Average Error: 58.4 → 0.3
Time: 29.2s
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

Original58.4
Target0.3
Herbie0.3
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  2. Initial simplification58.4

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. 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 2018365 +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: 19.2s)Debug log

start217.0ms

Algorithm
intervals

setup42.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.8b

localize12.0ms

Local error

Found 2 expressions with local error:

7.7b
(log (/ (- 1 eps) (+ eps 1)))
0.0b
(/ (- 1 eps) (+ eps 1))

rewrite5.0ms

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)))

series76.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

62.0ms
(log (/ (- 1 eps) (+ eps 1)))
14.0ms
(/ (- 1 eps) (+ eps 1))

simplify2.1s

Counts
54 → 68
Calls

54 calls. Slowest were:

352.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ eps 1) (+ eps 1)) (+ eps 1)))
239.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))
234.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))

prune535.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

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
Counts
2 → 23
Calls

2 calls. Slowest were:

2.0ms
(* (pow eps 5) -2/5)
0.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))

series73.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

41.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))
32.0ms
(* (pow eps 5) -2/5)

simplify1.1s

Counts
8 → 29
Calls

8 calls. Slowest were:

405.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
357.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
282.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))

prune394.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize9.0ms

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))))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

9.0ms
(* (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)))

series510.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

156.0ms
(cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))
126.0ms
(* (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))))
123.0ms
(cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))
105.0ms
(cbrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))

simplify5.9s

Counts
41 → 81
Calls

41 calls. Slowest were:

491.0ms
(* (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)))
446.0ms
(* (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)))
444.0ms
(+ (* 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))))))))

prune1.4s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 4 expressions with local error:

40.3b
(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
Counts
4 → 67
Calls

4 calls. Slowest were:

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)

series166.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

53.0ms
(* (* (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)))
42.0ms
(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))))
36.0ms
(* (pow eps 5) -2/5)
35.0ms
(* (pow eps 5) -2/5)

simplify5.3s

Counts
39 → 79
Calls

39 calls. Slowest were:

1.5s
(* (* (* (* (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))))
541.0ms
(+ (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))))
511.0ms
(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))))

prune1.2s

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0.0b

regimes86.0ms

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms