Average Error: 58.6 → 0.2
Time: 20.2s
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. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. 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 2019022 +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: 19.7s)Debug log

sample127.0ms

Algorithm
intervals
Results
68.0ms115×body1280valid
28.0ms83×body640valid
12.0ms246×body80nan
8.0ms37×body320valid
2.0ms16×body160valid
0.0ms5×body80valid

simplify31.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
31.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.0b

localize17.0ms

Local error

Found 2 expressions with local error:

6.4b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
18×times-frac
17×add-cube-cbrt
17×*-un-lft-identity
17×add-sqr-sqrt
15×log-prod
7×add-exp-log
4×associate-/r/
4×add-cbrt-cube
3×associate-/l*
3×associate-/r*
3×pow1
2×associate-/l/
2×add-log-exp
2×div-inv
2×log1p-expm1-u
2×flip-+
2×rem-log-exp
2×div-exp
2×flip3-+
2×expm1-log1p-u
1×flip--
1×log-pow
1×div-sub
1×flip3--
1×frac-2neg
1×clear-num
1×log-div
1×cbrt-undiv
Counts
2 → 62
Calls
2 calls:
Slowest
4.0ms
(/ (- 1 eps) (+ 1 eps))
2.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series91.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
72.0ms
(log (/ (- 1 eps) (+ 1 eps)))
18.0ms
(/ (- 1 eps) (+ 1 eps))

simplify1.6s

Counts
54 → 68
Calls
54 calls:
Slowest
270.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
211.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
154.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
148.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))
130.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune454.0ms

Pruning

3 alts after pruning (3 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))

rewrite3.0ms

Algorithm
rewrite-expression-head
Rules
3×add-cube-cbrt
3×associate-*l*
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
2×log1p-expm1-u
2×add-exp-log
2×add-cbrt-cube
2×pow1
2×expm1-log1p-u
1×fma-udef
1×*-commutative
Counts
2 → 23
Calls
2 calls:
Slowest
2.0ms
(* (pow eps 5) -2/5)
0.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))

series64.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
35.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))
30.0ms
(* (pow eps 5) -2/5)

simplify683.0ms

Counts
8 → 29
Calls
8 calls:
Slowest
254.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
214.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
207.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
2.0ms
(* -2/5 (pow eps 5))
2.0ms
(* -2/5 (pow eps 5))

prune204.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 4 expressions with local error:

42.1b
(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
(* (* (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)

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
18×pow1
9×add-exp-log
9×add-cbrt-cube
7×add-cube-cbrt
7×associate-*l*
7×*-un-lft-identity
7×add-sqr-sqrt
5×pow-prod-up
5×pow-plus
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×cbrt-unprod
3×*-commutative
3×prod-exp
3×pow-prod-down
3×associate-*r*
2×fma-udef
2×pow3
2×pow2
1×rem-cbrt-cube
1×pow1/3
1×distribute-lft-in
1×distribute-rgt-in
1×cbrt-prod
Counts
4 → 67
Calls
4 calls:
Slowest
9.0ms
(* (* (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))) (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))
6.0ms
(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))))
2.0ms
(* (pow eps 5) -2/5)
2.0ms
(* (pow eps 5) -2/5)

series92.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
28.0ms
(* (* (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))) (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))
24.0ms
(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))))
21.0ms
(* (pow eps 5) -2/5)
19.0ms
(* (pow eps 5) -2/5)

simplify4.8s

Counts
39 → 79
Calls
39 calls:
Slowest
1.2s
(* (* (* (* (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))) (* (* (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)))) (* (* (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))))
783.0ms
(+ (log (* (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)))) (log (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
379.0ms
(* (* (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))))
368.0ms
(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))))
288.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))

prune884.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize34.0ms

Local error

Found 4 expressions with local error:

6.2b
(log (- 1 eps))
6.2b
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
0.0b
(- (log (- 1 eps)) (log1p (pow eps 3)))
0.0b
(log1p (pow eps 3))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
19×*-un-lft-identity
17×add-sqr-sqrt
15×add-cube-cbrt
12×prod-diff
11×pow1
10×log-prod
7×log-pow
6×add-exp-log
6×log-div
5×add-log-exp
5×expm1-log1p-u
4×log1p-expm1-u
4×fma-neg
4×add-cbrt-cube
3×flip--
3×associate--l+
3×flip3--
2×distribute-lft-out--
2×rem-log-exp
2×diff-log
2×log1p-def
2×sub-neg
2×log1p-udef
2×associate--l-
1×difference-of-squares
1×distribute-lft-out
1×flip-+
1×log1p-expm1
1×flip3-+
Counts
4 → 85
Calls
4 calls:
Slowest
10.0ms
(- (log (- 1 eps)) (log1p (pow eps 3)))
8.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
2.0ms
(log (- 1 eps))
0.0ms
(log1p (pow eps 3))

series461.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
335.0ms
(log1p (pow eps 3))
70.0ms
(- (log (- 1 eps)) (log1p (pow eps 3)))
32.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
22.0ms
(log (- 1 eps))

simplify4.0s

Counts
65 → 97
Calls
65 calls:
Slowest
558.0ms
(log (- (* (* 1 1) (* 1 1)) (* (- (* eps eps) (* 1 eps)) (- (* eps eps) (* 1 eps)))))
317.0ms
(- (log -1) (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ 1 eps)))))
271.0ms
(- (+ (* 2 (log (/ 1 eps))) (log -1)) (+ (/ 1 eps) (* 1/2 (/ 1 (pow eps 2)))))
254.0ms
(- (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ -1 eps)))))
254.0ms
(- (+ (* 1/2 (pow eps 2)) (+ (* 4/3 (pow eps 3)) eps)))

prune583.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

regimes35.0ms

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms

end0.0ms

sample5.4s

Algorithm
intervals
Results
2.7s3411×body1280valid
1.2s2481×body640valid
871.0ms7987×body80nan
343.0ms1301×body320valid
98.0ms604×body160valid
16.0ms203×body80valid