Average Error: 30.9 → 0.4
Time: 50.4s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\frac{1}{\frac{\log base}{\log \left((e^{\log_* (1 + \sqrt{re^2 + im^2}^*)} - 1)^*\right)}}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log base}}\]
  3. Using strategy rm

  4. Applied expm1-log1p-u0.4

    \[\leadsto \frac{\log \color{blue}{\left((e^{\log_* (1 + \sqrt{re^2 + im^2}^*)} - 1)^*\right)}}{\log base}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\color{blue}{1 \cdot \log \left((e^{\log_* (1 + \sqrt{re^2 + im^2}^*)} - 1)^*\right)}}{\log base}\]

  7. Applied associate-/l*0.4

    \[\leadsto \color{blue}{\frac{1}{\frac{\log base}{\log \left((e^{\log_* (1 + \sqrt{re^2 + im^2}^*)} - 1)^*\right)}}}\]

  8. Final simplification0.4

    \[\leadsto \frac{1}{\frac{\log base}{\log \left((e^{\log_* (1 + \sqrt{re^2 + im^2}^*)} - 1)^*\right)}}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0))))

Details

Time bar (total: 48.7s)Debug log

sample1.1s

Algorithm
intervals

simplify46.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

46.0ms
(/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0)))

prune25.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize27.0ms

Local error

Found 2 expressions with local error:

0.3b
(/ (log (hypot re im)) (log base))
0.0b
(hypot re im)

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
2 → 43
Calls

2 calls. Slowest were:

9.0ms
(/ (log (hypot re im)) (log base))
0.0ms
(hypot re im)

series145.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

118.0ms
(/ (log (hypot re im)) (log base))
27.0ms
(hypot re im)

simplify434.0ms

Counts
37 → 49
Calls

37 calls. Slowest were:

187.0ms
(* -1 (/ (log (/ -1 re)) (- (log -1) (log (/ -1 base)))))
114.0ms
(/ (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (log base) (log base)) (log base)))
18.0ms
(/ (log (/ 1 re)) (log (/ 1 base)))

prune571.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 3 expressions with local error:

4.3b
(expm1 (log1p (hypot re im)))
0.3b
(/ (log (expm1 (log1p (hypot re im)))) (log base))
0.0b
(hypot re im)

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
3 → 54
Calls

3 calls. Slowest were:

11.0ms
(/ (log (expm1 (log1p (hypot re im)))) (log base))
0.0ms
(expm1 (log1p (hypot re im)))
0.0ms
(hypot re im)

series338.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

189.0ms
(/ (log (expm1 (log1p (hypot re im)))) (log base))
107.0ms
(expm1 (log1p (hypot re im)))
41.0ms
(hypot re im)

simplify751.0ms

Counts
40 → 63
Calls

40 calls. Slowest were:

181.0ms
(- (/ (log (- (exp (- (log (/ -1 re)))) 1)) (- (log -1) (log (/ -1 base)))) (+ (/ (exp (- (log (/ -1 re)))) (* (- (exp (- (log (/ -1 re)))) 1) (* (- (log -1) (log (/ -1 base))) re))) (* 1/2 (/ (pow (exp (- (log (/ -1 re)))) 2) (* (pow (- (exp (- (log (/ -1 re)))) 1) 2) (* (- (log -1) (log (/ -1 base))) (pow re 2)))))))
142.0ms
(- (* 1/2 (/ (pow (exp (- (log (/ 1 re)))) 2) (* (pow (- (exp (- (log (/ 1 re)))) 1) 2) (* (log (/ 1 base)) (pow re 2))))) (+ (/ (log (- (exp (- (log (/ 1 re)))) 1)) (log (/ 1 base))) (/ (exp (- (log (/ 1 re)))) (* (- (exp (- (log (/ 1 re)))) 1) (* (log (/ 1 base)) re)))))
129.0ms
(/ (* (* (log (expm1 (log1p (hypot re im)))) (log (expm1 (log1p (hypot re im))))) (log (expm1 (log1p (hypot re im))))) (* (* (log base) (log base)) (log base)))

prune876.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

4.3b
(expm1 (log1p (hypot re im)))
0.3b
(/ (log base) (log (expm1 (log1p (hypot re im)))))
0.2b
(/ 1 (/ (log base) (log (expm1 (log1p (hypot re im))))))
0.0b
(hypot re im)

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

4.0ms
(/ (log base) (log (expm1 (log1p (hypot re im)))))
2.0ms
(/ 1 (/ (log base) (log (expm1 (log1p (hypot re im))))))
0.0ms
(expm1 (log1p (hypot re im)))

series531.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

198.0ms
(/ 1 (/ (log base) (log (expm1 (log1p (hypot re im))))))
167.0ms
(/ (log base) (log (expm1 (log1p (hypot re im)))))
126.0ms
(expm1 (log1p (hypot re im)))
40.0ms
(hypot re im)

simplify1.7s

Counts
62 → 99
Calls

62 calls. Slowest were:

963.0ms
(* -1 (/ (- (log -1) (log (/ -1 base))) (log (/ -1 re))))
195.0ms
(* -1 (/ (log (/ -1 re)) (- (log -1) (log (/ -1 base)))))
145.0ms
(/ (* (* (log base) (log base)) (log base)) (* (* (log (expm1 (log1p (hypot re im)))) (log (expm1 (log1p (hypot re im))))) (log (expm1 (log1p (hypot re im))))))

prune1.2s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ (/ 1 (log base)) (/ 1 (log (hypot re im))))
0.2b
(/ 1 (log base))
0.2b
(/ 1 (log (hypot re im)))
0.0b
(hypot re im)

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

9.0ms
(/ (/ 1 (log base)) (/ 1 (log (hypot re im))))
2.0ms
(/ 1 (log (hypot re im)))
2.0ms
(/ 1 (log base))

series465.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

195.0ms
(/ 1 (log base))
165.0ms
(/ (/ 1 (log base)) (/ 1 (log (hypot re im))))
73.0ms
(/ 1 (log (hypot re im)))
31.0ms
(hypot re im)

simplify2.1s

Counts
65 → 102
Calls

65 calls. Slowest were:

1.1s
(/ (* (* (/ 1 (log base)) (/ 1 (log base))) (/ 1 (log base))) (* (* (/ 1 (log (hypot re im))) (/ 1 (log (hypot re im)))) (/ 1 (log (hypot re im)))))
188.0ms
(* -1 (/ (log (/ -1 re)) (- (log -1) (log (/ -1 base)))))
83.0ms
(/ (/ 1 (log base)) (/ 1 (log (hypot re im))))

prune1.4s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 0b

regimes403.0ms

Accuracy

0% (0.4b remaining)

Error of 0.4b against oracle of 0.1b and baseline of 0.4b

bsearch2.0ms

end0.0ms

sample36.5s

Algorithm
intervals