Average Error: 58.6 → 0.2
Time: 24.0s
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[(\varepsilon \cdot \left((\frac{-2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + -2)_*\right) + \left(\frac{-2}{5} \cdot {\varepsilon}^{5}\right))_*\]

Error

Bits error versus eps

Target

Original58.6
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

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

  2. Initial simplification58.6

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

  3. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\right)}\]

  4. Simplified0.2

    \[\leadsto \color{blue}{(\varepsilon \cdot \left((\frac{-2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + -2)_*\right) + \left({\varepsilon}^{5} \cdot \frac{-2}{5}\right))_*}\]

  5. Final simplification0.2

    \[\leadsto (\varepsilon \cdot \left((\frac{-2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + -2)_*\right) + \left(\frac{-2}{5} \cdot {\varepsilon}^{5}\right))_*\]

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))

Details

Time bar (total: 14.4s)Debug log

start276.0ms

Algorithm
intervals

setup17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.9b

localize10.0ms

Local error

Found 2 expressions with local error:

7.2b
(log (/ (- 1 eps) (+ eps 1)))
0.0b
(/ (- 1 eps) (+ eps 1))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 62
Calls

2 calls. Slowest were:

3.0ms
(/ (- 1 eps) (+ eps 1))
2.0ms
(log (/ (- 1 eps) (+ eps 1)))

series138.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

121.0ms
(log (/ (- 1 eps) (+ eps 1)))
17.0ms
(/ (- 1 eps) (+ eps 1))

simplify2.1s

Counts
54 → 68
Calls

54 calls. Slowest were:

370.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ eps 1) (+ eps 1)) (+ eps 1)))
261.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
201.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune550.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize33.0ms

Local error

Found 2 expressions with local error:

0.1b
(* (pow eps 5) -2/5)
0.0b
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 23
Calls

2 calls. Slowest were:

4.0ms
(* (pow eps 5) -2/5)
0.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))

series53.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

28.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))
25.0ms
(* (pow eps 5) -2/5)

simplify872.0ms

Counts
8 → 29
Calls

8 calls. Slowest were:

312.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
286.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
257.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))

prune427.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize18.0ms

Local error

Found 4 expressions with local error:

39.1b
(cbrt (* (* (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)) (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))) (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
0.1b
(* (pow eps 5) -2/5)
0.1b
(* (pow eps 5) -2/5)
0.1b
(* (pow eps 5) -2/5)

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 51
Calls

4 calls. Slowest were:

12.0ms
(cbrt (* (* (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)) (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))) (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
7.0ms
(* (pow eps 5) -2/5)
4.0ms
(* (pow eps 5) -2/5)

series97.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

34.0ms
(cbrt (* (* (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)) (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))) (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
23.0ms
(* (pow eps 5) -2/5)
22.0ms
(* (pow eps 5) -2/5)
19.0ms
(* (pow eps 5) -2/5)

simplify1.6s

Counts
19 → 63
Calls

19 calls. Slowest were:

591.0ms
(cbrt (* (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)) (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
318.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
318.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))

prune998.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize28.0ms

Local error

Found 4 expressions with local error:

7.0b
(log (cbrt (/ (- 1 eps) (+ eps 1))))
7.0b
(log (* (cbrt (/ (- 1 eps) (+ eps 1))) (cbrt (/ (- 1 eps) (+ eps 1)))))
0.0b
(+ (log (* (cbrt (/ (- 1 eps) (+ eps 1))) (cbrt (/ (- 1 eps) (+ eps 1))))) (log (cbrt (/ (- 1 eps) (+ eps 1)))))
0.0b
(cbrt (/ (- 1 eps) (+ eps 1)))

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 142
Calls

4 calls. Slowest were:

24.0ms
(+ (log (* (cbrt (/ (- 1 eps) (+ eps 1))) (cbrt (/ (- 1 eps) (+ eps 1))))) (log (cbrt (/ (- 1 eps) (+ eps 1)))))
9.0ms
(log (* (cbrt (/ (- 1 eps) (+ eps 1))) (cbrt (/ (- 1 eps) (+ eps 1)))))
3.0ms
(log (cbrt (/ (- 1 eps) (+ eps 1))))

series975.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

355.0ms
(+ (log (* (cbrt (/ (- 1 eps) (+ eps 1))) (cbrt (/ (- 1 eps) (+ eps 1))))) (log (cbrt (/ (- 1 eps) (+ eps 1)))))
328.0ms
(log (* (cbrt (/ (- 1 eps) (+ eps 1))) (cbrt (/ (- 1 eps) (+ eps 1)))))
221.0ms
(log (cbrt (/ (- 1 eps) (+ eps 1))))
70.0ms
(cbrt (/ (- 1 eps) (+ eps 1)))

simplify5.0s

Counts
110 → 154
Calls

110 calls. Slowest were:

448.0ms
(+ (log (* (/ (- 1 eps) (+ eps 1)) (/ (- 1 eps) (+ eps 1)))) (log (/ (- 1 eps) (+ eps 1))))
342.0ms
(- (+ (* 2/9 (* (/ 1 (pow eps 2)) (pow -1 1/3))) (pow -1 1/3)) (* 2/3 (* (/ 1 eps) (pow -1 1/3))))
335.0ms
(- (+ (* 4/3 (/ 1 eps)) (+ (* 4/15 (/ 1 (pow eps 5))) (* 4/9 (/ 1 (pow eps 3))))))

prune1.2s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes30.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch1.0ms