Average Error: 58.3 → 0.3
Time: 18.4s
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[(\varepsilon \cdot \left((\frac{-2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + -2)_*\right) + \left(\log \left(e^{\frac{-2}{5} \cdot {\varepsilon}^{5}}\right)\right))_*\]

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.3

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

  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. Using strategy rm

  5. Applied add-log-exp0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019016 +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: 17.8s)Debug log

sample225.0ms

Algorithm
intervals
Results
524×(pre true 80)
268×(body nan 80)
104×(body real 1280)
83×(body real 640)
41×(body real 320)
17×(body real 160)
11×(body real 80)

simplify21.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
20.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.3b

localize45.0ms

Local error

Found 2 expressions with local error:

6.1b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite10.0ms

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
5.0ms
(/ (- 1 eps) (+ 1 eps))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series90.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
72.0ms
(log (/ (- 1 eps) (+ 1 eps)))
17.0ms
(/ (- 1 eps) (+ 1 eps))

simplify2.1s

Counts
54 → 68
Calls
54 calls:
Slowest
329.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
244.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
230.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
213.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
176.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))

prune515.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

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))

series46.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
24.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))
22.0ms
(* (pow eps 5) -2/5)

simplify848.0ms

Counts
8 → 29
Calls
8 calls:
Slowest
328.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
296.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
208.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
4.0ms
(* -2/5 (pow eps 5))
3.0ms
(* -2/5 (pow eps 5))

prune250.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 3 expressions with local error:

2.3b
(log (exp (* (pow eps 5) -2/5)))
0.1b
(* (pow eps 5) -2/5)
0.0b
(fma eps (fma -2/3 (* eps eps) -2) (log (exp (* (pow eps 5) -2/5))))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
4×pow1
3×log1p-expm1-u
3×log-pow
3×add-exp-log
3×associate-*l*
3×add-cbrt-cube
3×log-prod
3×expm1-log1p-u
1×fma-udef
1×*-commutative
1×rem-log-exp
1×exp-to-pow
1×exp-prod
Counts
3 → 39
Calls
3 calls:
Slowest
4.0ms
(* (pow eps 5) -2/5)
4.0ms
(log (exp (* (pow eps 5) -2/5)))
0.0ms
(fma eps (fma -2/3 (* eps eps) -2) (log (exp (* (pow eps 5) -2/5))))

series86.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
37.0ms
(log (exp (* (pow eps 5) -2/5)))
27.0ms
(* (pow eps 5) -2/5)
22.0ms
(fma eps (fma -2/3 (* eps eps) -2) (log (exp (* (pow eps 5) -2/5))))

simplify884.0ms

Counts
15 → 48
Calls
15 calls:
Slowest
318.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
268.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
257.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
6.0ms
(log (* (cbrt (exp (* (pow eps 5) -2/5))) (cbrt (exp (* (pow eps 5) -2/5)))))
6.0ms
(log (sqrt (exp (* (pow eps 5) -2/5))))

prune571.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

38.6b
(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)

rewrite20.0ms

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)

series123.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
38.0ms
(* (* (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)))
36.0ms
(* (pow eps 5) -2/5)
25.0ms
(* (pow eps 5) -2/5)
23.0ms
(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))))

simplify5.0s

Counts
39 → 79
Calls
39 calls:
Slowest
1.3s
(* (* (* (* (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))))
607.0ms
(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))))
537.0ms
(+ (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))))
364.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
277.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))

prune859.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

regimes34.0ms

Accuracy

0% (0.2b remaining)

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

bsearch1.0ms

end0.0ms

sample6.0s

Algorithm
intervals
Results
15997×(pre true 80)
7996×(body nan 80)
3361×(body real 1280)
2520×(body real 640)
1258×(body real 320)
626×(body real 160)
236×(body real 80)