Average Error: 61.0 → 0.0
Time: 20.1s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\log_* (1 + (e^{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}} - 1)^*)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.0
Target0.4
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 61.0

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

  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 log1p-expm1-u0.0

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

  8. Final simplification0.0

    \[\leadsto \log_* (1 + (e^{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}} - 1)^*)\]

Reproduce

herbie shell --seed 2018362 +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: 8.0s)Debug log

start358.0ms

Algorithm
intervals

setup17.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 59.1b

localize20.0ms

Local error

Found 2 expressions with local error:

7.9b
(log (- 1 x))
0.0b
(/ (log (- 1 x)) (log1p x))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
2 → 53
Calls

2 calls. Slowest were:

10.0ms
(/ (log (- 1 x)) (log1p x))
2.0ms
(log (- 1 x))

series117.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

77.0ms
(/ (log (- 1 x)) (log1p x))
40.0ms
(log (- 1 x))

simplify2.2s

Counts
47 → 59
Calls

47 calls. Slowest were:

350.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
336.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
283.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune260.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 1 expressions with local error:

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

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

3.0ms
(/ (log1p (- x)) (log1p x))

series64.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

64.0ms
(/ (log1p (- x)) (log1p x))

simplify821.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

205.0ms
(- (+ (/ (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)))))))))))
202.0ms
(- (+ (* 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)))))
179.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))

prune274.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize10.0ms

Local error

Found 3 expressions with local error:

0.0b
(/ (log1p (- x)) (log1p x))
0.0b
(log1p (expm1 (/ (log1p (- x)) (log1p x))))
0.0b
(expm1 (/ (log1p (- x)) (log1p x)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
3 → 51
Calls

3 calls. Slowest were:

3.0ms
(/ (log1p (- x)) (log1p x))
0.0ms
(expm1 (/ (log1p (- x)) (log1p x)))
0.0ms
(log1p (expm1 (/ (log1p (- x)) (log1p x))))

series326.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

139.0ms
(expm1 (/ (log1p (- x)) (log1p x)))
107.0ms
(log1p (expm1 (/ (log1p (- x)) (log1p x))))
79.0ms
(/ (log1p (- x)) (log1p x))

simplify2.2s

Counts
36 → 60
Calls

36 calls. Slowest were:

504.0ms
(- (exp -1) (+ (* 1/12 (* (pow x 3) (exp -1))) (+ (* x (exp -1)) 1)))
252.0ms
(- (+ (/ (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)))))))))))
225.0ms
(- (+ (* 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)))))

prune573.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize13.0ms

Local error

Found 1 expressions with local error:

0.0b
(fma x (* -1/2 x) (- -1 x))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

0.0ms
(fma x (* -1/2 x) (- -1 x))

series20.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

20.0ms
(fma x (* -1/2 x) (- -1 x))

simplify516.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

190.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
179.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
146.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))

prune91.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes22.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms