Average Error: 30.6 → 0.4
Time: 16.2s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re^2 + im^2}^*\right)\right)\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.6

    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

  2. Initial simplification0.6

    \[\leadsto \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log 10}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.6

    \[\leadsto \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

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

    \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re^2 + im^2}^*\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

  6. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\sqrt{\log 10}}}\]
  7. Using strategy rm

  8. Applied div-inv0.4

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\sqrt{re^2 + im^2}^*\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\]

  9. Final simplification0.4

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re^2 + im^2}^*\right)\right)\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))

Details

Time bar (total: 12.3s)Debug log

start77.0ms

Algorithm
intervals

setup43.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.6b

localize16.0ms

Local error

Found 1 expressions with local error:

0.6b
(/ (log (hypot re im)) (log 10))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
1 → 33
Calls

1 calls. Slowest were:

12.0ms
(/ (log (hypot re im)) (log 10))

series201.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

201.0ms
(/ (log (hypot re im)) (log 10))

simplify567.0ms

Counts
34 → 36
Calls

34 calls. Slowest were:

147.0ms
(* -1 (/ (log (/ -1 re)) (log 10)))
145.0ms
(/ (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (log 10) (log 10)) (log 10)))
115.0ms
(* -1 (/ (log (/ 1 re)) (log 10)))

prune541.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.1b

localize10.0ms

Local error

Found 2 expressions with local error:

0.4b
(/ (log (hypot re im)) (sqrt (log 10)))
0.3b
(* (/ 1 (sqrt (log 10))) (/ (log (hypot re im)) (sqrt (log 10))))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
2 → 100
Calls

2 calls. Slowest were:

7.0ms
(* (/ 1 (sqrt (log 10))) (/ (log (hypot re im)) (sqrt (log 10))))
5.0ms
(/ (log (hypot re im)) (sqrt (log 10)))

series369.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

217.0ms
(/ (log (hypot re im)) (sqrt (log 10)))
152.0ms
(* (/ 1 (sqrt (log 10))) (/ (log (hypot re im)) (sqrt (log 10))))

simplify2.0s

Counts
105 → 106
Calls

105 calls. Slowest were:

412.0ms
(* (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10)))) (* (* (/ (log (hypot re im)) (sqrt (log 10))) (/ (log (hypot re im)) (sqrt (log 10)))) (/ (log (hypot re im)) (sqrt (log 10)))))
163.0ms
(/ (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (sqrt (log 10)) (sqrt (log 10))) (sqrt (log 10))))
134.0ms
(* (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10)))) (/ (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (sqrt (log 10)) (sqrt (log 10))) (sqrt (log 10)))))

prune1.6s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 2 expressions with local error:

0.3b
(* (/ 1 (sqrt (log 10))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
0.2b
(* (log (hypot re im)) (/ 1 (sqrt (log 10))))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
2 → 54
Calls

2 calls. Slowest were:

17.0ms
(* (/ 1 (sqrt (log 10))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
5.0ms
(* (log (hypot re im)) (/ 1 (sqrt (log 10))))

series347.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

183.0ms
(* (log (hypot re im)) (/ 1 (sqrt (log 10))))
164.0ms
(* (/ 1 (sqrt (log 10))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))

simplify2.4s

Counts
38 → 60
Calls

38 calls. Slowest were:

538.0ms
(* (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10)))) (* (* (* (log (hypot re im)) (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))))
301.0ms
(* (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10)))) (* (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10))))))
264.0ms
(* (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10)))))

prune863.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0b

localize5.0ms

Local error

Found 2 expressions with local error:

0.5b
(/ (* (/ 1 (sqrt (log 10))) (log (hypot re im))) (sqrt (log 10)))
0.2b
(* (/ 1 (sqrt (log 10))) (log (hypot re im)))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
2 → 60
Calls

2 calls. Slowest were:

13.0ms
(/ (* (/ 1 (sqrt (log 10))) (log (hypot re im))) (sqrt (log 10)))
5.0ms
(* (/ 1 (sqrt (log 10))) (log (hypot re im)))

series372.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

222.0ms
(* (/ 1 (sqrt (log 10))) (log (hypot re im)))
150.0ms
(/ (* (/ 1 (sqrt (log 10))) (log (hypot re im))) (sqrt (log 10)))

simplify1.6s

Counts
48 → 66
Calls

48 calls. Slowest were:

327.0ms
(* (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10)))) (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))))
149.0ms
(* -1 (/ (log (/ -1 re)) (log 10)))
123.0ms
(/ (* (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10)))) (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im)))) (* (* (sqrt (log 10)) (sqrt (log 10))) (sqrt (log 10))))

prune917.0ms

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0b

regimes222.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms