ 1 (* x (- (log -1) (log (/ -1 x))))) (+ (* 1/2 (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (* 1/2 (/ 1 (* (pow x 2) (- (log -1) (log (/ -1 x)))))))))))

prune1.7s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes24.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms