Average Error: 58.5 → 0.3
Time: 23.9s
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

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019004 +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: 22.8s)Debug log

sample186.0ms

Algorithm
intervals

simplify19.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

19.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.0b

localize10.0ms

Local error

Found 2 expressions with local error:

5.1b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 62
Calls

2 calls. Slowest were:

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

series92.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

82.0ms
(log (/ (- 1 eps) (+ 1 eps)))
10.0ms
(/ (- 1 eps) (+ 1 eps))

simplify2.0s

Counts
54 → 68
Calls

54 calls. Slowest were:

341.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
253.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
225.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune524.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize32.0ms

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))

series59.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

36.0ms
(* (pow eps 5) -2/5)
23.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))

simplify883.0ms

Counts
8 → 29
Calls

8 calls. Slowest were:

317.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
286.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
266.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))

prune325.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (sqrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))) (sqrt (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)
0.0b
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

10.0ms
(* (sqrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))) (sqrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
4.0ms
(* (pow eps 5) -2/5)
4.0ms
(* (pow eps 5) -2/5)

series102.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

34.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))
28.0ms
(* (sqrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))) (sqrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
20.0ms
(* (pow eps 5) -2/5)
19.0ms
(* (pow eps 5) -2/5)

simplify3.9s

Counts
38 → 80
Calls

38 calls. Slowest were:

600.0ms
(* (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)))
421.0ms
(* (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)))
339.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))

prune1.0s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize34.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))
0.6b
(cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))
0.6b
(cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))
0.5b
(* (cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))) (cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

10.0ms
(* (cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))) (cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
1.0ms
(cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))
1.0ms
(cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))

series569.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

155.0ms
(cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))
139.0ms
(* (cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))) (cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
138.0ms
(cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))
137.0ms
(cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))

simplify5.9s

Counts
41 → 81
Calls

41 calls. Slowest were:

497.0ms
(* (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)))
483.0ms
(+ (* 110/81 (/ (exp (* 1/3 (- (log -2/5) (* 5 (log (/ 1 eps)))))) (pow eps 4))) (+ (* 5/9 (/ (exp (* 1/3 (- (log -2/5) (* 5 (log (/ 1 eps)))))) (pow eps 2))) (exp (* 1/3 (- (log -2/5) (* 5 (log (/ 1 eps))))))))
466.0ms
(+ (* 110/81 (/ (exp (* 1/3 (- (log -2/5) (* 5 (log (/ 1 eps)))))) (pow eps 4))) (+ (* 5/9 (/ (exp (* 1/3 (- (log -2/5) (* 5 (log (/ 1 eps)))))) (pow eps 2))) (exp (* 1/3 (- (log -2/5) (* 5 (log (/ 1 eps))))))))

prune1.1s

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

regimes61.0ms

Accuracy

0% (0.2b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch1.0ms

end0.0ms

sample5.9s

Algorithm
intervals